{
"cells": [
{
"cell_type": "markdown",
"id": "fb6cab3f-4fb5-4575-8977-54096745d3a7",
"metadata": {},
"source": [
"# 1 | BASIC MATHEMATICS USING PYTHON"
]
},
{
"cell_type": "markdown",
"id": "afbdb4d3",
"metadata": {},
"source": [
"><b>Dr J H Klopper</b></p>\n",
">Department of Biostatistics and Bioinformatics</br>\n",
">Milken Institute School of Public Health</br>\n",
">George Washington University"
]
},
{
"cell_type": "markdown",
"id": "f7248074",
"metadata": {},
"source": [
"<p xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:dct=\"http://purl.org/dc/terms/\"><span property=\"dct:title\">This chapter of Algebra for Health Data Science</span> by <span property=\"cc:attributionName\">Dr JH Klopper</span> is licensed under <a href=\"http://creativecommons.org/licenses/by-nc-nd/4.0/?ref=chooser-v1\" target=\"_blank\" rel=\"license noopener noreferrer\" style=\"display:inline-block;\">Attribution-NonCommercial-NoDerivatives 4.0 International<img style=\"height:22px!important;margin-left:3px;vertical-align:text-bottom;\" src=\"https://mirrors.creativecommons.org/presskit/icons/cc.svg?ref=chooser-v1\"><img style=\"height:22px!important;margin-left:3px;vertical-align:text-bottom;\" src=\"https://mirrors.creativecommons.org/presskit/icons/by.svg?ref=chooser-v1\"><img style=\"height:22px!important;margin-left:3px;vertical-align:text-bottom;\" src=\"https://mirrors.creativecommons.org/presskit/icons/nc.svg?ref=chooser-v1\"><img style=\"height:22px!important;margin-left:3px;vertical-align:text-bottom;\" src=\"https://mirrors.creativecommons.org/presskit/icons/nd.svg?ref=chooser-v1\"></a></p>"
]
},
{
"cell_type": "markdown",
"id": "638e9cb9",
"metadata": {},
"source": [
"- Watch the pencil and paper video lecture [HERE](https://youtu.be/FQwnkxolP_4)\n",
"\n",
"- Watch the Python video lecture [HERE](https://youtu.be/wwBoL0wbxXg)"
]
},
{
"cell_type": "markdown",
"id": "49d45e7c",
"metadata": {},
"source": [
"## 1.1 Packages used in this chapter"
]
},
{
"cell_type": "markdown",
"id": "ac6a054a-878d-4683-9da8-0bc1f7750959",
"metadata": {},
"source": [
"While there are many functions in Python, we can extend the language with packages that contains even more functions."
]
},
{
"cell_type": "markdown",
"id": "68a5d2b7",
"metadata": {},
"source": [
"The `sympy` package, short for _symbolic Python_ is a Python package that contains functions for symbolic mathematics. This means that we can use Python to do mathematics in the same way that we would do it by hand. The `sympy` packages turns Python into a computer algebra system (CAS)."
]
},
{
"cell_type": "markdown",
"id": "ef48975e",
"metadata": {},
"source": [
"The import keyword is used to import packages into Python. It is typical to use an abbreviation for packages and use the code `import sympy as sym`. This means that we can use the abbreviation `sym` to refer to the `sympy` package and call functions in the package such as `sqrt` (the square root) by typing `sym.sqrt`. By not using an abbreviation, we will have to type `sympy.sqrt` every time we want to use the square root function. The choice is a matter of style."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "aea67e45-6af5-4ae5-8689-25806f67ca11",
"metadata": {},
"outputs": [],
"source": [
"import sympy"
]
},
{
"cell_type": "markdown",
"id": "5e2c4a56",
"metadata": {},
"source": [
"The `init_printing` function is used to make the output of the `sympy` package look like typeset mathematics. This is because it use LaTeX to print proper mathematical notation. The `init_printing` function has several arguments, which we will not use here."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "74a2803b-7527-4701-9091-c2590f174c81",
"metadata": {},
"outputs": [],
"source": [
"sympy.init_printing()"
]
},
{
"cell_type": "markdown",
"id": "02d8d89e",
"metadata": {},
"source": [
"Python has a built-in package for mathematics called `math`. The `math` package does not use symbolic mathematics, but rather numerical mathematics. This means that the `math` package will give us numerical answers, while the `sympy` package will give us symbolic answers."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "c4e7cfd6-68c0-47c8-948e-921748af4b7e",
"metadata": {},
"outputs": [],
"source": [
"import math"
]
},
{
"cell_type": "markdown",
"id": "085c2e0b",
"metadata": {},
"source": [
"The `matplotlib` package is a large Python packages used for generating plots. Here we use one of the models in the package called `pyplot`."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "35f0e1f3-d7e4-468b-bb21-5f1a5abf7605",
"metadata": {},
"outputs": [],
"source": [
"from matplotlib import pyplot"
]
},
{
"cell_type": "markdown",
"id": "cffdffff",
"metadata": {},
"source": [
"Monitors with high-resolution screen such as those found on Apple computer can print plotrs with more detail. To make use of the detail, we use a magic command as is shown below."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "ef9c4921",
"metadata": {},
"outputs": [],
"source": [
"%config InlineBackend.figure_format = 'retina'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"While not the focus of this course, we will perform some statistical tests as motivation for the algebra that we want to explore. The `stats` module in the `scipy` package contains functions for statistical tests. We will use the `ttest_ind` function from this module to perform a _t_-test."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"from scipy.stats import ttest_ind"
]
},
{
"cell_type": "markdown",
"id": "731a8219",
"metadata": {},
"source": [
"Lastly, we import the `numpy` package, short for numerical Python. This package is used for numerical mathematics and is used to create arrays. Arrays are similar to lists, but are more efficient for numerical calculations."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "a322d30e-ab5f-4a92-aead-07d8f11a031d",
"metadata": {},
"outputs": [],
"source": [
"import numpy"
]
},
{
"cell_type": "markdown",
"id": "a52500f8",
"metadata": {},
"source": [
"## 1.2 Introduction"
]
},
{
"cell_type": "markdown",
"id": "ced589f0",
"metadata": {},
"source": [
"We use numbers every day of our lives. We count objects and we consider costs and spending. While counting usually involves the use of whole numbers, we also consider parts of the whole when we cut a pizza into slices. Pizza slices are fractions of the whole and we pay for the pizza using decimal numbers. Numbers are everywhere in the life sciences. We count, add, multiply, calculate means and standard deviations. The list is endless. "
]
},
{
"cell_type": "markdown",
"id": "06e25fb3",
"metadata": {},
"source": [
"Computers and most computer languages can do mathematics. However, they do not do mathematics in the same way that we do mathematics by hand. Computers use numerical mathematics, which means that they use numbers to represent quantities. This is different from symbolic mathematics, which means that we use symbols to represent quantities. For example, we use the symbol $\\pi$ to represent the ratio of the circumference of a circle to its diameter and we use the symbols $x$ and $y$ to denote unknowns, and $f$ and $g$ to denote fuctions. With `numpy` and `sympy` Python can do both and we will explore how Python works with numbers and mathematical symbols."
]
},
{
"cell_type": "markdown",
"id": "60251f1a",
"metadata": {},
"source": [
"In this textbook, we explore mathematics, and algebra in particular. While being able to use a pencil and paper is part of the visceral experience and understanding of mathematics, we live in an era of powerful computers in our pockets and software on our computers. The calculations in these notebooks will be laid out in detail. We will, though, also make use of modern technology to gain a level of familiarity. We will use technology to verify our results or to simply do the calculations for us. We will use modern technology as a tool to learn about and understand the mathematics. Technology is a companion to our pencils and paper."
]
},
{
"cell_type": "markdown",
"id": "f6a21397",
"metadata": {},
"source": [
"Our technology partner will be the Python programming language. This is not a set of notebooks on Python and the material will not teach you how to become a Python programmer. It will simply serve as our calculator instead of the calculator on our phones. Along the way, though, we will come to realize how Python can help us build our intuition for the mathematics and how it helps us understand the mathematics. Python will be our teaching aid when we need it."
]
},
{
"cell_type": "markdown",
"id": "9a74401a",
"metadata": {},
"source": [
"In this chapter, we explore basic number types and arithmetic as a foundation to exploring the use of algebra in public health and biomedical research in later notebooks. A proper foundation in algebra can be an enjoyable and simple task. Being comfortable with algebra will allow everyone who goes on to study statistics, to concentrate on the understanding of data and its analysis instead of being _afraid of_ or _sparring with_ the mathematics."
]
},
{
"cell_type": "markdown",
"id": "879044e2-9ec5-4335-bab4-ae7d811a5bd2",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"## 1.3 Number types"
]
},
{
"cell_type": "markdown",
"id": "d91cfa83-a7e3-46d5-9a42-c04b5735d927",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.3.1 Computer number types"
]
},
{
"cell_type": "markdown",
"id": "94d7737c",
"metadata": {},
"source": [
"In Python we can refer to any item as an object. There are many object types in Python. Not all of these types are numbers."
]
},
{
"cell_type": "markdown",
"id": "8dcba368-3d2a-40d5-b166-72ffed01db57",
"metadata": {},
"source": [
"We use the `type` function to return the type of a Python object. Strings are not numbers or mathematical symbols. They are letter, words, sentences, and paragraphs of text. Strings are denoted by their enclosure in single or double quotes. Below we pass the string `'This is Python.'` to the `type` function."
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "cb8dbff1-182e-4006-bd1c-aae1fa5eaf2e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"str"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type('This is Python.')"
]
},
{
"cell_type": "markdown",
"id": "c785679e",
"metadata": {},
"source": [
"Python returns the value `str` to indicate that the object is a string. Next, we pass the number $1$ to the `type` function."
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "a6ccf7f8-cdfe-42ef-a85f-c52b5aed96c3",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"int"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(1)"
]
},
{
"cell_type": "markdown",
"id": "723f782b",
"metadata": {},
"source": [
"Python returns the value `int` to indicate that the object is an integer. Integers are whole numbers. We can also pass the number $1.0$ to the `type` function."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "05325bf8-5f7b-4897-a5a1-2a0b0ddb9f36",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"float"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(1.0)"
]
},
{
"cell_type": "markdown",
"id": "67448993",
"metadata": {},
"source": [
"We see that $1.0$ is of type `float`. This is short for floating point number. Floating point numbers are numbers with a decimal point. We can also pass the number $1j$ to the `type` function."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "f1dbb94a-1503-408b-9f15-12d7205b5ea1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"complex"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(1j)"
]
},
{
"cell_type": "markdown",
"id": "9275c257",
"metadata": {},
"source": [
"The `1j` symbol is a complex number. Complex numbers are numbers with an imaginary part. The imaginary part is denoted by the symbol $i$ or $j$. In Python, the symbol $j$ is used to denote the imaginary part of a complex number. The symbol $i$ is used to denote the imaginary part of a complex number in mathematics."
]
},
{
"cell_type": "markdown",
"id": "e46e1a79",
"metadata": {},
"source": [
"Boolean values are either `True` or `False`. We can pass the boolean values `True` and `False` to the `type` function."
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "df05d68c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"bool"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(True)"
]
},
{
"cell_type": "markdown",
"id": "cc5426c5",
"metadata": {},
"source": [
"Python returns the value `bool` to indicate that the object is a Boolean value."
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "9fe2b3c6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"bool"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(False)"
]
},
{
"cell_type": "markdown",
"id": "0290b34a",
"metadata": {},
"source": [
"`False` is also a Boolean value as indicated by the returned `bool` type."
]
},
{
"cell_type": "markdown",
"id": "13c6ed59",
"metadata": {},
"source": [
"`True` and `False` in Python are stored as $1$ and $0$ respectively. We can add `True` and `False` values together."
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "e366801b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVQoFW2RvRGCQBCFD7UAxhIgNXI0J8AO1BK0BEJItQMMTTU0w4DE1A6gBMYOzu+dg4PIzrzZ27dvf+7Os9aaLMsCY0wCZAvQgAT+KWIMJMghtlEU3cCpLMsZ3Bn/IK5HBAewB1+jQF1f4CJSohhUJHwRHbtz9uEDiRTUBKocMt9o8SGkaVoBUtaN+6um6xxSF3I31rgh08JXxEeX7I9iRA6KLv/TicodlVP8qtt+0gYk1pxD/KbDaa/P4iS06BLffk2rU2HjMV/qAui9+hZTGGqcBBJqn765D34Dzrdx4/BA7EwAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle 2$"
],
"text/plain": [
"2"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"True + True"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "547f6c37",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFXWS4Q2CMBCFhQkIbqAjqBvgBsoGOkf/GUbQFXQENjAyAhtA2AC/V3tGG2hyeXdfH71LSzKO48KWc64KeYeuiQrWiiVmBLyoL+hDG2iGiO3J2zTAM5qZKbAB1UdX1d6IHolGIFpP6kKnm7EA9JFJpZ8PLdIwy4TnD+U6MQ9IM82tb+s5g/GlTpyazQzWrdOM1lL3Fi9jn3tktyZWsYvaTqzteu7A7YRxA2vU1RtJboAePZiZXG1L4iT2+9ba0E8xEPopdoTe3r/YGx/SQ0OZAIYmAAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 0$"
],
"text/plain": [
"0"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"False + False"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "1db93f0e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZElEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUACBj0chKHhJQQLN0ZQZAGDGBRBIOACxKC4OQfE94B4NwDm+hiAOyllRAAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 1$"
],
"text/plain": [
"1"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"True + False"
]
},
{
"cell_type": "markdown",
"id": "26bb69c4",
"metadata": {},
"source": [
"We can also pass the value `None` to the `type` function."
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "aa4434a0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"NoneType"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"type(None)"
]
},
{
"cell_type": "markdown",
"id": "7e666d84",
"metadata": {},
"source": [
"`None` indicates that the object has no value and has the specific type `NoneType`. This is different from `False`, which is a Boolean value."
]
},
{
"cell_type": "markdown",
"id": "d739c2e3-425e-4194-8bd2-21bb73b8ea26",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.3.2 Mathematical number types"
]
},
{
"cell_type": "markdown",
"id": "356eae03",
"metadata": {},
"source": [
"In mathematics we put the elements of a set in curly braces, $\\{ 1,2,3,4, \\ldots \\}$. A <span style=\"color:lightgreen\">__set__</span> is any collection of objects, such as all the toppings on a pizza. With respect to numbers, the elememts are just that, numbers Each member (number) is separated by a comma. We also see a set of ellipses, $\\ldots$. Intuitively, we read this as a continuation along, or, following the pattern of elements, before the ellipses. This indicates a continuation of the numbers. The numbers never stop. The $\\ldots$ at the end of the list of elements indicate an infine extension of values. The numbers go to infinity. We refer to the number of elements in a set as the <span style=\"color:lightgreen\">__cardinality__</span> or <span style=\"color:lightgreen\">__size__</span> of the set."
]
},
{
"cell_type": "markdown",
"id": "9e2ec8e1",
"metadata": {},
"source": [
"In this section, we investigate the commonly used number system in mathematics."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consider a list of the smoking status of participants in a study. In the table below, we note $10$ participants. Each participant has either _never smoked_, _smoked in the past_, or _currently smokes_."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"| Participant | Smoking status |\n",
"|:------------|:-------------------|\n",
"| 1 | Currently smokes |\n",
"| 2 | Never smoked |\n",
"| 3 | Smoked in the past |\n",
"| 4 | Never smoked |\n",
"| 5 | Never smoked |\n",
"| 6 | Smoked in the past |\n",
"| 7 | Smoked in the past |\n",
"| 8 | Smoked in the past |\n",
"| 9 | Currently smokes |\n",
"| 10 | Currently smokes |"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can count the number of occurrences of each smoking status. We can count the number of participants who _never smoked_ by counting the number of times the smoking status is _never smoked_. We can count the number of participants who _smoked in the past_ by counting the number of times the smoking status is _smoked in the past_. Lastly, we can count the number of participants who _currently smoke_ by counting the number of times the smoking status is _currently smokes_. From the data we note that $3$ participants _never smoked_, $4$ participants _smoked in the past_, and $3$ participants _currently smoke_. We cannot have that three-and-half participants _never smoked_, nor can we have a negative number of people who _cuttenrly smoke_. Each count is a counting number."
]
},
{
"cell_type": "markdown",
"id": "3772d6ae",
"metadata": {},
"source": [
"__Definition 1.3.2.1__ The <span style=\"color:lightgreen\">__counting numbers__</span>, symbolized by $\\mathbb{N}$, are the set of numbers $\\{ 1,2,3,4,\\ldots \\}$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To accomodate the case where we count a zero number of participants for a specific smoking status, we add the number $0$ to the set of counting numbers. We call this set of numbers the natural numbers."
]
},
{
"cell_type": "markdown",
"id": "ce396c18",
"metadata": {},
"source": [
"__Definition 1.3.2.2__ The <span style=\"color:lightgreen\">__natural numbers__</span>, symbokized by $\\mathbb{N}_{0}$ are the set of numbers $\\{ 0,1,2,3,4,\\ldots \\}$."
]
},
{
"cell_type": "markdown",
"id": "4430e35a",
"metadata": {},
"source": [
"In many cases, we want to consider a range of natural numbers. Age is a good example in public health and biomedical research, that is, caputuring peoples’ age in years. In fancy terms, we would say the unit of measurement of age is years."
]
},
{
"cell_type": "markdown",
"id": "1344930b",
"metadata": {},
"source": [
"We consider a baby in its first year of life as being $0$ years old. Most humans do not live beyond $100$ years of age, let alone an infinite number of years. Maybe one day. As an android. Imagine that. Age can be expressed as a natural number in an interval. An interval (of ages) has a minimum and a maximum value. The minimum and maximum values for a set of age values are often quoted in research papers. In the set $\\{ 1,2,3, \\ldots , 10 \\}$ we use ellipses do denote the we follow the _pattern_ up until a maximum number. In this case, $10$."
]
},
{
"cell_type": "markdown",
"id": "ca305f1d",
"metadata": {},
"source": [
"To sound like mathematicians, we note that there is an order to the symbols that we use to denote numbers. We know that $5$ is larger than $2$. In fact ordering is an imporant and sophisticated topic in math. Even so, we have a natural intuition for order and make use of it all the time."
]
},
{
"cell_type": "markdown",
"id": "273f2378",
"metadata": {},
"source": [
"It is typical to use a lowercase letter as a symbol for an arbitrary number. For natural numbers, we typically use the symbols $i$, $j$, $k$, and $n$. In human research studies, we have a number of participants in a study. This number of people is referred to as the sample size. In many calculations, we would use the symbols $n$ to denote a sample size. The symbol $n$ takes on a specific value depending on how many people are in our study."
]
},
{
"cell_type": "markdown",
"id": "95f9ab31",
"metadata": {},
"source": [
"Finally, we visualize the natural numbers (at least the first few) as markers on the horizontal axis of the familiar Cartesian plane (mutually perpendicular lines), shown in __Figure 1.3.2.1__."
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "bd73b8b8",
"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
"<Figure size 800x200 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 197,
"width": 701
}
},
"output_type": "display_data"
}
],
"source": [
"x = range(10)\n",
"y = numpy.repeat(0, 10)\n",
"\n",
"pyplot.figure(figsize=(8, 2))\n",
"pyplot.scatter(\n",
" x=x,\n",
" y=y,\n",
" s=60,\n",
" marker='o',\n",
" alpha=.8,\n",
" lw=1\n",
")\n",
"pyplot.grid(color='0.5')\n",
"for pos in ['right', 'top', 'bottom', 'left']:\n",
" pyplot.gca().spines[pos].set_visible(False)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It may be useful to include what we refer to as additive inverses to the natural numbers. The additive inverse of a number is the number that we add to the number to get $0$. For example, the additive inverse of $3$ is $-3$ because $3+(-3)=0$. The additive inverse of $-3$ is $3$ because $-3+3=0$. We can extend the natural numbers to include the additive inverses of the natural numbers. We call this set of numbers the integers."
]
},
{
"cell_type": "markdown",
"id": "bfa03ef8",
"metadata": {},
"source": [
"__Definition 1.3.2.3__ The <span style=\"color:lightgreen\">__integers__</span>, symbolized by $\\mathbb{Z}$ are the whole numbers $\\{ \\ldots , -3,-2,-1,0,1,2,3, \\ldots \\}$. The integers is the set of all natural numbers and their additive inverses. The additivive inverse of a natural number $n$ is $-n$ such that $n + (-n) = 0$."
]
},
{
"cell_type": "markdown",
"id": "c8a2b88a",
"metadata": {},
"source": [
"Arbitrary integers are usually denoted by lowercase letters such as $a$, $b$, $c$, $p$, and $q$."
]
},
{
"cell_type": "markdown",
"id": "41739c8e",
"metadata": {},
"source": [
"As with natural numbers, there are an infinite number of integers, and we can also specify an interval or intervals of integers."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consider the case where we have $100$ participants in a study and $34$ of them suffer from hypertension. We might want to express the proportion of participants who have a diagnosis of hypertensions. To do this, we introduce rational numbers."
]
},
{
"cell_type": "markdown",
"id": "9e118e17",
"metadata": {},
"source": [
"__Definition 1.3.2.4__ The <span style=\"color:lightgreen\">__rational numbers__</span>, symbolized by $\\mathbb{Q}$, are defined in __Equation 1.3.2.1__."
]
},
{
"cell_type": "markdown",
"id": "75802cf2",
"metadata": {},
"source": [
"$$\n",
"\\mathbb{Q} \\text{ is the set of all numbers of the form } \\frac{a}{b} \\text{ where } a,b \\in \\mathbb{Z} \\text{ and } b \\neq 0.\n",
"\\tag{Equation 1.3.2.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "36766f3b",
"metadata": {},
"source": [
"The top number $a$ is the <span style=\"color:lightgreen\">__numerator__</span> and the bottom number, $b$, is the <span style=\"color:lightgreen\">__denominator__</span>. Rational numbers are fractions and are often used when we express ratios, shown in __Equation 1.3.2.2__."
]
},
{
"cell_type": "markdown",
"id": "ce6d1db9",
"metadata": {},
"source": [
"$$\n",
"\\frac{a}{b} = \\frac{\\text{part}}{\\text{whole}} = \\frac{\\text{numerator}}{\\text{denominator}} = \\text{numerator} \\div \\text{denominator}\n",
"\\tag{Equation 1.3.2.2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the case of our $100$ participants where $34$ have hypertensions, we could state that the proportion of hypertensive participants is $34 \\div 100 =0.34$. We confirm this calculation using code below."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.34$"
],
"text/plain": [
"0.34"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"34 / 100"
]
},
{
"cell_type": "markdown",
"id": "502bd6a2",
"metadata": {},
"source": [
"The word fraction is typically reserved for cases in which the numerator is smaller than the denominator. We consider a fraction of the whole. If we have are investigating $100$ people and $53$ of them are ill, we can concern ourlselves with the fraction of the whole (all $100$ people) that are ill. Fractions are very, very usefull. In resaerch, we often use these fractions as proportions."
]
},
{
"cell_type": "markdown",
"id": "8bef0ebf",
"metadata": {},
"source": [
"The `Rational` function in the `sympy` package creates a rational number. The numerator and the denominator of the rational number are passed to the `Rational` function as arguments, separated by a comma."
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "c99b2580-3acb-42c0-ae33-d43ea39beb37",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{3}{4}$"
],
"text/plain": [
"3/4"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(3, 4)"
]
},
{
"cell_type": "markdown",
"id": "81b7e59b",
"metadata": {},
"source": [
"An integer can also be expressed as a rational number, shown in __Equation 1.3.2.3__."
]
},
{
"cell_type": "markdown",
"id": "36de9dd9",
"metadata": {},
"source": [
"$$\n",
"3 = \\frac{3}{1} = \\frac{6}{2} = \\frac{9}{3} = \\ldots\n",
"\\tag{Equation 1.3.2.3}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "95ecd51e",
"metadata": {},
"source": [
"Rational numbers can be expressed as decimal numbers and the latter can be expressed as a finite decimal number or a repeating decimal number. We see three such rational numbers in __Equation 1.3.2.4__, which is then expressed in code. They are treated as numerical value by Python when we do not use the `Rational` function from `sympy`."
]
},
{
"cell_type": "markdown",
"id": "3df8a711",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"& \\frac{3}{4} = 0.75 \\text{ (finite decimal digits)} \\\\ \\\\\n",
"& \\frac{1}{3} = 0.\\overline{3} \\text{ (repeating pattern of decimal digits)} \\\\ \\\\\n",
"& \\frac{1}{7} = 0.\\overline{142857} \\text{ (repeating pattern of decimal digits)}\n",
"\\end{align*}\n",
"\\tag{Equation 1.3.2.4}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "a0198f68",
"metadata": {},
"source": [
"A bar is drawn over the repeating sequences of numbers. The bar is called a <span style=\"color:lightgreen\">__vinculum__</span>. The vinculum is used to indicate that the sequence of numbers repeats indefinitely. A finite numbert of decimal digits or an infinite number of repeating decimal digits are both hallmarks of rational numbers."
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "0b84d949",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.75$"
],
"text/plain": [
"0.75"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Example of a finite decimal representation of a rational number\n",
"3/4"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "5f21a274",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.333333333333333$"
],
"text/plain": [
"0.3333333333333333"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Example of a repeating decimal number\n",
"1/3"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "6100fb20",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.142857142857143$"
],
"text/plain": [
"0.14285714285714285"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Example of a longer repeating decimal number (142857 142857)\n",
"1/7"
]
},
{
"cell_type": "markdown",
"id": "67f2fbf2",
"metadata": {},
"source": [
"In the case of the repeating decimal expansion, the repeating part is truncated in the output. As such, the decimal number is an approximation of the exact rational number as shown in __Equation 1.3.2.5__."
]
},
{
"cell_type": "markdown",
"id": "c11bd11e",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\text{Exact value: } \\frac{1}{3} \\\\ \\\\\n",
"&\\text{Numerical approximation: } 0.\\overline{3}\n",
"\\end{align*}\n",
"\\tag{Equation 1.3.2.5}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "beaa0ff6",
"metadata": {},
"source": [
"The set of rational numbers also contain the primes numbers."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"While we typically do not concern ourselves with prime nunmbers in the day-to-day analysis of data, we have to mention the prime numbers."
]
},
{
"cell_type": "markdown",
"id": "996001c3",
"metadata": {},
"source": [
"__Definition 1.3.2.5__ A <span style=\"color:lightgreen\">__prime number__</span> is an integer that is divisible only by itself AND $1$."
]
},
{
"cell_type": "markdown",
"id": "7442144d",
"metadata": {},
"source": [
"__Definition 1.3.2.6__ A number $a$ is <span style=\"color:lightgreen\"><b>divisible</b></span> by a number $b \\ne 0$ if there exists an integer $c$ such that $a = bc$. In other words, there is no remainder after division, shown in __Equation 1.3.2.6__."
]
},
{
"cell_type": "markdown",
"id": "e7608091",
"metadata": {},
"source": [
"$$\n",
"\\frac{a}{b} = c \\text{ (with no remainder)}\n",
"\\tag{Equation 1.3.2.6}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "8d8a2b49",
"metadata": {},
"source": [
"By definition, $1$ is not a prime number. The `isprime` function in the `sympy` package returns `True` if the input is a prime number and `False` otherwise."
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "f4655590",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Use sympy to detrmine if 1 is a prime number\n",
"sympy.isprime(1)"
]
},
{
"cell_type": "markdown",
"id": "b625a6e7",
"metadata": {},
"source": [
"In the code cell below, we use list comprehension to print the prime numbers between $1$ and $100$."
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "ae9eda67",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left[ 2, \\ 3, \\ 5, \\ 7, \\ 11, \\ 13, \\ 17, \\ 19, \\ 23, \\ 29, \\ 31, \\ 37, \\ 41, \\ 43, \\ 47, \\ 53, \\ 59, \\ 61, \\ 67, \\ 71, \\ 73, \\ 79, \\ 83, \\ 89, \\ 97\\right]$"
],
"text/plain": [
"[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 7\n",
"3, 79, 83, 89, 97]"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Use list comprehension and the isprime function to determine the prime numbers less than 100\n",
"[p for p in range(100) if sympy.isprime(p)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This brings us to our first exmaple problem."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.3.2.1__ Determine if $997$ is prime."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `isprime` function in `sympy` and pass the number $997$ as an argument."
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.isprime(997)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see that $997$ is indeed a prime number."
]
},
{
"cell_type": "markdown",
"id": "f865a9c3",
"metadata": {},
"source": [
"__Homework 1.3.2.1__ Determine if $1001$ is prime."
]
},
{
"cell_type": "markdown",
"id": "aa6cac40",
"metadata": {},
"source": [
"__Defintion 1.3.2.7__ A <span style=\"color:lightgreen\">__composite number__</span> is an integer that is not prime number, and is therefor divisible by at least one other number other than itself or $1$."
]
},
{
"cell_type": "markdown",
"id": "142b2904",
"metadata": {},
"source": [
"The number $16$ is a composite number as it is divisible by $2$ and $8$ and $4$, as well as $1$ and $16$."
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "077fd696",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Determine if 16 is a prime number\n",
"sympy.isprime(16)"
]
},
{
"cell_type": "markdown",
"id": "d4eeee10",
"metadata": {},
"source": [
"The `factorint` function in the `sympy` package returns the prime factors of a number and how many times each prime factor occurs."
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "080a7ac1",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\left\\{ 2 : 4\\right\\}$"
],
"text/plain": [
"{2: 4}"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Determine the prime factorization of 16\n",
"sympy.factorint(16)"
]
},
{
"cell_type": "markdown",
"id": "343b790e",
"metadata": {},
"source": [
"The `factorint` function returns a Python dictionary object (key-value pairs). The keys are the prime factors and the values are the number of times they occur in the prime factorization of the given number. This is termed the __multiplicities__ of a prime factor. For $16$ there is only a single prime. The prime factor is $2$ and there are four $2$’s in the prime facotrization of $16$. We write $16=2^{4}$."
]
},
{
"cell_type": "markdown",
"id": "aa97fd38",
"metadata": {},
"source": [
"__Defintition 1.3.2.8__ The <span style=\"color:lightgreen\">__prime factorization__</span> of a number is the product of its prime factors."
]
},
{
"cell_type": "markdown",
"id": "b97feb9f",
"metadata": {},
"source": [
"__Defintion 1.3.2.9__ The <span style=\"color:lightgreen\">__multiplicity__</span> of a prime factor is the number of times the prime factor occurs in the prime factorization of a number."
]
},
{
"cell_type": "markdown",
"id": "63f10cf4",
"metadata": {},
"source": [
"__Homework 1.3.2.2__ Determine the prime factors of $23452$ and their multiplicities."
]
},
{
"cell_type": "markdown",
"id": "604b57df",
"metadata": {},
"source": [
"__Definition 1.3.2.10__ The set of <span style=\"color:lightgreen\">__irrational numbers__</span> are the set of numbers that canot be expressed as a ratio of two integers."
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "c3c8ad2b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\sqrt{2}$"
],
"text/plain": [
"√2"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Example of an irrational number\n",
"sympy.sqrt(2)"
]
},
{
"cell_type": "markdown",
"id": "194d0439",
"metadata": {},
"source": [
"The decimal approximation of $\\sqrt{2}$ is calculated below using the `evalf()` method."
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "f3371984",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 1.4142135623731$"
],
"text/plain": [
"1.41421356237310"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Decimal approximation of square root of 2\n",
"sympy.sqrt(2).evalf()"
]
},
{
"cell_type": "markdown",
"id": "c230b857",
"metadata": {},
"source": [
"There is no termination to, and no repeating pattern in, the decimal expansion of $\\sqrt{2}$, making it an irrational number. If a decimal expansion of a number does not terminate or repeat, then it is irrational."
]
},
{
"cell_type": "markdown",
"id": "4438e93c",
"metadata": {},
"source": [
"The number $\\pi$ is also an irrational number. The `pi` function from the `sympy` package returns the value of $\\pi$. The `evalf()` method returns the decimal approximation of $\\pi$ to the specified number of digits."
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "5de0beef",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAJCAYAAAAGuM1UAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtUlEQVQYGXWQsQ3CMBREE2AAZmCECHpLkI45YARat1BSpqUCsQEpPAElZVpaOkrzTvKPTKR86XT/7t+345QxxkLlvT9COzCXHqmm1EIKK/MAWliBK1gm3cLKPadwBb6Is3OuCyFs0Rf0i35N3yrI7E1fzCRgwarGOyWhmxobiCe5IKjbPplX4eX6f4HgHugdVgtrjPsbOEmP1V+62VCMv8l1v4CpcDf8BLx6bEGDQz6kvw908QNYwThArcyywAAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle \\pi$"
],
"text/plain": [
"π"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Print the number pi\n",
"sympy.pi"
]
},
{
"cell_type": "code",
"execution_count": 33,
"id": "15880e23",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.1415926535897932384626433832795028841971693993751$"
],
"text/plain": [
"3.1415926535897932384626433832795028841971693993751"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Print the value of pi with 50 digits\n",
"sympy.pi.evalf(50)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So much of the data that we collected are real numbers, which we define in __Definition 1.3.2.9__."
]
},
{
"cell_type": "markdown",
"id": "005faa5c",
"metadata": {},
"source": [
"__Defintion 1.3.2.11__ The <span style=\"color:lightgreen\">__real numbers__</span>, $\\mathbb{R}$, are the union (combination) of the rational and irrational numbers."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"While we are often constrained by the accuracy of our measuring instruments, we can consider the real numbers to be continuous. So, if we state that a person is $6$ foot and $6$ inches tall, it is only an approximation. The person (well me, since I am $6' \\, 6''$ tall) is not exactly $6$ foot and $6$ inches tall. The person is $6$ foot and $6$ inches tall to the accuracy of our measurement accuracy."
]
},
{
"cell_type": "markdown",
"id": "f5d4ee97",
"metadata": {},
"source": [
"The real numbers can be represented as a point on the number line. There are an infinite number of real numbers. In many cases, most of the statistical analysis that we do in public health and in biomedical research involve real numbers. We generate some pseudo-random numbers below and assign them to the computer variable `x`"
]
},
{
"cell_type": "code",
"execution_count": 34,
"id": "0d4e967a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1.62434536, -0.61175641, -0.52817175, -1.07296862, 0.86540763,\n",
" -2.3015387 , 1.74481176, -0.7612069 , 0.3190391 , -0.24937038])"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Seed the pseudorandom number generator with the integer 1\n",
"numpy.random.seed(1)\n",
"\n",
"# Create a variable called x that holds an array of 10 real numbers from the standard normal distribution\n",
"x = numpy.random.randn(10)\n",
"x"
]
},
{
"cell_type": "markdown",
"id": "dca93f79",
"metadata": {},
"source": [
"__Figure 1.3.2.1__ shows the number line with the real numbers $x$."
]
},
{
"cell_type": "code",
"execution_count": 35,
"id": "3fc6e532",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
"<Figure size 1000x200 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 219,
"width": 860
}
},
"output_type": "display_data"
}
],
"source": [
"# Plot the values in the array x as markers on the x-axis\n",
"pyplot.figure(figsize=(10, 2))\n",
"pyplot.scatter(\n",
" x,\n",
" [0,0,0,0,0,0,0,0,0,0],\n",
" color='red',\n",
" marker='o',\n",
" s=100,\n",
" alpha=0.5,\n",
" label='x'\n",
")\n",
"pyplot.grid()\n",
"pyplot.xlim(-3, 3)\n",
"pyplot.hlines(0, -3, 3, color='black')\n",
"pyplot.gca().spines['top'].set_visible(False)\n",
"pyplot.gca().spines['right'].set_visible(False)\n",
"pyplot.gca().spines['bottom'].set_visible(False)\n",
"pyplot.gca().spines['left'].set_visible(False)\n",
"pyplot.title('Figure 1.3.2.1');"
]
},
{
"cell_type": "markdown",
"id": "5b207b00-dca4-4352-ae48-db2bee0cfae2",
"metadata": {
"tags": []
},
"source": [
"## 1.4 Simple arithmetic"
]
},
{
"cell_type": "markdown",
"id": "e02e1d89-68ee-41de-bb56-78c427c04437",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.1 Rounding"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We often find that the results of statistical calculations contain values with many decimal places and we need to round these. Look at the _p_ value for the equal variance _t_ test below."
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.0285355177361664$"
],
"text/plain": [
"0.02853551773616643"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Seed the numpy pseudorandom number generator with the integer 2\n",
"numpy.random.seed(2)\n",
"\n",
"# Create a variable named groupA that holds an array of 100 real numbers from a normal distribution with a mean of 100 and a standard deviation of 10\n",
"groupA = numpy.random.normal(100, 10, 100)\n",
"\n",
"# Create a variable named groupB that holds an array of 100 real numbers from a normal distribution with a mean of 101 and a standard deviation of 9\n",
"groupB = numpy.random.normal(101, 9, 100)\n",
"\n",
"# Conduct an equal variance t test comparing the values in groupA to those in group Band show the p-value\n",
"ttest_ind(groupA, groupB, equal_var=True)[1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We would typically round this _p_ value to two decimal places and write $0.03$."
]
},
{
"cell_type": "markdown",
"id": "54bebecb",
"metadata": {},
"source": [
"The `round` function rounds a number to a specified number of decimal places. By default, the number fo decimal places is zero, but the function will add a single decimal place with a value of $0$ to the output."
]
},
{
"cell_type": "code",
"execution_count": 37,
"id": "ce7fb77b-e6f7-4001-b5e6-9702a8961337",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOCAYAAADNGCeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABGUlEQVQ4EZWT3Q2CQBCElVgAkQ60A386wA7UDtQ2eDOWoFZgtAPtwEAJdKChA/yG7Bkl93BsMuze3O6wHLf9uq57zrIsmxBfwZS4cvyvhz/Y+oUfgwNcKW5AEOPP4A1mYAS8Rm7Oxh5/U4LV5vgFKCMeFViBHfsXJfmM/S18jG+ElEOs7rU+ah3pEWgr8gpP7hMuRTjuIpZSpKNoW3NekGmQmN7aVvCsh0FiFA6t2PuHba/TZ3qa+aOS0M58Z+WUXNevIDG7Air2nZ3jyiAxa+GBH1n861xnjy5iGjNNSNumEIW6b4sllune9i0k+cTijV86klifuAYbcX0NOqTeKtPFVIJuui7jnT2JNGbFGvQKaNDnQLPaTMYHN8BYMp9me8UAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle 10$"
],
"text/plain": [
"10"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Round a number to the nearest whole number\n",
"round(10.4)"
]
},
{
"cell_type": "code",
"execution_count": 38,
"id": "c924fe22-296f-4b13-9d91-e9c93e37e6d6",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAOCAYAAAAi2ky3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfElEQVQoFWP8//8/Aww0NDQYAdmrgdgYyP4AE0ensaljAQoKABXOBuJ3QGwCxEpAjAEIqQMZBLI5FKQTyC4DUiBXYQBC6pgwdJApMGoQ4YAbDSPSw0gYqkWIgFYMdYygTAtM/qCMCgIuQAzKe+eA+B4Q7wbKzQLSYIBPHQB5ZC0BptgXqwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 11$"
],
"text/plain": [
"11"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Round a number to the nearest whole number\n",
"round(10.51)"
]
},
{
"cell_type": "markdown",
"id": "ee68b190",
"metadata": {},
"source": [
"A decimal digit with a value $5$ and up leads to the preceding digit being rounded up and a decimal value with a value of $4$ and down leads to the preceding decimal digit being rounded down."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Below, we round the _p_ value from our equal variance _t_ test to two decimal places."
]
},
{
"cell_type": "code",
"execution_count": 39,
"id": "998dab6c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.03$"
],
"text/plain": [
"0.03"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Round the number 10.59 to the nearest tenth (a single decimal place)\n",
"round(0.0285355177361664, 2)"
]
},
{
"cell_type": "markdown",
"id": "77d1336c",
"metadata": {},
"source": [
"__Homework 1.4.1.1__ Round $3.666666666667$ to two decimal places."
]
},
{
"cell_type": "markdown",
"id": "b6eaa0a5",
"metadata": {},
"source": [
"In our solution the value $0.0285355177361664$ was printed to the screen. The decimal digits $0$ and $2$ are in the _first two places_. The next digit is an $8$, which is $5$ or more. As such, we round the $2$ up to a $3$ and the output is $0.03$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Rounding to the nearest integer is not always desired. In many statistical tests we calculate the degrees of freedom. Below, we calculate the degrees of freedom for an unequal variance _t_ test. (Ignore the code and consider the results.)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"TtestResult(statistic=0.01605247831794392, pvalue=0.9872145027924664, df=145.58823529411765)"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Seed the numpy pseudorandom number generator with the integer 2\n",
"numpy.random.seed(2)\n",
"\n",
"# Create a variable named groupC that holds an array of 100 real numbers from a normal distribution with a mean of 101 and a standard deviation of 20\n",
"groupC = numpy.random.normal(101, 20, 100)\n",
"\n",
"# Conduct an unequal variance t test comparing the values in groupA to those in group C and show the degrees of freedom\n",
"ttest_ind(groupA, groupC, equal_var=False)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We note that the degrees of freedom is $145.58823529411765$. In the case of degrees of freedom, we want to round down."
]
},
{
"cell_type": "markdown",
"id": "ec141f9d",
"metadata": {},
"source": [
"The `math` package has a `floor` function that rounds a number down to the nearest integer. The `ceil` function rounds a number up to the nearest integer."
]
},
{
"cell_type": "code",
"execution_count": 41,
"id": "7ae43f6a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 145$"
],
"text/plain": [
"145"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Round down the number 145.58823529411765to the nearest whole number\n",
"math.floor(145.58823529411765)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We would state that the _T_ statistic has $145$ degrees of freedom."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As example, we also show the `ceil` function for the sake of completeness."
]
},
{
"cell_type": "code",
"execution_count": 42,
"id": "b01fbbb8",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAOCAYAAAAi2ky3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfElEQVQoFWP8//8/Aww0NDQYAdmrgdgYyP4AE0ensaljAQoKABXOBuJ3QGwCxEpAjAEIqQMZBLI5FKQTyC4DUiBXYQBC6pgwdJApMGoQ4YAbDSPSw0gYqkWIgFYMdYygTAtM/qCMCgIuQAzKe+eA+B4Q7wbKzQLSYIBPHQB5ZC0BptgXqwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 11$"
],
"text/plain": [
"11"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Round up the number 10.1 to the nearest whole number\n",
"math.ceil(10.1)"
]
},
{
"cell_type": "markdown",
"id": "afc146a1",
"metadata": {},
"source": [
"__Homework 1.4.1.2__ Determine the ceiling an the floor of $\\pi$."
]
},
{
"cell_type": "markdown",
"id": "4b2992ed-e66f-41ce-b8c5-02831e524de0",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.2 Addition"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Arithmetic is the branch of mathematics that deals with the study of numbers and their operations. The four basic operations are addition, subtraction, multiplication, and division. We will explore each of these operations in turn."
]
},
{
"cell_type": "markdown",
"id": "41d73f76",
"metadata": {},
"source": [
"The `+` symbol placed between two numbers add the numbers in Python. Below we calculate $3+4$."
]
},
{
"cell_type": "code",
"execution_count": 43,
"id": "2207114a-65cb-4563-92c8-ff3d9158c018",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAu0lEQVQoFYWR0Q3CMAxEU8QAsEJHQLABbAAzMAKfyW9WATaADZAYgRGgG5R3CoSkWKqlq93z+RI5jfe+dc6twTGE0JHNaBBu6ZzMbiI7DOZT6hW4gkfiq69OOoiR0DGxUS4DTldS76ws4U2FERHRLvN937shuHcEi5K3RC2iSylSPcnWvyJSCnWUkzjJ7VVy33rouMfGWtPf0Vr+sz4z/WVHVjGD0u7MZ8xCBMs0O+KISI6KUcf7R2S+1Bs1T3OSPIJvGwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 7$"
],
"text/plain": [
"7"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Adding 3 and 4\n",
"3 + 4"
]
},
{
"cell_type": "markdown",
"id": "7fe4ecd9",
"metadata": {},
"source": [
"More than two numbers can be added. We calculate $3+4+5$."
]
},
{
"cell_type": "code",
"execution_count": 44,
"id": "a848bae4-30a0-498b-8435-a691c50f64ab",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOCAYAAADNGCeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQ4EaWT7Q2CMBCG0TgA0Q10Az82wA10BR3Bv/yVDdAJjG4gK8gGOoJxA3zepm0Q1IBeUt7rfbzt3dFOURSBkziOx+hH1gT94ewOsQ3RN3Y/Be/aY89l66GE4J4lhwKUUBNLlIJz50Tfol9kY2VdPg/WkrXGcXCBb1CJivFCjm6pClRN0NWnoUTEXSFQJWXJ2ITYh23IlHQjqdZLyxz2ykd80yFZfvBraAH+vM3NalwQiEgDMxP+iwwSNf4EaQK2GoDivUCQslEPffk/3QyCFUR90P9zOqXxABQsgWABjEB/I3T1rV2ZJKnhM9A9KXFIdMC9erOBcVEC+PI/2dPV8Axd/SpLhC0xZCjmOeCNbMQR2w39DO6s7QyqHPWrKuahPwGbS2D0goaxHgAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 12$"
],
"text/plain": [
"12"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Adding 3 and 4 and 5\n",
"3 + 4 + 5"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.2.1__ Calculate $3+4+5+6+7+8+9+10$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `+` symbol to add the numbers."
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 52$"
],
"text/plain": [
"52"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3 + 4 + 5 + 6 + 7 + 8 + 9 + 10"
]
},
{
"cell_type": "markdown",
"id": "60d0df52",
"metadata": {},
"source": [
"__Homework 1.4.2.1__ Calculate $1+2+3+4+5+6+7+8+9+10$."
]
},
{
"cell_type": "markdown",
"id": "8389d79a-ec4d-4d2d-979f-614112b54402",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.3 Subtraction"
]
},
{
"cell_type": "markdown",
"id": "1abd1b0c",
"metadata": {},
"source": [
"The `-` symbol placed between two numbers subtracts the second number from the first number in Python. We calculate $3-4$."
]
},
{
"cell_type": "code",
"execution_count": 46,
"id": "518ce30f-e78a-4a72-8a26-907c85c15524",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAj0lEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhpwFwtBgFIIFJyM1CjtgPIIKOBBwAWJQYgElgHtAvBsAO3sgJ/LwrLIAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle -1$"
],
"text/plain": [
"-1"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Subtracting 4 from 3\n",
"3 - 4"
]
},
{
"cell_type": "markdown",
"id": "0b37d8d2",
"metadata": {},
"source": [
"More than one number can be subtracted. We calculate $23-6-10$."
]
},
{
"cell_type": "code",
"execution_count": 47,
"id": "b34f746f-bcab-4b76-a9d3-9fd57dcf87ac",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAu0lEQVQoFYWR0Q3CMAxEU8QAsEJHQLABbAAzMAKfyW9WATaADZAYgRGgG5R3CoSkWKqlq93z+RI5jfe+dc6twTGE0JHNaBBu6ZzMbiI7DOZT6hW4gkfiq69OOoiR0DGxUS4DTldS76ws4U2FERHRLvN937shuHcEi5K3RC2iSylSPcnWvyJSCnWUkzjJ7VVy33rouMfGWtPf0Vr+sz4z/WVHVjGD0u7MZ8xCBMs0O+KISI6KUcf7R2S+1Bs1T3OSPIJvGwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 7$"
],
"text/plain": [
"7"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Subtracting 6 and 10 from 23\n",
"23 - 6 - 10"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.3.1__ Calculate $23-6-10-4-5-6-7-8-9-10$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `-` symbol to subtract the numbers."
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle -42$"
],
"text/plain": [
"-42"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"23 - 6 - 10 - 4 - 5 - 6 - 7 - 8 - 9 - 10"
]
},
{
"cell_type": "markdown",
"id": "5af0b1c9",
"metadata": {},
"source": [
"__Homework 1.4.3.1__ Calculate $100-1-2-3-4-5-6-7-8-9-10$."
]
},
{
"cell_type": "markdown",
"id": "874c838b-47a6-44cf-855b-ae1229aec0e5",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.4 Multiplication"
]
},
{
"cell_type": "markdown",
"id": "72101ef7",
"metadata": {},
"source": [
"The `*` symbol between two numbers multiplies the numbers in Python. We calculate $3 \\times 4$."
]
},
{
"cell_type": "code",
"execution_count": 49,
"id": "f1bbbf86-07f0-45a5-845c-cfffd0bb0187",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOCAYAAADNGCeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQ4EaWT7Q2CMBCG0TgA0Q10Az82wA10BR3Bv/yVDdAJjG4gK8gGOoJxA3zepm0Q1IBeUt7rfbzt3dFOURSBkziOx+hH1gT94ewOsQ3RN3Y/Be/aY89l66GE4J4lhwKUUBNLlIJz50Tfol9kY2VdPg/WkrXGcXCBb1CJivFCjm6pClRN0NWnoUTEXSFQJWXJ2ITYh23IlHQjqdZLyxz2ykd80yFZfvBraAH+vM3NalwQiEgDMxP+iwwSNf4EaQK2GoDivUCQslEPffk/3QyCFUR90P9zOqXxABQsgWABjEB/I3T1rV2ZJKnhM9A9KXFIdMC9erOBcVEC+PI/2dPV8Axd/SpLhC0xZCjmOeCNbMQR2w39DO6s7QyqHPWrKuahPwGbS2D0goaxHgAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 12$"
],
"text/plain": [
"12"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Mutiplying 3 and 4\n",
"3 * 4"
]
},
{
"cell_type": "markdown",
"id": "9757f634",
"metadata": {},
"source": [
"More than two numbers can be multiplied. We calculate $3 \\times 4 \\times 2$."
]
},
{
"cell_type": "code",
"execution_count": 50,
"id": "ad66ba09-5ae7-4ebc-aff2-e94f347cc0ea",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 24$"
],
"text/plain": [
"24"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Multiplying 3 and 4 and 2\n",
"3 * 4 * 2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.4.1__ Calculate $3 \\times 4 \\times 2 \\times 5 \\times 6 \\times 7 \\times 8 \\times 9 \\times 10$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `*` symbol to multiply the numbers."
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3628800$"
],
"text/plain": [
"3628800"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3 * 4 * 2 * 5 * 6 * 7 * 8 * 9 * 10"
]
},
{
"cell_type": "markdown",
"id": "239755ea",
"metadata": {},
"source": [
"__Homework 1.4.4.1__ Calculate $1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6 \\times 7 \\times 8 \\times 9 \\times 10$."
]
},
{
"cell_type": "markdown",
"id": "956de93a-2bb8-4343-ae08-988f64bbf1a8",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.5 Division"
]
},
{
"cell_type": "markdown",
"id": "4df6473c",
"metadata": {},
"source": [
"The forward slash `/` symbol between two numbers divides the first number by the second number in Python. We calculate $10$ divided by $2$."
]
},
{
"cell_type": "code",
"execution_count": 52,
"id": "fe46066e-ce4c-452b-963c-9ce0d45f065a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAPCAYAAAARZmTlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABjElEQVQ4Ea2U723CQAzFU9QBItgg3QDaDdIN2m5QukH5mHyrwgYwQmED0gkQbAAbgNgg/b3kjEKUP1VUSy/ne2efz/blvCzLvCiKxhrLgPNBUOb66vdeIT9xHPuoezeXLpkUQ/cX/8RZnRgfQAJ3FDfQBzmDCxgDBViDCUbiOgW7HUZbxhmYo8/ABj3InVUCyrLqWwp8p+BQ9YdLwEa8ZZIH7Pl5xc/KXN5iyyQkG/8/goRspnJXJe8HZGiN94g4hVA/RkC1/IKrOyFLhbAu+y4ZWiYy/sZpDtQ0YYeuU7bJ0C22XZCiXGz2DK6G6Eo1BYu2CH9cG1kmdfYKFBBQpWuSul6YrWV5GrCJ7rPueZM01h0/y77OxrijMnkERpQD5Sdho9bm46Cy1mVrmaQKsmQjPQNVUdO1QZesMNBBq6Inaa9sFWSBctNg5p/OQz9aLnA+yMBNaZkvMTgzvjhTD12VeQPv4u7020MqXV1bidJUQ/UOXUSYMD+gp4wfxmlkrk31QMpeD+QTuP5nv4JEA14+QuryAAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 5.0$"
],
"text/plain": [
"5.0"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Dividing 10 by 2\n",
"10 / 2"
]
},
{
"cell_type": "markdown",
"id": "23952c3d",
"metadata": {},
"source": [
"Note that we can write division using the $\\div$ symbols as $10 \\div 2$. We can also use the notation in __Equation 1.4.5.1__."
]
},
{
"cell_type": "markdown",
"id": "2385ce91",
"metadata": {},
"source": [
"$$\n",
"\\frac{10}{2}\n",
"\\tag{Equation 1.4.5.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "4c5ac2ce",
"metadata": {},
"source": [
"In the notation used in __Equation 1.4.5.1__, the number above the line is called the <span style=\"color:lightgreen\">__numerator__</span> and the number below the line is called the <span style=\"color:lightgreen\">__denominator__</span>."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.5.1__ Calculate $10$ divided by $2$ divided by $5$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `/` symbol to divide the numbers."
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 1.0$"
],
"text/plain": [
"1.0"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"10 / 2 / 5"
]
},
{
"cell_type": "markdown",
"id": "d6b0db8e",
"metadata": {},
"source": [
"__Homework 1.4.5.1__ Calculate $100$ divided by $2$ divided by $5$."
]
},
{
"cell_type": "markdown",
"id": "df6477c2-e535-48af-b21b-2579abb04d05",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.6 Powers"
]
},
{
"cell_type": "markdown",
"id": "47282ba3",
"metadata": {},
"source": [
"We use short-hand notation when we want to multiply a number by itself a number of times. Instead of writing $2 \\times 2 \\times 2$, which is $2$ multiplied by itself $3$ times, we write $2^{3}$, placing the $3$ as a superscript after the $2$. This short-hand notation shows $2$ raised to the power $3$, which simply means $2$ multiplied by itself $3$ times."
]
},
{
"cell_type": "markdown",
"id": "6d1e5251",
"metadata": {},
"source": [
"Th double star symbol `**` between two numbers raises the first number to the power of the second number in Python. We calculate $2$ raised to the power of $3$ or $2 \\times 2 \\times 2$."
]
},
{
"cell_type": "code",
"execution_count": 54,
"id": "dd5f1145-c508-4fe0-9b37-2917c6efffab",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVQoFW2SsXHCQBBFD2jALdhOiXADCnAHogSbEhgiKXUJuAW5AxMo95gO7NQZ0IF4b+E00oz/zGrv7/z7u3enSdd1SdR1fUfaBrl+5A31fVCFRlVVu7zOmVpDlPKpana9kr5j5/jzAo0uIYQ8Es9jzZhl4RflEudPwtky3ljsJBP7CwQNqQyS0oZsFzd+WMuOClfwd4tApyVxkIihULczoZNX8kD8YBBdojXEUz+R1+TATeA4br7PjrZyrh4InU13D7ec3RbU6+GrUE6pKIpT27Zzln9TBFqfyc70H3Tc59ae2He12APu7F7R7/AeFxRtf+yVg5/iAl5gbKb5R2kWAAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 8$"
],
"text/plain": [
"8"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# 2 to the power of 3\n",
"2 ** 3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.6.1__ Calculate $2$ raised to the power of $4$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Equation 1.4.6.1__ shows the general form of the power in this problem."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"2^{4} = 2 \\times 2 \\times 2 \\times 2 = 16\n",
"\\tag{Equation 1.4.6.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "a89e2fea",
"metadata": {},
"source": [
"__Homework 1.4.6.1__ Calculate $2$ raised to the power of $5$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.6.2__ Calculate $3$ raised to the power of $2$ raised to the power of $3$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Equation 1.4.6.2__ shows the mathematical notations for this problem."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"3^{2^{3}} = 3^{2 \\times 2 \\times 2} = 3^{\\left( 2^{3} \\right)} = 3^{8} = 6561\n",
"\\tag{Equation 1.4.6.2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `**` symbol to raise the first number to the power of the second number."
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 6561$"
],
"text/plain": [
"6561"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3**2**3"
]
},
{
"cell_type": "markdown",
"id": "e35b6704-9849-42f3-a8d4-b579cbd06895",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.7 Square root"
]
},
{
"cell_type": "markdown",
"id": "d07ca4eb",
"metadata": {},
"source": [
"Consider the number resulting form $3 \\times 3$. The result is $9$. We state that the square roots of $9$ is $3$."
]
},
{
"cell_type": "markdown",
"id": "619c3b08",
"metadata": {},
"source": [
"__Definition 1.4.7.1__ The <span style=\"color:lightgreen\">__square root__</span> of a number $a$ is the number $b$ such that $b \\times b = a$. We write $b = \\sqrt{a}$."
]
},
{
"cell_type": "markdown",
"id": "bf9ad35a",
"metadata": {},
"source": [
"We can also write $\\sqrt{a}$ as $a^{\\frac{1}{2}}$."
]
},
{
"cell_type": "markdown",
"id": "8c0f33f1",
"metadata": {},
"source": [
"Below, we calculate the square root of $9$ using the calculation in __Equation 1.4.7.1__."
]
},
{
"cell_type": "markdown",
"id": "3586c786",
"metadata": {},
"source": [
"$$\n",
"\\sqrt{9} = 9^{\\frac{1}{2}}\n",
"\\tag{Equation 1.4.7.1}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 56,
"id": "895c08c0-194e-4059-86fe-da681791b63e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.0$"
],
"text/plain": [
"3.0"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Positive square root of 9\n",
"9 ** (1/2)"
]
},
{
"cell_type": "markdown",
"id": "87f4ea5c",
"metadata": {},
"source": [
"We can also use the `sqrt` function in the `math` package to calculate the square root of $9$."
]
},
{
"cell_type": "code",
"execution_count": 57,
"id": "f0b747b8-0d7e-4027-bd68-99658cdbe9e5",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.0$"
],
"text/plain": [
"3.0"
]
},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Positive square root of 9 using the sqrt function from numpy\n",
"math.sqrt(9)"
]
},
{
"cell_type": "markdown",
"id": "9bfbfb2f",
"metadata": {},
"source": [
"Only the positive square root value is returned. We do remember that $\\left(-3\\right)^{2}$ is also equal to $9$."
]
},
{
"cell_type": "code",
"execution_count": 58,
"id": "ec3449fd",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQoFWWR3XHCQAyEbU8KcKCCUEKcElxCWmDogGe/Jh14KIGU4FSQnw5wBzB0YL7PPvFwaGbRam/vJORymqYiouu6Pjh5BbZoV7VSI0UN/wM9/NMD8ivpGzTwsVIkDmAVJgX4P+kXzF3C+I4wgjw0t1yqK3/S6SV3UZ+T9qZxHhbB4fNYJ2ETrb8QNrmL2j9k1GHcWvF6azbgmqLbOK8nHTjrBzgBW/4Au6g1dyPFQ/Cqpj14jtYPpiTYfuDC9UkB4h5d+oti0hzFmRvreNFZ8j0e0XZcdOnLt5YgOI/hS8YRbVhoUdwArdZQb6DcVHgAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle 9$"
],
"text/plain": [
"9"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that -3 squared is 9\n",
"(-3) ** 2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The `sympy` package has a `sqrt` function that calculates the square root of a number analytically. We calculate the square root of $10$ using the `sqrt` function."
]
},
{
"cell_type": "code",
"execution_count": 59,
"id": "9325c6ad",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\sqrt{10}$"
],
"text/plain": [
"√10"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Symbolic computation of the square root of 10\n",
"sympy.sqrt(10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that the exact solution $\\sqrt{10}$ is returned. We can also use the `evalf` method to return the decimal approximation of the square root of $10$ in `sympy`."
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.16227766016838$"
],
"text/plain": [
"3.16227766016838"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Symbolic computation of the square root of 10 using numerical approximation\n",
"sympy.sqrt(10).evalf()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.7.1__ Calculate the square root of $81$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use the `sqrt` funtion in the `sympy` package to calculate the square root of $81$."
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6UlEQVQoFWWR3XHCQAyEbU8KcKCCUEKcElxCWmDogGe/Jh14KIGU4FSQnw5wBzB0YL7PPvFwaGbRam/vJORymqYiouu6Pjh5BbZoV7VSI0UN/wM9/NMD8ivpGzTwsVIkDmAVJgX4P+kXzF3C+I4wgjw0t1yqK3/S6SV3UZ+T9qZxHhbB4fNYJ2ETrb8QNrmL2j9k1GHcWvF6azbgmqLbOK8nHTjrBzgBW/4Au6g1dyPFQ/Cqpj14jtYPpiTYfuDC9UkB4h5d+oti0hzFmRvreNFZ8j0e0XZcdOnLt5YgOI/hS8YRbVhoUdwArdZQb6DcVHgAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle 9$"
],
"text/plain": [
"9"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.sqrt(81)"
]
},
{
"cell_type": "markdown",
"id": "644c225b",
"metadata": {},
"source": [
"__Homework 1.4.7.1__ Calculate the square root of $100$."
]
},
{
"cell_type": "markdown",
"id": "9707b45a",
"metadata": {},
"source": [
"The `sympy` package can manage the square root of negative numbers. We calculate the square root of $-9$."
]
},
{
"cell_type": "code",
"execution_count": 62,
"id": "94cd78e7",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3 i$"
],
"text/plain": [
"3⋅ⅈ"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.sqrt(-9)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see a complex number returned. The `sympy` package uses the symbol $i$ to denote the imaginary part of a complex number."
]
},
{
"cell_type": "markdown",
"id": "cacb46ec",
"metadata": {},
"source": [
"### 1.4.8 Higher roots"
]
},
{
"cell_type": "markdown",
"id": "1a5ccead",
"metadata": {},
"source": [
"We can also calculate higher roots. Below we calculate the cube root of $27$, using the calculation in __Equation 1.4.8.1__."
]
},
{
"cell_type": "markdown",
"id": "3d6cdc5b",
"metadata": {},
"source": [
"$$\n",
"\\sqrt[3]{27} = 27^{\\frac{1}{3}}\n",
"\\tag{Equation 1.4.8.1}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 63,
"id": "4d599310",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.0$"
],
"text/plain": [
"3.0"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Cube root of 27\n",
"27 ** (1/3)"
]
},
{
"cell_type": "markdown",
"id": "da871569",
"metadata": {},
"source": [
"Next we calculate the root in __Equation 1.4.8.2__."
]
},
{
"cell_type": "markdown",
"id": "02b537aa",
"metadata": {},
"source": [
"$$\n",
"\\sqrt[4]{16} = 16^{\\frac{1}{4}}\n",
"\\tag{Equation 1.4.8.2}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 64,
"id": "8d0d56b3",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 2.0$"
],
"text/plain": [
"2.0"
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Fourth root of 16\n",
"16 ** (1/4)"
]
},
{
"cell_type": "markdown",
"id": "1e8cc6cb",
"metadata": {},
"source": [
"Obviously $-2$ is also a solution to $\\sqrt[4]{16}$ since $(-2) \\times (-2) \\times(-2) \\times(-2) =16$. We confirm this result using code below."
]
},
{
"cell_type": "code",
"execution_count": 65,
"id": "6f2ce83f",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 16$"
],
"text/plain": [
"16"
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that (-2) to the power of 4 is 16\n",
"(-2) ** 4"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.8.1__ Calculate the cube root of $64$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We use power notation, `**`, in Python to calculate the cube root of $64$."
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 4.0$"
],
"text/plain": [
"3.9999999999999996"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"64**(1/3)"
]
},
{
"cell_type": "markdown",
"id": "3dd2355e",
"metadata": {},
"source": [
"The `sympy` package contains the `root` function that calculates the $n$-th root of a number. We calculate the cube root of $64$ using the `root` function."
]
},
{
"cell_type": "code",
"execution_count": 67,
"id": "1651af1c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArklEQVQoFWP4//8/AzZcX19vBMQzYXJMDLjBaqCUEEwaq8KGhoYymAIYjaEQqMgIKPkBimHqGDAUAmXCgYpnwVVAGSgKoVbORFcE4sMVAhUpAfkfgPQ9vAqBkunYrIRpApsIVJAGFMBqJVwh1EoBXFbCFDICQ98FyEmHCSDRIUA2KJj2APE9rNEHijagAe+BeDUxUSgANAmEwYARpAMZAN0K8hQoqEBOAoE1QHwaAIFUXZ4aMJQ9AAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 4$"
],
"text/plain": [
"4"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Calculate the cube root of 64 using sympy\n",
"sympy.root(64, 3)"
]
},
{
"cell_type": "markdown",
"id": "df4aa610",
"metadata": {},
"source": [
"__Homework 1.4.8.1__ Calculate the cubic root of $1000$. Use pure Python (the exponentiation operator `**`), the `math` package, and the `sympy` package. Try to explain the difference in the results."
]
},
{
"cell_type": "markdown",
"id": "75fad03f",
"metadata": {},
"source": [
"### 1.4.9 Absolute values"
]
},
{
"cell_type": "markdown",
"id": "20512157",
"metadata": {},
"source": [
"The absolute value of a number is the distance of the number from zero on the number line. The absolute value of a number is always positive. It is written as $\\lvert a \\rvert$."
]
},
{
"cell_type": "markdown",
"id": "3dc195d1",
"metadata": {},
"source": [
"The `abs()` function from the `numpy` package returns the absolute value of a number. We calculate $\\lvert -3 \\rvert$ below."
]
},
{
"cell_type": "code",
"execution_count": 68,
"id": "e010e622",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVQoFXWS4Q2CQAyFwTiArOAIRjdgBF1BRjD+4y8j6Ao6AiOoq8gG+H0ndwGiLylt3732rg153/eZqOt6jatCkmUrvHkD38rlCkk8kIxCuT3cDTsQ3xcE4qgNh4HgEzrhzxJR+CLuBpO3o3nC0gjS6iKxX86rxcVP7GicQGFJ0mAV8dWD0NFAQG5winaYz3lgAWHqmIw9RZOp/wotQvzGuboivNErNQ9niFeXcZgngidiq38iDuPOWoST3cFth6o2Ck/zNhQ5jDe4oi4NQ+JaDqOCNXH6KT7nF1G9okQFwwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 3$"
],
"text/plain": [
"3"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Asbolute value of -3\n",
"abs(-3)"
]
},
{
"cell_type": "markdown",
"id": "6d0ce451",
"metadata": {},
"source": [
"__Defintion 1.4.9.1__ The <span style=\"color:lightgreen\">__absolute value__</span> of a number $a$ is defined in __Equation 1.4.9.1__."
]
},
{
"cell_type": "markdown",
"id": "4f71e796",
"metadata": {},
"source": [
"$$\n",
"\\lvert a \\rvert = \\begin{cases} a & \\text{if } a \\ge 0 \\\\ -a & \\text{if } a < 0 \\end{cases}\n",
"\\tag{Equation 1.4.9.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "6bbd7303",
"metadata": {},
"source": [
"Form the defintion we have see why $\\lvert -3 \\rvert = - (-3) = 3$."
]
},
{
"cell_type": "markdown",
"id": "1968ce6e-4f32-4e3a-a07e-f7cde055ff8d",
"metadata": {
"tags": []
},
"source": [
"### 1.4.10 Logarithm"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Logarithms are very frequenctly used in health data science. In __Figure 1.4.10.1__ we show a histogram of the age of participants in a study. (Ignore the code and consider the results.) We see that the distribution of age is skewed to the right. We can transform the data using a logarithm to make the distribution more symmetric, shown in __Figure 1.4.10.2__."
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"data": {
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",
"text/plain": [
"<Figure size 1000x200 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 219,
"width": 837
}
},
"output_type": "display_data"
}
],
"source": [
"# Create an array of 100 values for a variable named \"age\" that is right-skewed\n",
"age = numpy.random.gamma(2, 2, 100)\n",
"\n",
"# Create a histogram of the values in the array Age\n",
"pyplot.figure(figsize=(10, 2))\n",
"pyplot.hist(\n",
" age,\n",
" color='red',\n",
" alpha=0.5,\n",
" bins=20\n",
")\n",
"pyplot.grid()\n",
"pyplot.xlim(0, 20)\n",
"pyplot.gca().spines['top'].set_visible(False)\n",
"pyplot.gca().spines['right'].set_visible(False)\n",
"pyplot.gca().spines['bottom'].set_visible(False)\n",
"pyplot.gca().spines['left'].set_visible(False)\n",
"pyplot.title('Figure 1.4.10.1');"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
"<Figure size 1000x200 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 219,
"width": 833
}
},
"output_type": "display_data"
}
],
"source": [
"# Recreate the histogram but use a logarithm transformation of the values in the array age\n",
"pyplot.figure(figsize=(10, 2))\n",
"pyplot.hist(\n",
" numpy.log(age),\n",
" color='red',\n",
" alpha=0.5,\n",
" bins=20\n",
")\n",
"pyplot.grid()\n",
"pyplot.xlim(0, 3)\n",
"pyplot.gca().spines['top'].set_visible(False)\n",
"pyplot.gca().spines['right'].set_visible(False)\n",
"pyplot.gca().spines['bottom'].set_visible(False)\n",
"pyplot.gca().spines['left'].set_visible(False)\n",
"pyplot.title('Figure 1.4.10.2');"
]
},
{
"cell_type": "markdown",
"id": "86daf3d0-d83a-4a50-839f-b4944ddbf44f",
"metadata": {},
"source": [
"The logarithm of a number is defined by a base. As an example, we consider a base $10$. We can raise $10$ to any power. As an example we have that $10^{2}=100$. For the number $100$ and the base $10$, we would state that the logarithm base $10$ of $100$ is $2$, since it is $10$ raised to the power $2$ that is $100$. For the values $a,b,c$, we state the logarithm in __Equation 1.4.10.1__."
]
},
{
"cell_type": "markdown",
"id": "44f528fa-bda9-4780-8c13-99e7b6296ef1",
"metadata": {},
"source": [
"$$\n",
"\\text{If } a^{b}=c, \\text{ then } \\log_{a}{c}=b, \\, \\text{ for all } a>0\n",
"\\tag{Equation 1.4.10.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "de84933f",
"metadata": {},
"source": [
"From the defintion we note that $c>0$, since a positive value raised to any real-valued power is positive."
]
},
{
"cell_type": "markdown",
"id": "e75dacdb",
"metadata": {},
"source": [
"The `log()` function from the `numpy` package calculates the logarithm of a number. The default base is $e$, Euler's number."
]
},
{
"cell_type": "code",
"execution_count": 71,
"id": "5886b3c3-0008-4f4f-8c79-27ae6022245a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 6.90775527898214$"
],
"text/plain": [
"6.907755278982137"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Natural log of 1000\n",
"numpy.log(1000)"
]
},
{
"cell_type": "markdown",
"id": "0234e0ea",
"metadata": {},
"source": [
"The `sympy` package can also calculate logarithms. The `log()` function from the `sympy` package calculates the logarithm of a number. The default base is $e$. Note that `sympy` returns an exact value."
]
},
{
"cell_type": "code",
"execution_count": 72,
"id": "2879237a",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\log{\\left(1000 \\right)}$"
],
"text/plain": [
"log(1000)"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.log(1000)"
]
},
{
"cell_type": "markdown",
"id": "3c237185",
"metadata": {},
"source": [
"To return a number approximation in `sympy`, we use the `evalf` method."
]
},
{
"cell_type": "code",
"execution_count": 73,
"id": "77dbec94",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 6.90775527898214$"
],
"text/plain": [
"6.90775527898214"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.log(1000).evalf()"
]
},
{
"cell_type": "markdown",
"id": "6dae8ba9-6f0a-49ef-9474-477b6f0bb5c0",
"metadata": {},
"source": [
"Euler's number, $e$, is approximated below using the `numpy` function `exp`. We pass the argument value $1$, since any number raised to the power $1$ is itself and so $e^{1}=e$."
]
},
{
"cell_type": "code",
"execution_count": 74,
"id": "e28672f6-36e4-4dab-9a39-56efedf45fb3",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 2.71828182845905$"
],
"text/plain": [
"2.718281828459045"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Express Euler's number e as a decimal number\n",
"numpy.exp(1)"
]
},
{
"cell_type": "markdown",
"id": "6b801ebe",
"metadata": {},
"source": [
"Once again, `sympy` returns an exact value."
]
},
{
"cell_type": "code",
"execution_count": 75,
"id": "81c9b195",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAJCAYAAADgkQYQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAl0lEQVQYGWP8//8/Aww0NDQIANmVQPwWKqYMpMsZYYqAClyAAjOBOBTIPgdSBKSNgFQ4C5SjBKR3A3E6TAFIHAhAGneDFQEZIBNAQACoqAzCZBAG0ruB/D0wRSAda4ACXVAFKBQTEu80EhuFCVN0DygKMh4DAE0PgVnXCZRNR1YBlAQFB0i8EzkIQA4GmXYXiEEA5AmwGwEapS5+LAtwgwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle e$"
],
"text/plain": [
"ℯ"
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.exp(1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.4.10.1__ Calculate the logarithm base $e$ of $43545$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The `log` function from the `numpy` package is used to calculate this natural logarithm."
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 10.6815501651268$"
],
"text/plain": [
"10.68155016512684"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"numpy.log(43545)"
]
},
{
"cell_type": "markdown",
"id": "938b2f5a",
"metadata": {},
"source": [
"The `log10` function from the `numpy` package calculates the logarithm of a number with a base of $10$."
]
},
{
"cell_type": "code",
"execution_count": 77,
"id": "1435b237",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.0$"
],
"text/plain": [
"3.0"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Log base 10 of 1000\n",
"numpy.log10(1000)"
]
},
{
"cell_type": "markdown",
"id": "9d883ac5",
"metadata": {},
"source": [
"__Homework 1.4.10.1__ Calculate the logarithm base $10$ of $43545$."
]
},
{
"cell_type": "markdown",
"id": "9e1fa159",
"metadata": {},
"source": [
"__Homework 1.4.10.2__ Calculate the natural logarithm of $43545$."
]
},
{
"cell_type": "markdown",
"id": "8e1d3748-15d5-441d-bb0b-1c8e120ae867",
"metadata": {
"tags": []
},
"source": [
"### 1.4.11 Exponent"
]
},
{
"cell_type": "markdown",
"id": "e65092d2-9874-42ba-a968-a9895ef99b56",
"metadata": {},
"source": [
"Euler's number is useful when modeling growth or decay. For a simple model, we may have exponential growth at every time step $t$. The number of cases may then be as shown in __Equation 1.4.11.1__ with the number of cases at $t=0$ of $1$."
]
},
{
"cell_type": "markdown",
"id": "3cd00505-459a-44e7-bbec-759b5cbaa97c",
"metadata": {},
"source": [
"$$\n",
"\\text{Number of organisms at time }t = e^{t}\n",
"\\tag{Equation 1.4.11.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "a3e9d5d5-ff96-4cf7-89f1-2b31241008f8",
"metadata": {},
"source": [
"At time $t=5$, we have $e^{5}$ organisms, which is $e \\times e \\times e \\times e \\times e$, calculated below using the `exp` function in `numpy`."
]
},
{
"cell_type": "code",
"execution_count": 78,
"id": "f7dd814e-01cd-442c-84e5-4343dc4c87eb",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 148.413159102577$"
],
"text/plain": [
"148.4131591025766"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"numpy.exp(5)"
]
},
{
"cell_type": "markdown",
"id": "1113f585-ef0c-4bae-b426-dfcc9ab19bae",
"metadata": {},
"source": [
"We can use the `floor` function to round the result down."
]
},
{
"cell_type": "code",
"execution_count": 79,
"id": "33191421-3865-48ea-af3b-d9ae1ca6e815",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 148$"
],
"text/plain": [
"148"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Round exp(5) to 3 digits\n",
"math.floor(numpy.exp(5))"
]
},
{
"cell_type": "markdown",
"id": "5f2db65c-0c27-4f44-9fa3-9d26b3457485",
"metadata": {},
"source": [
"Starting with $e^{0}=1$ organism at $t=0$, we have that there will be $148$ organisms as $t=5$. __Figure 1.4.11.1__ demonstrates the growth."
]
},
{
"cell_type": "code",
"execution_count": 80,
"id": "8107a5ed-35bc-4c42-b04e-1a72afae0e3e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 454,
"width": 571
}
},
"output_type": "display_data"
}
],
"source": [
"fig, ax = pyplot.subplots()\n",
"\n",
"ax.plot([0, 1, 2, 3, 4, 5],\n",
" [numpy.exp(t) for t in [0, 1, 2, 3, 4, 5]],\n",
" \"b--\")\n",
"ax.grid()\n",
"ax.set_title(\"Figure 1.4.11.1\")\n",
"ax.set_xlabel(\"Time\")\n",
"ax.set_ylabel(\"Number of cases\")\n",
"pyplot.hlines(0, 0, 5, color='black')\n",
"pyplot.vlines(0, 0, 150, color='black')\n",
"pyplot.gca().spines['top'].set_visible(False)\n",
"pyplot.gca().spines['right'].set_visible(False)\n",
"pyplot.gca().spines['bottom'].set_visible(False)\n",
"pyplot.gca().spines['left'].set_visible(False);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will learn more about exponents in later chapters."
]
},
{
"cell_type": "markdown",
"id": "e867a661-9fc2-4c01-9616-b301a2723644",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"### 1.4.12 Order of arithmetic operations"
]
},
{
"cell_type": "markdown",
"id": "8b77a317-0290-48f0-8adf-0391dc9e2bd3",
"metadata": {},
"source": [
"We will commonly use the following order of arithmetic operations, expressed using the acronymn PEMDAS.\n",
"\n",
"1. __P__ arentheses\n",
"2. __E__ xponents (powers and roots)\n",
"3. __M__ ultiplication and __D__ ivision\n",
"4. __A__ ddition and __S__ ubtraction"
]
},
{
"cell_type": "markdown",
"id": "c9d412d4",
"metadata": {},
"source": [
"We start by calculating $3+2 \\times 4$, which is $3 + (2 \\times 4) = 3 + 8 = 11$."
]
},
{
"cell_type": "code",
"execution_count": 81,
"id": "1a7a4740-696d-44d3-893c-9e95ba2c3d65",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAOCAYAAAAi2ky3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfElEQVQoFWP8//8/Aww0NDQYAdmrgdgYyP4AE0ensaljAQoKABXOBuJ3QGwCxEpAjAEIqQMZBLI5FKQTyC4DUiBXYQBC6pgwdJApMGoQ4YAbDSPSw0gYqkWIgFYMdYygTAtM/qCMCgIuQAzKe+eA+B4Q7wbKzQLSYIBPHQB5ZC0BptgXqwAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 11$"
],
"text/plain": [
"11"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Expression with addition and multiplication\n",
"3 + 2 * 4"
]
},
{
"cell_type": "markdown",
"id": "5a132b1c",
"metadata": {},
"source": [
"We can use parenthesis to first add $3+2$ and then multiply by $4$. This is $(3+2) \\times 4 = 5 \\times 4 = 20$."
]
},
{
"cell_type": "code",
"execution_count": 82,
"id": "ff5aee81-a966-4f63-99d6-26ce7fcaaac1",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 20$"
],
"text/plain": [
"20"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Using parenthesis to change the order of operations (addition before multiplication)\n",
"(3 + 2) * 4"
]
},
{
"cell_type": "markdown",
"id": "566c6637",
"metadata": {},
"source": [
"In the examples below, we simplificy the indicated expressions, where the symbol $\\cdot$ indicates multiplication."
]
},
{
"cell_type": "markdown",
"id": "73000764",
"metadata": {},
"source": [
"__Problem 1.4.12.1__ Calculate the expression in __Equation 1.4.12.1__."
]
},
{
"cell_type": "markdown",
"id": "98722c58",
"metadata": {},
"source": [
"$$\\left( 3 \\cdot 2 \\right)^{2} - 4 (6+2) \\tag{Equation 1.4.12.1}$$"
]
},
{
"cell_type": "code",
"execution_count": 83,
"id": "ab2df1fc",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArklEQVQoFWP4//8/AzZcX19vBMQzYXJMDLjBaqCUEEwaq8KGhoYymAIYjaEQqMgIKPkBimHqGDAUAmXCgYpnwVVAGSgKoVbORFcE4sMVAhUpAfkfgPQ9vAqBkunYrIRpApsIVJAGFMBqJVwh1EoBXFbCFDICQ98FyEmHCSDRIUA2KJj2APE9rNEHijagAe+BeDUxUSgANAmEwYARpAMZAN0K8hQoqEBOAoE1QHwaAIFUXZ4aMJQ9AAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 4$"
],
"text/plain": [
"4"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(3 * 2)**2 - 4 * (6 + 2)"
]
},
{
"cell_type": "markdown",
"id": "c622cdc9",
"metadata": {},
"source": [
"__Problem 1.4.12.2__ Calculate the expression in __Equation 1.4.12.2__."
]
},
{
"cell_type": "markdown",
"id": "80e95b03",
"metadata": {},
"source": [
"$$\\frac{5^{2} - 4}{7} - \\sqrt{11-2} \\tag{1.4.12.2}$$"
]
},
{
"cell_type": "code",
"execution_count": 84,
"id": "fbbdb2b1",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOCAYAAAAxDQxDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJUlEQVQ4EZ2U/W3CMBDFE9QBUNmgjNAyQWGDlg3oHPmvYoSyQjsCG/AxAhsUZYPwe1FeiopjW5x0Ovt87z3nDlw2TVPYqqpad+tf4hRfkzv5PBVj+NJCFB0g+iT+iJA4Jii3YJ0UoyaKH3WkH8Qxxa1Il6uJ2n9pHzNwSXwrBMk7fgyQ7cjNIdLXxSyJt9AclnOAyS3TecyS+FHGbSXwOKSSi9cXmUQzGbJY67Lwbt2QgPMTL+6MEwmFZmM+31b/qyHLwmtGblmoPc75R3Ejlot367YwPN2w/M1P5zFL4i30DctLgOmZ3PHq1oGSNpXEt0IQbSg/E9/MxFptW+Kr6xz5Btdz0xv7JP6hry4K3V6P6Iyo4Su+su9fDNY1rnnt8f8WxV8AAUqEua7v5XIAAAAASUVORK5CYII=",
"text/latex": [
"$\\displaystyle 0.0$"
],
"text/plain": [
"0.0"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"((5**2 - 4) / 7) - numpy.sqrt(11 - 2)"
]
},
{
"cell_type": "markdown",
"id": "a345bd91",
"metadata": {},
"source": [
"__Problem 1.4.12.3__ Calculate the expression in __Equation 1.4.12.3__."
]
},
{
"cell_type": "markdown",
"id": "83ea4d64",
"metadata": {},
"source": [
"$$6 - \\lvert 5 - 8 \\rvert + 3(4 - 1) \\tag{1.4.12.3}$$"
]
},
{
"cell_type": "code",
"execution_count": 85,
"id": "2294e2e6",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOCAYAAADNGCeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQ4EaWT7Q2CMBCG0TgA0Q10Az82wA10BR3Bv/yVDdAJjG4gK8gGOoJxA3zepm0Q1IBeUt7rfbzt3dFOURSBkziOx+hH1gT94ewOsQ3RN3Y/Be/aY89l66GE4J4lhwKUUBNLlIJz50Tfol9kY2VdPg/WkrXGcXCBb1CJivFCjm6pClRN0NWnoUTEXSFQJWXJ2ITYh23IlHQjqdZLyxz2ykd80yFZfvBraAH+vM3NalwQiEgDMxP+iwwSNf4EaQK2GoDivUCQslEPffk/3QyCFUR90P9zOqXxABQsgWABjEB/I3T1rV2ZJKnhM9A9KXFIdMC9erOBcVEC+PI/2dPV8Axd/SpLhC0xZCjmOeCNbMQR2w39DO6s7QyqHPWrKuahPwGbS2D0goaxHgAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 12$"
],
"text/plain": [
"12"
]
},
"execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"6 - abs(5 - 8) + 3 * (4 - 1)"
]
},
{
"cell_type": "markdown",
"id": "2af81de6",
"metadata": {},
"source": [
"__Homework 1.4.12.1__ Calculate the expression in __Equation 1.4.12.4__."
]
},
{
"cell_type": "markdown",
"id": "5b5bb7a1",
"metadata": {},
"source": [
"$$\n",
"\\frac{3\\pi - \\lvert 1 - \\sqrt{2} \\rvert}{27 \\times 3 - 1}\n",
"\\tag{Equation 1.4.12.4}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "f1ca36ca",
"metadata": {},
"source": [
"## 1.5 Properties of real numbers"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are many useful properties of real numbers. We will explore some of these properties in this section."
]
},
{
"cell_type": "markdown",
"id": "1c70a1d0",
"metadata": {},
"source": [
"__Defintion 1.5.1__ The <span style=\"color:lightgreen\">__commutative property of addition__</span> states that the order of the numbers does not matter. We have that $a+b=b+a$ for two real numbers $a$ and $b$"
]
},
{
"cell_type": "markdown",
"id": "e3d5fbc0",
"metadata": {},
"source": [
"We use the comparison operator `==` to test if two numbers are equal. The output is a boolean value, either `True` or `False`."
]
},
{
"cell_type": "markdown",
"id": "670534e1",
"metadata": {},
"source": [
"__Problem 1.5.1__ Determine if $3+4=4+3$."
]
},
{
"cell_type": "code",
"execution_count": 86,
"id": "82f68eab",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 3 + 4 = 4 + 3\n",
"3 + 4 == 4 + 3"
]
},
{
"cell_type": "markdown",
"id": "853b78b5",
"metadata": {},
"source": [
"The result is the boolean value `True`, which indicates that indeed $3+4=4+3$."
]
},
{
"cell_type": "markdown",
"id": "8eb7e9fe",
"metadata": {},
"source": [
"__Defintion 1.5.2__ The <span style=\"color:lightgreen\">__commutative property of multiplication__</span> states that the order of the numbers does not matter. We have that $a \\cdot b = b \\cdot a$ for two real numbers $a$ and $b$."
]
},
{
"cell_type": "markdown",
"id": "eb9e81c9",
"metadata": {},
"source": [
"__Problem 1.5.2__ Determine if $3 \\times 4 = 4 \\times 3$."
]
},
{
"cell_type": "code",
"execution_count": 87,
"id": "9ace59bd",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 3 x 4 = 4 x 3\n",
"3 * 4 == 4 * 3"
]
},
{
"cell_type": "markdown",
"id": "566f75df",
"metadata": {},
"source": [
"Neither subtraction nor division is commutative. The `!=` comparison operator tests if two numbers are not equal."
]
},
{
"cell_type": "markdown",
"id": "02dfa4c9",
"metadata": {},
"source": [
"__Problem 1.5.3__ Determine if $3-4=4-3$."
]
},
{
"cell_type": "code",
"execution_count": 88,
"id": "33b6b9a4",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 3 - 4 is not equal to 4 - 3\n",
"3 - 4 != 4 - 3"
]
},
{
"cell_type": "markdown",
"id": "81d91652",
"metadata": {},
"source": [
"By using the _is equal to_ `==` comparison operator, we see that $3-4 \\neq 4-3$, by considering that the result is `False`. The $\\neq$ symbol means that the tow sides are not equal to each other."
]
},
{
"cell_type": "code",
"execution_count": 89,
"id": "008a7f17",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Alternative way to show that 3 - 4 is not equal to 4 - 3\n",
"3 - 4 == 4 - 3"
]
},
{
"cell_type": "markdown",
"id": "ae084221",
"metadata": {},
"source": [
"__Problem 1.5.4__ Determine if $10 \\div 5 = 5 \\div 10$."
]
},
{
"cell_type": "markdown",
"id": "49758550",
"metadata": {},
"source": [
"We use the `/` symbol for division and consider the cases using `!=` and `==`."
]
},
{
"cell_type": "code",
"execution_count": 90,
"id": "90428cd6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 10 divided by 5 is not equal to 5 divided by 10\n",
"10 / 5 != 5 / 10"
]
},
{
"cell_type": "code",
"execution_count": 91,
"id": "e8d3c593",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Alternative way to show that 10 divided by 5 is not equal to 5 divided by 10\n",
"10 / 5 == 5 / 10"
]
},
{
"cell_type": "markdown",
"id": "abb98f6d",
"metadata": {},
"source": [
"__Defintion 1.5.3__ The <span style=\"color:lightgreen\">__associative property of addition__</span> holds such that $a + (b + c) = (a + b) + c$, for three real numbers $a,b,c$."
]
},
{
"cell_type": "markdown",
"id": "a26649d1",
"metadata": {},
"source": [
"__Problem 1.5.5__ Determine if $3 + (4 + 5) = (3 + 4) + 5$."
]
},
{
"cell_type": "code",
"execution_count": 92,
"id": "10c50681",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that (3 + 4) + 5 is equal to 3 + (4 + 5)\n",
"(3 + 4) + 5 == 3 + (4 + 5)"
]
},
{
"cell_type": "markdown",
"id": "630ad94f",
"metadata": {},
"source": [
"__Definition 1.5.4__ The <span style=\"color:lightgreen\">__associative property of multiplication__</span> holds such that $a \\cdot (b \\cdot c) = (a \\cdot b) \\cdot c$, for three real numbers $a,b,c$."
]
},
{
"cell_type": "markdown",
"id": "ea686654",
"metadata": {},
"source": [
"__Problem 1.5.6__ Determine if $\\left( 3 \\times 4 \\right) \\times 5 = 3 \\times \\left( 4 \\times 5 \\right)$."
]
},
{
"cell_type": "code",
"execution_count": 93,
"id": "070de0b8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that (3 x 4) x 5 is equal to 3 x (4 x 6)\n",
"(3 * 4) * 5 == 3 * (4 * 5)"
]
},
{
"cell_type": "markdown",
"id": "b6e840a0",
"metadata": {},
"source": [
"Associativity does not hold for subtraction or division. We see an example of subtraction below, where the `==` logical operator tests if two numbers are equal and returns a value of `False` when comparing $(10-3)-2$ and $10 - (3 - 2)$."
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 94,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that (10 - 3) - 2 is not equal to 10 - (3 - 2)\n",
"(10 - 3) - 2 == 10 - (3 - 2)"
]
},
{
"cell_type": "markdown",
"id": "74674cf0",
"metadata": {},
"source": [
"__Defintion 1.5.5__ The <span style=\"color:lightgreen\">__distributive property__</span> holds such that $a(b + c) = ab + ac$, for three real numbers $a,b,c$. That is to say that multiplication distributes over addition. The distributive property also holds for subtraction. It does, however, not hold for division, nor for addition or subtraction over multiplication."
]
},
{
"cell_type": "markdown",
"id": "487703d4",
"metadata": {},
"source": [
"__Problem 1.5.7__ Determine if $3 \\left( 2 + 7 \\right) = \\left( 3 \\times 2 \\right) + \\left() 3 \\times 7 \\right)$."
]
},
{
"cell_type": "code",
"execution_count": 95,
"id": "c6ad59b2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 3(2 + 7) is equal to 3 x 2 + 3 x 7\n",
"3 * (2 + 7) == 3 * 2 + 3 * 7"
]
},
{
"cell_type": "markdown",
"id": "6340de00",
"metadata": {},
"source": [
"There is a special case of distributivity that holds for subtraction. We have that $a - (b + c) = a - b - c$, for three real numbers $a,b$ and $c$."
]
},
{
"cell_type": "markdown",
"id": "728b7110",
"metadata": {},
"source": [
"__Problem 1.5.8__ Determine if $3 - \\left( 2 + 1 \\right) = 3 - 2 - 1$."
]
},
{
"cell_type": "code",
"execution_count": 96,
"id": "a723524f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 3 - (2 + 1) is equal to 3 - 2 - 1\n",
"3 - (2 + 1) == 3 - 2 - 1"
]
},
{
"cell_type": "markdown",
"id": "544c36b7",
"metadata": {},
"source": [
"__Definition 1.5.6__ The number $0$ is the <span style=\"color:lightgreen\">__additive identity__</span> of real numbers. We have that $a + 0 = a$ for any real number $a$."
]
},
{
"cell_type": "markdown",
"id": "8a3e20a6",
"metadata": {},
"source": [
"__Problem 1.5.9__ Determine if $5 + 0 = 5$."
]
},
{
"cell_type": "code",
"execution_count": 97,
"id": "d43c18ce",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show to 5 + 0 = 5\n",
"5 + 0 == 5"
]
},
{
"cell_type": "markdown",
"id": "c265031b",
"metadata": {},
"source": [
"__Definition 1.5.7__ The number $1$ is the <span style=\"color:lightgreen\">__multiplicative identity__</span> of real numbers. We have that $a \\cdot 1 = a$ for any real number $a$."
]
},
{
"cell_type": "markdown",
"id": "0d4dbdfa",
"metadata": {},
"source": [
"__Problem 1.5.10__ Determine if $5 \\times 1 = 5$."
]
},
{
"cell_type": "code",
"execution_count": 98,
"id": "8a787178",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 98,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 5 x 1 i s equal to 5\n",
"5 * 1 == 5"
]
},
{
"cell_type": "markdown",
"id": "c533bff1",
"metadata": {},
"source": [
"__Defintion 1.5.8__ The <span style=\"color:lightgreen\">__additive inverse__</span> of a real number $a$ is the number $-a$ such that $a + (-a) = 0$."
]
},
{
"cell_type": "markdown",
"id": "6a2deb62",
"metadata": {},
"source": [
"__Problem 1.5.11__ Determine if $5 + \\left( -5 \\right) = 0$."
]
},
{
"cell_type": "code",
"execution_count": 99,
"id": "dad94fe9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 99,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 5 + (-5) is equal to 0\n",
"5 + (-5) == 0"
]
},
{
"cell_type": "markdown",
"id": "b72208fe",
"metadata": {},
"source": [
"__Problem 1.5.12__ Show that $3$ is the additive inverse of $-3$."
]
},
{
"cell_type": "markdown",
"id": "006a6079",
"metadata": {},
"source": [
"If $3$ is the additiive inverse of $-3$ then $3 + \\left( -3 \\right)$ should be equal to $0$."
]
},
{
"cell_type": "code",
"execution_count": 100,
"id": "e95c3964",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 100,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 3 is the additive inverse of -3\n",
"(-3) + 3 == 0"
]
},
{
"cell_type": "markdown",
"id": "0e9c923b",
"metadata": {},
"source": [
"Since the additive inverse of any number $a$ is $-a$ we have that the additive inverse of $-3$ is $- \\left( -3 \\right)$, which is $3$."
]
},
{
"cell_type": "markdown",
"id": "304fdebd",
"metadata": {},
"source": [
"__Definition 1.5.9__ The <span style=\"color:lightgreen\">__multiplicative inverse__</span> of a real number $a$ is the number $\\frac{1}{a}$ such that $a \\cdot \\frac{1}{a} = 1$ and $a \\ne 0$."
]
},
{
"cell_type": "markdown",
"id": "920aa9ba",
"metadata": {},
"source": [
"__Problem 1.5.13__ Show that $\\frac{1}{5}$ is the multiplicative inverse of $5$."
]
},
{
"cell_type": "code",
"execution_count": 101,
"id": "558ca8bf",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 101,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Show that 5 x (1/5) is equal to 1\n",
"5 * (1/5) == 1"
]
},
{
"cell_type": "markdown",
"id": "a0196442",
"metadata": {},
"source": [
"## 1.6 The uses of division"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In health data science we often have to consider values that are fractions. Fractions involve division, which we also commonly use in ratios."
]
},
{
"cell_type": "markdown",
"id": "751e805b",
"metadata": {},
"source": [
"### 1.6.1 Fractions and ratios"
]
},
{
"cell_type": "markdown",
"id": "44af3bea",
"metadata": {},
"source": [
"A <span style=\"color:lightgreen\">__fraction__</span> is a part of a whole and is calculated using division. A <span style=\"color:lightgreen\">__ratio__</span> similarly involves division."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We see both, and the fact that we accomplish both calculations by division, in __Equation 1.6.1.1__. In either case, $a$ and $b$ are integers and we always have that $b \\ne 0$. We also omit the case where $a=b$, which is simply $1$. We also remeber that if $a=0$, we have that our fraction or ratio is $0$."
]
},
{
"cell_type": "markdown",
"id": "e02c456f",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\text{Fraction: } \\frac{a}{b}, \\text{ where } b \\ne 0 \\\\ \\\\\n",
"&\\text{Ratio: } a:b , \\text{ where we calculate } \\frac{a}{a+b} \\text{ and } \\frac{b}{a+b} \\\\ \\\\\n",
"&\\;\\;\\;\\text{ and } a+b \\text{ is never equal to } 0\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.1.1}"
]
},
{
"cell_type": "markdown",
"id": "fd1bebfd",
"metadata": {},
"source": [
"It is important to understand the basics of fractions and ratios. The basics include the _mechanics_ of the parts of a fraction or a ratio and the comparison of different fractions and ratios."
]
},
{
"cell_type": "markdown",
"id": "644f80a4",
"metadata": {},
"source": [
"We have seen that a fraction or a ratio has a numerator and a denominator. Note that in the case of ratios we also use the terms divided or antecedent for the numerator and the term consequent for the denominator. A fraction describes a count of parts of a whole. By parts of the whole, we are referring to the denominator. The numerator describes how many of those parts we count."
]
},
{
"cell_type": "markdown",
"id": "788e53da",
"metadata": {},
"source": [
"Consider any shape, such as a square or a circle. We can divide a shape into equal parts. That would be $b$ equal parts. If $b=2$ then we have divided the shape into halves. If $b=4$, then we have divided the shape into $4$ equally sized parts called quarters. This is the value of the denominator."
]
},
{
"cell_type": "markdown",
"id": "197123d2",
"metadata": {},
"source": [
"The numerator, $a$, counts the number of $b$-parts of the whole. In __Figure 1.6.1.1__, we have divided the whole (of a circle) into $b=4$ equal parts (the denominator) and we consider $a=3$ of those $b=4$ parts (the numerator)."
]
},
{
"cell_type": "code",
"execution_count": 102,
"id": "0a798bd3",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 410,
"width": 389
}
},
"output_type": "display_data"
}
],
"source": [
"pyplot.pie(\n",
" [1, 1, 1, 1],\n",
" colors=['red', 'red', 'red', 'lightgray'],\n",
" wedgeprops={\"edgecolor\":\"k\",'linewidth': 1}\n",
")\n",
"pyplot.title('Figure 1.6.1.1')\n",
"pyplot.show()"
]
},
{
"cell_type": "markdown",
"id": "ad70a428",
"metadata": {},
"source": [
"In __Figure 1.6.1.2__, we visualize the difference between three eights and seven eights, seen in __Equation 1.6.1.2__. We keep $b=8$ fixed and compare $a=3$ to $a=7$."
]
},
{
"cell_type": "markdown",
"id": "714d2560",
"metadata": {},
"source": [
"$$\n",
"\\frac{3}{8} < \\frac{7}{8}\n",
"\\tag{Equation 1.6.1.2}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 103,
"id": "16391d9c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"text/plain": [
"<Figure size 640x480 with 2 Axes>"
]
},
"metadata": {
"image/png": {
"height": 266,
"width": 515
}
},
"output_type": "display_data"
}
],
"source": [
"fig, ax = pyplot.subplots(1, 2)\n",
"ax[0].pie(\n",
" [1, 1, 1, 1, 1, 1, 1, 1],\n",
" colors=['red', 'red', 'red', 'lightgray', 'lightgray', 'lightgray', 'lightgray', 'lightgray'],\n",
" wedgeprops={\"edgecolor\":\"k\",'linewidth': 1}\n",
")\n",
"\n",
"ax[1].pie(\n",
" [1, 1, 1, 1, 1, 1, 1, 1],\n",
" colors=['red', 'red', 'red', 'red', 'red', 'red', 'red', 'lightgray'],\n",
" wedgeprops={\"edgecolor\":\"k\",'linewidth': 1}\n",
")\n",
"ax[0].set_title('Figure 1.6.1.2')\n",
"pyplot.show()"
]
},
{
"cell_type": "markdown",
"id": "159c3cb4",
"metadata": {},
"source": [
"With a good understanding of the parts of a fraction or a ratio, we consider the _mechanics_. By the mechanics we mean, changes in the values of $a$ or $b$, where we consider a change in one, while keeping the other constant."
]
},
{
"cell_type": "markdown",
"id": "9c57a1bf",
"metadata": {},
"source": [
"If we keep $b$ constant and we increase $a$, we are counting more of the parts. The actual _overall_ value that the fraction or ratio describes gets bigger. If we decrease $a$, then we have fewer parts, and the overall value gets smaller."
]
},
{
"cell_type": "markdown",
"id": "dc196b47",
"metadata": {},
"source": [
"In __Figure 1.6.1.3__, we visualize the difference between three eighths (on the left) and three fifths (on the right), seen in __Equation 1.6.1.3__."
]
},
{
"cell_type": "markdown",
"id": "53dbcde0",
"metadata": {},
"source": [
"$$\n",
"\\frac{3}{8} < \\frac{3}{5}\n",
"\\tag{Equation 1.6.1.3}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 104,
"id": "4137c5f4",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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zfPly1q9fn+V7br75Zp577rk8va+IiIiISG4pHBCjypUrx+zZs7nuuutYuXIlAGvWrGHNmjVZnnfPPffw+uuvZ7nL6lx48ESff/45V199tf3hcPXq1axevTrLc+6++26GDx/O1VdfHag/zkk6derE/Pnz6d27N4sXLwZg69atbN269bTfU6NGDapUqZLn9961axd//fXXGZ+zatWqU35foAwePJgSJUrwwgsvkJKSwu7du5k4ceJJz6tUqRJjxozh8ssvD9h7r1+//ox//tTU1FN+PTExMU/v6+xESUtLY+XKlfbf7VMpWrQozz//PP379ydfvnx5em8RERERkdxSOCDG1axZkyVLljBq1Ci+/PJL/vnnHxITE6lYsSLNmzfn3nvv5corrwSw5+J7PB6KFSt22tcsVaoU8+fPt1/z77//zvKa9913n/2awda4cWMWLVrEzJkz+eabb/jtt9/YsWMHhw4dIj4+nrJly1KnTh1atWrF1VdfTevWrfF4PCGpLdg8Hg9PPvkkt956K6NHj2bGjBls3bqVI0eOULp0aRo2bEjnzp3p3bs3BQsWNF1uQPTp04f27dvz448/8scff/D333/z77//cujQIQCKFStGxYoVadKkCVdccQU333yzPf1ERERERMQUjzc7e8OJhIF169bZi9zVrVv3pE4AERERERERyR0tSCgRw9mO3rx5c4OViIiIiIiIRBeFAxIRtmzZwptvvmk/7tatm8FqREREREREoovCATGuS5cuTJkyxd7r/UQ//PADl1xyiT1nu2HDhnTo0CGEFYqIiIiIiEQ3rTkgxtWoUYMtW7ZQtGhRmjZtSo0aNShQoAD79+/nzz//5N9//7WfW7hwYX799ddTbkEnIiIiIiIiuaNwQIzzhwNnU6tWLSZOnMiFF14YgqpERERERETcQ+GAGLdp0ya++uor5s2bx8aNG9m3bx/79+8nLi6OsmXLcuGFF9KpUye6d+9O/vz5TZcrIiIiIiISdRQOiIiIiIiIiLicFiQUERERERERcTmFAyIiIiIiIiIup3BARERERERExOUUDoiIiIiIiIi4nMIBEREREREREZdTOCAiIiIiIiLicgoHRERERERERFxO4YCIiIiIiIiIyykcEBEREREREXE5hQMiIiIiIiIiLqdwQERERERERMTlFA6IiIiIiIiIuJzCARERERERERGXUzggIiIiIiIi4nIKB0RERERERERcTuGAiIiIiIiIiMspHBARERERERFxOYUDIiIiIiIiIi6ncEBERERERETE5RQOiIiIiIiIiLicwgERERERERERl1M4ICIiIiIiIuJyCgdEREREREREXE7hgIiIiIiIiIjLKRwQERERERERcTmFAyIiIiIiIiIup3BARERERERExOUUDoiIiIiIiIi4nMIBEREREREREZdTOCAiIiIiIiLicgoHRERERERERFxO4YCIiIiIiIiIyykcEBEREREREXE5hQMiIiIiIiIiLqdwQERERERERMTlFA6IiIiIiIiIuJzCARERERERERGXUzggIiIiIiIi4nIKB0RERERERERcTuGAiIiIiIiIiMspHBARERERERFxOYUDIiIiIiIiIi6ncEBERERERETE5RQOiIiIiIiIiLicwgERERERERERl1M4ICIiIiIiIuJyCgdEREREREREXE7hgIiIiIiIiIjLKRwQERERERERcTmFAyIiIiIiIiIup3BARERERERExOViTRcgIjmTkZHB0aNHOXLkCEeOHCE5OZmMjAwyMjJYuXIl6enpxMfHU79+fWJiYoiJiSE+Pp6iRYtStGhRChUqREyMckERERE5Pa/XS1JSEgcPHsTr9eb4+z0eD8WLF6dIkSJ4PJ4gVCgigaZwQMSgjIwM9u3bx44dO9i5c2eWXzt27GD//v12CJCYmMiRI0c4evRont7T4/FQuHBhihYtSpEiRezQoGzZslSsWNH+ValSJXtcunRp/WAXERGJYMnJyezbty9Hv44fP57n942Li6NMmTKULVuWMmXKZOtXgQIFAvAnFpGc8nhzEwWKSLYlJyezceNG1q5dy5o1a1i7di1r165l48aN7N69m7S0NNMlnlX+/PmpWLEitWvXpk6dOll+1axZk/z585suUURExPW8Xi+7du1i1apVrF692v599erVbNu2zXR52VahQgXq169PvXr1qFevnj2uUqWKblaIBJHCAZEASUtL459//mHx4sUsW7bMDgE2b95MRkZGrl83Fijq+FXkhHE8kA9rAZGPgRTf+H4gA0gHjgNHgMQTfvf/Ss91dZAvXz5q1apFnTp1qFu3LhdccAEXXnghderUIV++fHl4ZRERETmVtLQ0Nm7ceMoQ4PDhw3l67VigjO9XWaAk1nVGTqUDh4B9jl8peaoMihQpQt26de2wwP/7OeecQ1xcXB5fXUQUDojkQmpqqh0E+H/99ddfOWq/KwtUASr6flVyjP2PywI5aawrifWDOA5Izub3eLHCgz3ADmCn79eJ423AgRzUUrhwYTso8P+qW7euAgMREZEc8Hq9rF+/nnnz5jFv3jz+/PNP1q1bR2pqarZfoyRQF6hA5gf/0/0qBgTj3rwX6+bEvrP82g2sBfbm4LXz5ctH7dq1adasGZdeeilt2rShfv366jIQySGFAyLZkJiYyG+//cbPP//ML7/8wpIlS0hOPvvH76JAHcevur7fz8X64RtouQkHcuIAsA7rh/aJv5Ky8f2FCxfmwgsvpG3btrRt25ZWrVpRsGDBIFQqIiISmdLT01m5ciVz5861A4Fdu3Zl63urA/WA+if8XpbgfOAPpv3AasevVb7fN2F1Rp5NmTJluOSSS2jTpg1t2rThggsuIDZWy62JnInCAZFTSExMZP78+fz888/8/PPPLFy48KxrA5wDXOj71RQ4HyhPaH8YBzscOB0vVnfBSmCx49eWs3xffHw8rVq1yhIWaBEiERFxk5SUFBYtWsS8efOYO3cuv/322xmnBuTHutlwYgBQBygcioINO451o8IfFvh/X+372ukUKVKE1q1b06ZNGy699FJatGihGxQiJ1A4IILVsrd06VKmTp3KzJkzzxoGnAM0IzMMuAAoEZJKz8xUOHA6+4AlZIYFizhzYOAPCzp06EDnzp0577zz1BIoIiJRJSMjg4ULF/L9998zd+5cfv/99zNOSywKXAy08f1qTs6mHLpFKtY1x1xgHvArcPAMz8+fPz/Nmzfn0ksv5ZprruHiiy/W1EdxPYUD4lrHjx9nzpw5TJ06le+++47t27ef9rn1gba+X5dhdQSEo3ALB05lK/AL8LPv14YzPLdWrVp07tyZ6667jjZt2mhXBBERiUjJycnMnj2bb7/9lqlTp7Jz587TPrcsVghwqe/3Rmjv8dzIAP7GCgrmYYUGO87w/DJlytCpUyduuOEGrrzySgoVKhSKMkXCisIBcZU9e/Ywffp0vvvuO2bOnMnRo0dP+TxnGHAp1gI+kSASwoETZTcsKF68ONdccw2dO3fmmmuuoWTJkiGqUEREJOcSEhKYNm0a33zzDTNmzCAxMfGUz6tBZlfApVjTA9QzF3herPUKnGHButM8t2DBglSvXp1WrVoxaNAg6tSpE6oyRYxSOCBRLzExka+//pqxY8cya9asU24rGA+0B64DOgGVQ1xjoERiOHCijcB3vl+/AKea3JE/f36uvfZa7rjjDjp16qR1CkREJCwcPXqUadOmMWHCBL7//vtTLl5cALgSuB64Cqga4hol0y7gR+Bb4H9YuymcyjXXXEPXrl25/vrrKV68eMjqEwk1hQMSlVJTU5k1axZjx47lm2++ISnp5LX0y2IFAdcBVwBFQlxjMERDOOB0COuH9XfA977HJypWrBi33HILd9xxB5dddhkxMTEhrFBERNzu2LFjzJgxgwkTJjBt2rRTXnOUwrrmuAErEHDDwoGR5jgwGyso+Apr3aQTxcXFcc0119ClSxc6d+5MkSLRcPUokknhgEQNr9fLwoULGTNmDOPHj2fv3pN3yK0OdMUKBFoC0bbsTLSFA06pWIsLfQtMwEr7T1SlShW6devGHXfcQaNGjUJan4iIuMvatWv54IMPGD16NIcOHTrp6+WAW3y/2qB1AyLJrcBk37gYkHCK5xQuXJg77riDvn376ppDoobCAYl4hw4d4osvvmDkyJH8/fffJ329JNAFuB24CIjm+8rRHA44pQNzgDHAFE7dBtisWTMeeOABunbtSuHCukcjIiJ5l5aWxrRp0xg+fDizZs066eulgJuxrjsuQ4FAJErC6i5Nwlr74SjWLggTgEmc+ubExRdfTL9+/bjpppuIj48PWa0igaZwQCLWokWLGD58OOPHj+fYsWNZvlYA6AzcAXTA+rDsBm4JB5ySsKYdjAVmcPIaBcWKFePOO++kT58+nH/++SGvT0REIt+uXbsYNWoUI0eOZNu2bVm+Fg/cBnTHWr9I++pEtq+wAh6wFodc4/haOtZihl8C4zj55kS5cuW45557uP/++6lWrVrQaxUJNIUDElGSk5OZNGkSQ4cO5Y8//jjp65cAdwM3AW5cLsaN4YDTPmAi8DFWyn+iyy+/nIceeojOnTsTG6v7OSIicnper5dff/2V4cOHM2XKFFJTU7N8vRbQB+gFlDZRoATF7Vgf/AFeAp4+zfMSsDoYhwH/nPC1mJgYOnXqRL9+/bjiiiu0HpJEDIUDEhH279/P0KFDGT58OHv27MnyteLAXcD9gNvvC7s9HHBaBHyAle6fuDRU1apVefjhh7n//vspWrRo6IsTEZGwdeTIEcaMGcPw4cNZuXJllq95sBYW7Iu1sKA+8kWXZKy1IvxrDBzh7AtWe7G6CYZhdR2c2MF4zjnn0KdPH3r27EmpUqUCWa5IwCkckLC2fft23n77bUaOHMnRo0ezfK0R8CBWG59mlFsUDpzsMPAZMJST9zMuWbIkDz74IA8//DBlypQJfXEiIhI2tm7dypAhQ/j0009JTMzaMF4WuAe4D6hhoDYJjelY4Q9YnSEbcvj9O4FRwEhg+wlfK1CgAN27d+eJJ57g3HPPzVuhIkGicEDC0rp16xgyZAifffZZlja+fFjzwB7EmkLgMVRfuFI4cHoZwCzgfaxtEZ0nvkKFCnHvvffy6KOPUrWqdpwWEXGTHTt28Nprr/Hhhx+SkpKS5WsXAf2wrj20zFz0uxsY7Rs/CbySy9dJw1oPaRjw0wlfy5cvH3fddRfPPPMMNWvWzOU7iASHwgEJK0uXLuX1119n0qRJOP9qFgB6A/9Fif2ZKBzInlXAEKy5gs72v/z583PHHXfw+OOPU7duXTPFiYhISOzevZvBgwczYsQIjh8/bh8vhLWgcR+giaHaJPRSgfLAQd/jA1jXVXm1Gmua46dY3Yx+sbGx3H333Tz11FNavFDChsIBCQsrVqzgySefZNq0aVmOF8NK7PtjnbDlzBQO5My/wJtYLYDO/S48Hg9du3blpZdeonbt2maKExGRoNi3bx9vvPEGQ4cOJSkpc1WawsDDwKNogUE3moW1jgRAVaxrhEA6AryHdd1xyHE8Li6Oe++9l0GDBlG5cuUAv6tIzigcEKM2b97Ms88+y5gxY7J0CpQDHsFa8MeNuw7klsKB3NmD9QN7KCen+vfddx/PPPMMFSpUMFOciIgExMGDB3nrrbd49913s6wpUBDrRsRjWGsLiDs9gLVWAMB/gLeD9D6Hgf/z/UpwHI+Pj6dPnz48/vjjuuYQYxQOiBF79+7llVdeYcSIEVnm91UGBmHN+SpoqrgIpnAgbxKAEVip/j7H8cKFC/Of//yHgQMHUqxYMTPFiYhIrhw+fJh33nmHt99+m4SEzI9j8Vg7HQ0C9FHM3dKBSlg3C8BaWDDYfycOAG8B7wLOJbcLFizIgw8+yGOPPabFkiXkFA5ISCUmJvL222/z5ptvcuTIEft4SayFX/qhUCAvFA4ERgLWD+y3yPoDu3Tp0jz11FP06dOHAgUKmClORESyJTExkffff5833niDgwcP2sfzY+088CRQxVRxElbmApf5xhWBHSF8773AG1jdi84pjkWKFKF///4MGDBAWyBKyCgckJBIT0/n448/5plnnmHPnj328YJY0wceA0qYKS2qKBwIrN3Ay1hthqmO49WqVWPw4MF06dIFj0d7ZoiIhBOv18ukSZP4z3/+w44dmR/z8gG9gKeB6qaKk7DUH2t6IVjTC0YYqGEX8DrW4oXOa7hSpUoxZMgQevXqRUxMjIHKxE0UDkjQLVy4kL59+7Jo0SL7WD6s1P5ZrDYuCQyFA8GxEXgGGHfC8csvv5yhQ4dy3nnnGahKREROtH79evr168fMmTPtYzHAnVjncS0xKyfKAKoB232PN2M2PNoOvAp8RNYbExdddBEjRoygUaNGZgoTV1D8JEGzb98+7rvvPlq2bJklGLgF+AcrGVUwIJGgFjAWWAp0cByfM2cOjRs3ZuDAgVmmyYiISGgdP36cF154gQYNGmQJBjphXXN8ioIBObU/yQwGymK+q6QyMAxYD3R3HJ8/fz5Nmzbl0Ucf1TWHBI3CAQm49PR0Ro4cSd26dfnoo4/sXQjOB34GJgF1DNYnkltNgBnAVKCm71haWhpvvvkm9erVY/z48agZS0QktGbNmkXDhg15/vnnSU62+uaqAl9jna/rmixOwt4Ux7ijsSpOVg3rxsSPZF43p6en8/bbb1O/fn2mTJmiaw4JOE0rkID6888/6devX5ZOgaLAC8CDWIsASfBoWkHoHAOGAK+R9b/15Zdfzvvvv8/5559vpjAREZfYsWMHAwYMYMKECfaxWGAA1rTFwqYKk4jhxeoo2eR7vBY411w5p5WMtWjhK8Bxx/Frr72W999/n1q1apkpTKKOOgckIJKSkhgwYACtWrXKEgzcDqzB2i9WwYBEk4LAc1jtqp0dx+fMmUOTJk148cUXSU1NPfU3i4hIrqWlpfHee+9Rr169LMFAG6zpX4NRMCDZs4zMYKAU4RkMgLXt5tPA38A1juPff/89559/Pi+//LLdNSOSF+ockDybP38+PXv2ZN26dfaxBlhbslx22u+SYFDngDnTgIfJvMgAuOCCC/j000+1eJCISID8+eefPPDAAyxdutQ+VgbrrmoPQPvHSE48jXU3HqAr8KXBWrLLC3yFtcPCdsfxunXrMnz4cNq1a2emMIkK6hyQXDt27BiPPvool1xyiR0MxGO1Wi9BwYC4SyesRP8prN04AJYuXUqzZs146aWX1EUgIpIHx44d46GHHqJVq1ZZgoF7gNVATxQMSM54gcmOx4NMFZJDHuBmYBXWFBr/NceaNWto3749d911F4cPHzZWn0Q2dQ5IrixYsICePXuydu1a+1grYDRQz1hVos6B8LAI60L1b8expk2b8umnn9KwYUMzRYmIRKiVK1fStWtX/v4786zaCGsv+ouMVSWR7m+sTleAYkCkfpxeDjwALHAcq1mzJuPGjaNVq1aGqpJIpc4ByZFjx44xcOBALrnkEjsY8HcL/IqCARGAZsBi4EkyE/0lS5Zw4YUX8sorr6iLQEQkG7xeLyNGjKB58+Z2MFAQeAvrHKtgQPLCuUtBe2NV5F0jrGvwUUBx37FNmzZxySWX8Nprr5GRkWGuOIk46hyQbFu1ahVdunRhxYoV9rGWWHsHKxQID+ocCD+n6iK46KKL+PLLL6lWrZqZokREwtyBAwfo3bs333zzjX2sITAeOM9UURJVGmPddQf4HeuaNtJtAboD8x3H2rVrxxdffEGlSpUMVSWRRJ0Dki2fffYZzZo1s4OBeKzVgH9DwYDImfi7CAaRecKdP38+TZo0YerUqeYKExEJU3PnzqVx48ZZgoEHgT9RMCCBsY7MYKAw0REMAFQHfgGeIXMNjtmzZ9OoUSOmTZtmrjCJGAoH5IwSExO566676NmzJ0lJSYA1P2sx8BiZLdMicnrxwKtYSX4N37GDBw9y/fXX88gjj2j7IRERrC0Kn3vuOS6//HK2bdsGWNvLfQu8DxQwWZxEFeeUgmhbQDsWeBGYDVT2Hdu/fz+dO3emf//+uuaQM9K0Ajmtv/76iy5durBmzRr72L3AO0AhU0XJGWlaQfg7hLW6tvPC5MILL2TChAnUrl3bTFEiIoZt2bKF22+/nd9++80+1hYYQ+YHHJFAaY417Q+sD9GXG6wlmPYDvbECNr/GjRszfvx46tVT76+cTJ0DchL/AkAtW7a0g4EiwDjgQxQMiORFCWASMAwrxAFYvHgxF1xwARMmTDBWl4iIKVOmTKFJkyZ2MJAPeBn4EQUDEnhbyAwGChK9wQBAaeBrYChWFyNYN/8uvPBCPv74Y3SPWE6kcECySEpKonv37vTt29duO7oAWAJ0M1qZSPTwAH2BP4BzfceOHDlC165deeihh7SbgYi4wvHjx7n//vu55ZZbOHToEGDNmZ4LPIWmLkpwfOUYu2HHCw/Qj6xrdiQlJXHPPffQtWtXEhISzBUnYUfTCsS2detWbrjhBpYsWWIfewh4g8y0UcKbphVEniNAH2Cs41i7du2YOHEipUuXNlSViEhw7dmzhxtuuIEFCzJ3Z78NGInVYSUSLJdgLagN8B3QyWAtoZYEDMD6d+bXoEEDpk2bRvXq1Q1VJeFE4YAAsGDBAm688UZ2794NWNMIPgNuMlqV5JTCgcjkBT7B6iZI8R2rVasWU6dO5fzzzzdXmIhIEPzzzz906tSJTZs2AVZr9/vA3WSusC4SDDuAKlg/d918rTQFa/2jQ77H5cuXZ+rUqbRo0cJcURIWNK1AGD16NG3btrWDgZrAAhQMiISKB2vBoDlAed+xjRs30qpVK213KCJR5ccff+Siiy6yg4FKWHdxe6NgQILva6xgAMDNH4Nvxppm4J/auHv3bi677DImT55ssCoJBwoHXCwtLY0BAwZw9913k5Ji3a+8HFiItV2hiITWRVj//pr6HicmJnLDDTfw6quvatEgEYl4o0aN4pprruHw4cOAtabRn77fRULBuVPQQ8aqCA/nYt0MvNT3+Pjx49x6660MHjxY1xwupmkFLnXw4EG6du3KzJkz7WP9gP8D8hurSvJK0wqiQxJWe61z74IuXbrwySefUKiQ9gsRkciSkZHBoEGDGDJkiH2sM9YuSEWMVSVusxeoAGQAsVjXSbpLav13uA/43HGsd+/ejBgxgvz59anAbfRvwoW2bdtGmzZt7GAgFvgAa5sTnQJEzCsEfAm84jg2YcIErrzySg4cOGCoKhGRnEtKSuLWW2/NEgw8gtXerWBAQulbrGAArA49fQiyxAOfYm0f6vfxxx/ToUMHDh48aKYoMUb/Llxm1apVXHTRRfz999+Atf/pj8D9RqsSkRN5gCeBb8i8gJ4/fz5t2rRh27ZtxuoSEcmunTt3ctlll/HVV9bmcTHAMKwuRW1TKKHmnFLQx1gV4cmDtX3ol2TuUDZ79mwuuugiNm7caK4wCTlNK3CRP/74g2uvvda+81gL+AE4x2hVEkiaVhCdlgLXALt9j6tWrcoPP/xA/fr1DVYlInJ6K1asoGPHjmzduhWAosBEoIPRqsStDmIt+JuKFUyloDukp7MAuB5rGgZAmTJl+Pbbb7nooosMViWhon8XLjFjxgzatWtnBwNNsFYHVjAgEv4uwPr3Wtv3eOvWrVxyySVZ9gcXEQkXM2bM4OKLL7aDgWpY5zAFA2LKd1jBAEAj9AHoTFoDvwP+2w/79u2jXbt2fPnllwarklDRvw0X+Pzzz+ncuTNJSUkAtAN+wVqURUQiQ22si2v/TgYHDhygffv2TJ8+3WBVIiJZTZo0ic6dO3PkyBEAmgN/AA2NViVu55xScI+xKiJHLWA+0N73ODk5me7du/PBBx8YrEpCQeFAlHvzzTfp0aMH6enpANwKfA8UM1qViORGeWAOmT+sjx07xvXXX8+nn35qrigREZ9JkybRrVs3+5rjZuBndDNCzDqCNY0WrA8+9xmsJZKUAGaQNUzp06ePAoIop3AgSnm9Xp5//nkGDhxoH+tH1oVGRCTyFAOmA7f5Hqenp9OrVy9GjBhhsCoRcbsTg4G7sdYY0OarYtr3ZK7DVB9rly7JnvzAh8ATjmMKCKKbwoEo5PV6eeaZZ3jhhRfsYy8C76PVgUWiQTxW0PeQ41jfvn15//33DVUkIm42ceLEk4KBj9BFpoQH55SCnqaKiGAe4FVODgh0UyI6abeCKJORkUGrVq1YuHChfewdoL+xiiSUtFuBu3ixth56zXGsU6dOfPfdd4YqEhG3mThxIt27d1cwIGEpCSjr+93j+72A0Yoilxdri+XXHceGDx9Onz7aGDKa6NwdRTIyMmjZsmWWYGAoCgZEopUHeAV4xnFs2rRpdO7c2VBFIuImJwYDvVEwIOHlB6xAAOBcFAzkhb+DYJDjWN++fdVBEGV0/o4SXq+Xp59+mkWLFmU5vug0zxeR6ODh5Dm906ZN0xQDEQmqCRMmnBQMfIguLCW8OKcU3G6siujhvylxYkAwfPhwQxVJoGlaQZR4/vnns6wx4NQTGB3SasQUTStwn9fJ+kPaSe1+IhIMEyZM4Pbbb1cwIGEtGSgHJPgeHwGKmCsnqpxqWuOwYcPo27evoYokUHQejwKDBw/OEgw8+eST3HnnnfbjT4FeoS9LRILsxGDgarJOMejbt6+2ORSRgDoxGLgHBQMSnn4iMxioiYKBQPJ3EDzpONavXz91EEQB7eYR4T799FOeeCJz/dDHH3+cbt264W8I+eKLL6zn+b6uDgKR6HCqYOB/WGl+KpkLBt1zzz2ULVuWjh07hrhCEYk2pwoGRqJgQMKTc0pBF2NVRC8P8LJv/Krv9379+gGogyCCaVpBBJs+fTrXX3+9/UP6oYce4r777rO/7vV6eeONN+yAADTFINppWoE7nC4Y8PNiLUTqX3WgYMGC/PTTT7Ru3TpEFYpItPnhhx/o2LGjggGJCKlABeCA7/F+oJS5cqKaF3iazIAAYMyYMdx+u1Z5iEQKByLUggULaN++PceOHQOge/fuPPHEE3g8nizPU0DgLgoHot/ZggG/DKAbMNH3uGTJkvz666+cd955Qa5QRKLNihUruPjiizly5AigYEDC34/Alb5xVeBfg7W4wYkBQVxcHD/99BOXXHKJwaokN3Rej0D//PMPHTt2tIOBq6++mscff/ykYADA4/EwcOBArUEgEgWyGwyAdXL/HGjve3zw4EGuvvpqtm7dGsQKRSTa7Nq1i06dOtnBwI0oGJDw55xScLOxKtzDP8XA37+ckpLCDTfcwPr16w1WJbmhzoEIs3XrVi666CK2bdsGQMuWLRk+fDhxcXFn/D51ELiDOgeiV06CAacE4HJgie/xeeedx7x58yhVSg2WInJmSUlJtG3bloULFwLQDPiFk7dPFQkn6UBlYLfv8U6sKQYSfKlAR2CW73GdOnVYsGCBrjkiiILfCHLw4EE6dOhgBwP169fnnXfeOWswAOogEIlkuQ0GAIoB3wO1fY//+ecfOnfuTFJSUgArFJFok5GRwZ133mkHA1WBqSgYkPA3n8xgoAIKBkIpPzAJ8E9gXLt2LTfffDMpKSkGq5KcUDgQIdLS0rj11lv5559/AKhatSrDhw+nSJHsb8yigEAk8uQlGPArD/zg+x1g/vz59O7dGzWOicjpDBo0iK+++gqAosA0oKLRikSyxzml4HpjVbhXcWA6UM73+Oeff+b+++/XNUeEUDgQIR599FF++uknAEqVKsXIkSMpU6ZMjl9HAYFI5AhEMOBXG5hB5j7P48eP5/XXXz/Dd4iIW3300UcMGTIEsC4UJwCNjFYkkj0ZZA0HBp3uiRJUNbA6jQr4Hn/66ae89tpr5gqSbNOaAxFg1KhR3HvvvQDExsby8ccf07Rp0zy9ptYgiE5acyB6BDIYcPoWuME39ng8fPPNN1x33XUBeGURiQY//vgjHTp0sLcsHAZox3KJFH8ArXzjssAeg7UITAZudTweP348Xbp0MVWOZIM6B8Lcr7/+St++mT+Wn3nmmTwHA6AOApFwFqxgAKwWy5d9Y6/Xy+23387ff/8doFcXkUj2zz//cMstt9jBwCMoGJDI4uwauNZYFeJ3C+DsF+jRowcLFiwwVY5kgzoHwti///5Ls2bN2Lt3LwC33347TzzxREDfQx0E0UWdA5EvmMGAnxfohtUqDFCrVi3+/PNPSpcuHeB3EpFIsXv3blq1asXmzZsB6Ax8DeQzWZRIDniBc4CNvsergbrmyhEfL3AP8InvcdmyZfn999+pVauWwarkdNQ5EKaOHj3K9ddfbwcDrVq14r///W/A30cdBCLhIxTBAFj7EX8C+HuQNm7cyK233kpqamoQ3k1Ewt2xY8e44YYb7GDgAmAcCgYksvxFZjBQEgUD4cIDjMDaVhlg7969dOzYkUOHDpkrSk5L4UAY8nq99OrVi2XLlgFQrVo13nzzTWJjY4PyfgoIRMwLVTDgVwj4hswdDObMmcOAAQOC+I4iEq4efvhhfv/9d8DaH/47MhcvFYkUkx3jq41VIacShzXlo57v8erVq+nRo4d2MAhDCgfC0Pvvv8+kSZMAKFy4MO+//z7FixcP6nsqIBAxJ9TBgF9VrLbhON/joUOHMnHixBC8s4iEi3HjxjFq1CgACmIFA5WNViSSO9qlILyVxNri0D+BcerUqbz77rsGK5JT0ZoDYWbx4sW0bt3abu99//33adu2bcjeX2sQRDatORB5TAUDTqOAe33jYsWKsWTJEmrXrh3iKkQk1NasWUOzZs1ITEwErJ/1PY1WJJI7/wDn+8bFgMMGa5Ez+x7o6Bvnz5+f3377jebNm5ssSRzUORBGEhIS6NKlix0M9OjRI6TBAKiDQCSUwiEYAOgN3O4b+89DycmKl0Si2bFjx7jtttvsYKAHCgYkcjm7BtoZq0Ky41pgoG+cmprKbbfdpvUHwojCgTDh9Xq577772LBhAwANGzakf//+RmpRQCASfOESDEDmYkHn+h4vXrw44DujiEh4ufPOO1m+fDkA9YFhZssRyRNnOPCYsSoku14BWvvGmzdv5u6779b6A2FC4UCYGDVqFBMmWBuLFS1alCFDhpA/f35j9SggEAmecAoG/IoCE4F43+N33nmHqVOnGqxIRILloYceYsoU6+NUAax/+4WNViSSe+uxdioA6+9x6zM8V8JDfmA81nRYgK+//pqhQ4carEj8FA6EgRUrVvDwww/bj1944QWqVKlisCKLAgKRwAvHYMCvCfC243HPnj35999/DVUjIsHw+++/Z7kIzwC2mytHJM+cXQOXGqtCcqoa8Jnj8cCBA1mxYoWpcsRH4YBhR48e5bbbbuP48eMAdO3alSuvvNJwVZkUEIgETjgHA359gJt944MHD9KtWzfS0tJMliQiAZKens7jjz+e5VgK0AG4BysoEIk0znDgv8aqkNzoDDziGycnJ3P77bfbn4nEDIUDhj3++OOsXr0agHr16vHf/4bfaU0BgUjeRUIwANb6A6OAGr7H8+fPZ/DgweYKEpGAeeONN5g7dy5gbWVaz/G1j4FawFYDdYnk1r/AQt+4AFqMMBK9BjT0jVesWMGTTz5pshzX01aGBs2ZM4d27azTWIECBZg0aRI1atQwW9QZaJvD8KetDMNTpAQDTn8AFwPpWFsNLV68mIYNG57lu0QkXC1ZsoSWLVuSlpaGB/gZuBDrrt0ox/NigQ+wdjERCXfvAP/xjdsBP5krRfJgBdCczGvXWbNmccUVVxisyL3UOWBIYmIid999t/34kUceCetgANRBIJIbkRgMALQkc8Xn1NRUevbsaW+zKiKRJSkpie7du9tThJ7AmptdGPgImAAU9z03DWuKQQffWCScOacUmNnjSwKhIdb1kl+PHj3Yv3+/qXJcTeGAIY8//jibN28G4MILL6Rbt25mC8omBQQi2RepwYDfc8D5vvGSJUs0vUAkQj322GOsWbMGsLoFnj/h67cBy8i6yvsPQDlgafDLE8mVncBvvnEccJ3BWiTvHgb8q67t2LGDPn36mCzHtTStwIDZs2fTvn17AAoWLMiUKVOoWrWq4apyRlMMwpOmFYSPSA8G/BYBrcicXrBo0SIaNWpkuCoRya4//viD1q1b4/V6KQQsAeqe5rlpwAtYe5D7Lw49wLOcHCiImDYc6OcbXwLMM1iLBMYOoBHg7xmYNm0aHTt2NFiR+6hzIMQSExPp3TtzJl///v0jLhgAdRCInEm0BAMAzQD/2uaaXiASWdLS0njggQfw3wd6mdMHA2CtN/AS1rztSr5jXqzA4AIgIXiliuSYc0rBg8aqkECqBLzrePzggw+SlJRkqhxXUjgQYo899pg9naBZs2YRM53gVBQQiJwsmoIBv2eBBr7x0qVLef3118/0dBEJE8OHD2fZsmUANAYeyub3XQ78RdY27WVABWBG4MoTybV9wC++cSxwq8FaJLC6k7nrxObNm3n11VdNluM6mlYQQnPnzuWyyy4DInc6waloikH40LQCs6IxGPBbjLVIoX96wbJlyzjvvPMMVyUip7Njxw7q1avHkSNHAJhP1jUFssOL1br9KFl/pvTE2vpQd5jElI+xFs4Ea5X7Pw3WIoG3Gmt6QSrWNcfy5cupV6/eWb5LAkHn9RBJTU2lX79+9uNHHnkkKoIBUAeBCER3MADWImZP+Mapqak8+OCDKFsWCV8DBgywg4F7yXkwANZ6A/2wPnjVdxz/FKiBtce8iAnOKQUPGKtCgqUeWXdM6tu3r645QkThQIgMGzaMlStXAnD++efTpUsXwxUFlgICcbNoDwb8ngJq+sZz5sxhwoQJJssRkdOYNWuW/e+zDPBaHl+vEdbipPc5jm0FamNthSgSSoeAH33jfFidLBJ9nsQKIcG65hg3bpzBatxD0wpCYOfOndStW5cjR47g8XgYO3YsDRs2NF1WUGiKgVmaVhB6bgkG/KYBnX3jSpUqsXr1aooWLWqyJBFxOH78OA0bNmT9+vUAfEJgg/rJWJ0IhxzHrsQ6N8QF8H1ETucL4C7fuAnabjOaTQc6+cbly5dn9erVlChRwmBF0U+dAyHw2GOP2a19N910U9QGA6AOAnEXtwUDYP2Q9v+g3rFjBy+++KLJckTkBEOGDLGDgUuAHgF+/VuwFie82HFsFlAeq7tAJNicUwp6n/ZZEg06Ajf6xrt37+bpp582WY4rqHMgyJyLEBYvXpzvvvuOkiVLGq4q+NRBYIY6B0LHjcGA30bgPKy/Y7Gxsfz1119anFAkDKxfv54GDRqQnJxMPqwP8Q3O8j25lYa17eHLQIbvmAd4GlBkKMGSCJQFjmPd4TyGOlai3VasNU+OYt2E/PPPP2nWrJnhqqKXOgeCKC0tjQcfzNx59eGHH3ZFMADqIJDo5uZgAKAWmYsT+s9zyplFzPJ6vTz44IMkJ1vR8H8IXjAA1vZxLwCzgcr+GrACg8ZknXYgEijfYwUDYC1ap2Ag+lUFnvONvV4vffr0IT093WRJUU3hQBANGzaMFStWAFC/fn1uvvlmwxWFlgICiUZuDwb8HkeLE4qEkylTpvDDDz8AWS+mg+0y4C/gesex5UBFrHUIRALJOaWgp6kiJOQeITPsXLRoESNHjjRYTXTTtIIgOXjwILVq1eLQoUMAjB07lkaNGpktyhBNMQgdTSsILgUDWX0HXOcbV6tWjbVr1xIfH2+yJBFXSkxMpF69emzfvh2Ar8icpxsqXmAEMICsP3/uwvp5r7tRklfHsKYUHMWawpIEFDBakYTSr0Ab37h48eKsWbOG8uXLmywpKulcHSSDBw+2g4HrrrvOtcEAqINAooOCgZN1xvrvAPDvv/8yYsQIk+WIuNa7775rBwMdgRsM1OAB+gILsdYk8fscqA5sMVCTRJcfsIIBgHNQMOA2l5D52eHw4cO8+uqrJsuJWuocCILt27dzzjnncPz4ceLi4pg2bRoVK1Y0XZZx6iAIPnUOBIeCgdNbBlzgG5cpU4YNGzZQrFgxgxWJuMvhw4epUaMGhw4dIgZYBdQxXFMS8CjwgeNYPmAYcL+RiiQa3AmM8Y2fJ3RTZyR87MZa9ygJiI+PZ/369VSpUsVwVdFFnQNB8MILL3D8uLVcSteuXRUM+KiDQCKRgoEzawJ084337dvHm2++abAaEfd555137E7FuzAfDAAUwppiMAUo4TuWDjwAXAGkmClLIlgK1lQ2v/+YKkSMKg/4l3pPTk7mtddeM1lOVFLnQICtWbOG888/n/T0dIoUKcKMGTMoUaKE6bLCijoIgkedA4GlYCB7NmCtGp0GFC5cmA0bNmgeoEgIHDhwgJo1a5KQkEAssAbrrlo4+Re4HWu+sF9xYBbQ3EhFEolmANf6xjWATeZKEcP2YS2InAjkz5+fdevWUb16dcNVRQ91DgTYU089ZW+v0atXLwUDp6AOAokECgayrzaZrcJHjx7lpZdeMlmOiGu8/fbbJCQkANbP0HALBgCqAXOw2sD9F52HgZbAk4Zqksjj3KWgq7EqJByUAfr7xqmpqbzyyismy4k66hwIoD///JOWLVsC1tzb6dOnU6hQIcNVhS91EASeOgcCQ8FAzu3GCgmOArGxsaxevZratWsbrkokeu3bt4+aNWuSmJhIfmAd1sJ/4WweVhfBVsexhsDPQCkTBUlESAMqAPt9j/ejvy9udwCreyAB65pjzZo11KoVjvFo5FHnQAANGpT5ceKBBx5QMHAW6iCQcKRgIHfKY21hBpCWlsYzzzxjshyRqPfGG2+QmJgIwD2EfzAA1jZky8i6zeIKoDIw1URBEhF+ITMYqIKCAbH+DjivOdSxGDjqHAiQX3/9lTZtrN03q1WrxjfffEP+/PkNVxUZ1EEQOOocyBsFA3mTgNU9sA8r/Fu1ahV169Y1XJVI9Nm9eze1atUiKSmJeGA91oemSOEFPgQeAY47jt+OtfWh7lyJU1+sBS4BHgbeNViLhI/DWOtPHAJiYmJYvXo15557rtGaooHOvwHiXC3zvvvuUzCQA+ogkHCgYCDvigH/9Y29Xi9DhgwxWY5I1BoyZAhJSUmAtd5HJAUDAB6suhcBDRzHx2KtUaDF5sQvA/ja8fhxU4VI2ClO5jVHRkYGL774oslyooY6BwLgr7/+okmTJgBUrFiR6dOnKxzIBXUQ5J06B3JHwUDgJGBd3B/GWkV448aN2oNYJIB27NhB7dq1OX78OAWAjUAkb5h8DOsCf7jjWD7gPaw7xuJuv2JNRwFr3YGdBmuR8HMEa+2B/Vg3G//++2/q169vuKrIps6BAHj99dftcc+ePRUM5JI6CMQEBQOBVQzo5xunpqby1ltvmSxHJOq8/vrrHD9uNeP3JbKDAYCCwDCsu8MlfcfSsc4j7bD2txf3cu5ScL2xKiRcFQUe8429Xi8vvPCCyXKigjoH8mj9+vXUrVuXjIwMSpUqxf/+9z8KFixouqyIpg6C3FPnQM4oGAiOPViLox0HChUqxJYtWyhTpozhqkQi37Zt26hduzYpKSkUwmq/L2e6qADaCtwBzHUcKw78gLX1obiLF+tniX93i81ExsKbElpHsbZx3YN1o/Gvv/6iYcOGhquKXOocyKM33niDjIwMAO644w4FAwGgDgIJBQUDwVMOa/V0gKSkJN5//32T5YhEjcGDB5OSYt1Lf4joCgYAqgKzgRfJvEA9DLRGc83daCGZwUAZFAzIqRUGnvCNvV6vdi7II3UO5MGOHTuoWbMmKSkpFC5cmJkzZ1KsWDHTZUUNdRDknDoHskfBQPBtAc7B2p+6ZMmSbNmyhaJFixquSiRyJSQkULlyZRITEymE9W8smvtxfsXaveBfx7HzsboKtJWdOzwO+Je1vQv4zGAtEt6OYa09sBvIly8fW7ZsoXLlyoarikzqHMiD//u//7MT/C5duigYCDB1EEgwKBgIjepYF/YABw8e5MMPPzRZjkjEGzNmDImJiQDcSXQHAwCXAMuAmx3H/gYqkXX1eolOXrKuN/DE6Z4ogrV2yf2+cXp6Oh999JHJciKaOgdyKTExkcqVK5OQkEBcXBw//PCD5tQGiToIsk+dA2emYCC0VmHd6fMC1apVY8OGDcTGxhquSiTyeL1eGjZsyN9//w1YH5obG60odLzAKKA/1t1Bv27AGHSXK1r9BTTxjUsCB8yVIhFiG1ADa0HTihUrsmXLFi0Snws6p+bSF198QUJCAgCdOnVSMBBE6iCQQFAwEHr1gWt943///ZfvvvvOZDkiEWvevHl2MHAx7gkGADzAvcAiwLnE2JdAFWCDiaIk6CY7xlcZq0IiSRUyd7TYuXMn3377rclyIpbCgVzwer0MHTrUftytWzeD1biDAgLJCwUD5jzoGDvPmyKSfcOHD7fHfQ3WYdJ5wJ9kPafsBOoCWvI0+mhKgeSG8/zoPG9K9mlaQS7Mnj2b9u3bA9C0aVM++0xLpISKphicmaYVnEzBgFkZQD1gne/xypUrOf/88w1WJBJZdu3aRdWqVUlLS6Ms1urt8aaLMuxb4G6ytppfhnVuL2CkIgmkVVhhEFj72CcYrEUiixera3GN7/E///xD/fr1DVYUedQ5kAvOJEpdA6GlDgLJCQUD5sUA/RyPP/jgA1OliESkUaNGkZaWBlhbhLo9GACrdXg50NZx7BegPLDAREESUM6ugXbGqpBI5AH6OB6PGDHCVCkRS50DObRz506qVq1Keno6ZcqUYebMmVrswgB1EJyaOgcyKRgIH4exVhhPAooVK8aOHTsoXLiw4apEwl9aWho1a9Zk27ZteIBNaK93p3Ssc/1zvjFYHw4eBd4wVZTk2QVYi24C/AZcZK4UiUCHsK45jmFdc2zfvp0iRYqYLSqCqHMghz755BPS060fQTfeeKOCAUPUQSBnomAgvBQHuvrGCQkJTJgwwWQ5IhFj2rRpbNu2DYBOKBg4UT7gKWAumf9tvMCbWG3p+wzVJbm3gcxgoDAKBiTnSgDdfeOEhATGjRtnsJrIo3AgB9LT0+29uj0eD7fccovhitxNAYGcioKB8PSAY6ypBSLZo4UIs+cirA+UtzqOrcJavXzKqb5Bwpbz/1cbY1VIpHOeL4cNG4Ya5bNP4UAO/PTTT/z7778AXHLJJVSqVMlwRaKAQJwUDISvZkBT33jhwoUsX77cZDkiYW/t2rXMmjULgFpoO7ezKQFMAEYBBX3HkoFbsDqXMsyUJTnkDAf+a6wKiXRNgZa+8fLly1mwQKuRZJfCgRwYM2aMPb7xxhsNViJOCggEFAyEOw/Q2/F47NixpkoRiQjODps+6IItO/znmcVAY8fxCUBlMndNkfC0FWu7SrB2nWhvsBaJfNrWMHe0IGE2HT16lAoVKpCYmEjRokWZM2cO8fFaMzicaJFC9y5IqGAgMuwDKgJpQJUqVdiyZQsxMfrII3Ki48ePU7FiRQ4dOkQ8sB0obbqoCHMceBx4z3EsBngb6G+kIjmbd4FHfOPLgdnmSpEocBwrFDwAxMXFsX37dsqUKWO4qvCnq7Jsmjp1KomJiQBcddVVCgbCkDoI3EnBQOQoA3Twjbdt28bcuXNNliMStn744QcOHToEWPPoFQzkXAGsD5tTyfzvl4H14fNSrA8OEl6cUwoeMVWERI0CZH4GSElJ4euvvzZZTsRQOJBNzhbYjh07GqxEzkQBgbsoGIg8tzvGzqlaIpJpypTMj0ldz/A8ObvOwF9Yd6L95gHlsLbJk/CwC/jVN44DrjNYi0SPLo6x87wqp6dpBdmwd+9eKlasSHp6OhUqVOCHH35QK2yYc+sUAzdNK1AwEJmSgPJAIlC8eHF27dpFgQIFDFclEj5SUlIoV64chw8fphiwB1CvYt6lA0OAZ3xjv/9gTTUQs0aQOUf8YjKDApG88AI1gH+B2NhY9uzZQ8mSJc0WFeb0CTcbJk6cSHq69aPk2muvVTAQAdRBEN0UDESuQsBNvvHhw4eZPn26yXJEws7s2bM5fPgwYN31VjAQGPmwfm78ivVhwe//gPpYa6KIOc57uv2MVSHRxgPc7BunpaXx3XffmSwnIuhTbjY4W187depksBLJCQUE0UnBQOS7wzHW1AKRrJytrzef4XmSO62AZWRtN16NtXDZRBMFCfuBn33jWLL+vxHJK+d5dPLkycbqiBSaVnAWGzdupHbt2gDUqVNH81UikJumGET7tAIFA9EhHaiCNcc0f/787NmzhxIlSpgtSiQMpKWlUbFiRfbt20chYC9Wt40EnhfrhsGDWNOd/G7B2vpQd89C5xMyt7ptTuZ2hiKBkIF1zbETiI+PZ+/evRQtWtRwVeFL576z+Pbbb+3xNddcY7ASyS11EEQHBQPRIx9wm2+cmprKjBkzTJYjEjbmzZvHvn1Wg/u1KBgIJg/WdcASoInj+GSgErDGQE1u5bztdr+xKiRaxQA3+sbJycmazngWCgfOwjk3pV27dgYrkbxQQBDZFAxEn+sdY80BFLE4W141pSA06gK/k3XrvN3AeWihwlA4BMzyjfMBPcyVIlHMeT5VF/iZaVrBGRw8eJCyZcuSnp5OtWrVmDZtGh6Px3RZkgfRPsUgGqcVKBiITqlAWeAw1q4Fe/fuJX/+/IarEjEnIyODKlWqsHPnTuKxphSo8TW0pmNdEzgXJ7wYmIm6OIJlDOC/bdMYaz0IkUBLAypi/dsuVKgQe/fupVAh/as+FXUOnMGMGTPsXQouu+wyBQNRQB0EkUXBQPTKj9U2DdauBb/+qo2rxN0WLFjAzp07Aetcp2Ag9DoCy4H2jmO/YW2/qjNUcDjv4fY+7bNE8iYWuME3TkpK4ocffjBYTXhTOHAGzlbXyy+/3GAlEkgKCCKDgoHo19kxnjp1qrE6RMKBdikIDxWxOgVex/pAAZAItAH6myoqSiWS+XPdg9YbkODSrgXZo2kFp5GSkkK5cuU4fPgwRYsW5ZdfflHLa5SJxikG0TKtQMGAOxwEymG1+9WuXZt169apQ0tcyev1UrNmTbZs2UIssAfrfC5m/QF0AzY5jtUB5mGduyRvJpG5OO15wN8Ga5Hol4L17/YwUKxYMfbs2UN8fLzhqsKPOgdOY968eRw+fBiANm3aKBiIQuogCE8KBtyjJHCpb7xhwwZWrVplshwRYxYvXsyWLVsAq6VdwUB4aAksxQoI/NYCVbG2O5S8cU4p0EKEEmxxwHW+cUJCAj/++KPJcsKWwoHT0JQCd1BAEF4UDLiPphaIwLRp0+yxphSEl+LAWKxrg8K+YylAV6z/Vxlmyop4xwD/33oP8KDBWsQ9bnGMneddyaRw4DT+9z/rI0lsbCwXX3yx4WokmBQQhAcFA+50nWPsP++KuM3cuXPtcQeDdcipebDubC8BLnAc/wprjQL1POXcTOCob3wO2g1CQqMd1paZYHWJy8kUDpzCzp07WbNmDQANGjSgaFGtGRztFBCYpWDAvWr5fgH8/vvvHD9+3GQ5IiGXkpLC77//DkB1rJZ1CU91gAXAAMexPUAD4A0jFUUu55SC7saqELcpAjT1jf/++2/2799vspywpHDgFH7++Wd73Lx5c3OFSEgpIDBDwYC09f2enJxsf0gScYvFixdz7NgxIHMNDglf8cBbwPdAWd+xDOAx4CIgyVBdkSQFcE4iG3C6J4oEgfM8q22UT6Zw4BSc4UCzZs3MFSIhp4AgtBQMCGSGA5D1/CviBs7W1jYG65CcuQZYDlzpOLYAazX0X4xUFDlmY60YD1ADKGauFHEh53nWOaVLLAoHTsF/cRobG0uTJk2M1iKhp4AgNBQMiN9ljrHCAXEb58WpOgciSwWsn1tDgFjfsaNYgedDhmqKBM4pBbed9lkiwXGJY6xw4GQer9frNV1EONmxYweVK1cGoEmTJnzxxReGKxJTvF4vb7zxRpa/Az2B0cYqOruSwCGs7VqSzZZyRgoG5ES1gY1AfHw8hw4dokCBAqZLEgm69PR0SpcuzeHDhykH7MJa/E4iz0KsLQ83OI6dC8zFChHEkob138M/03svUMZcOeJSDYC/gXz58nHw4EGtL+egzoET/PJLZjOY1htwN3UQBIeCATmVtr7fte6AuMnKlSs5fNhqsG6DgoFI1hxrN4PbHcfWAdWAcUYqCk9zyQwGKqNgQMzwd2mlp6ezYMECo7WEG4UDJ9BihOKkgCCwFAzI6bR1jDW1QNzC2dKq9QYiXzFgDPA51qroAKlYgcENWAsXup1zSsFNxqoQt3Oeb7WlYVYKB07gXG+gcePGZouRsKCAIDAUDMiZaN0BcSPnRanWG4ged2J1EVzoOPYtVjv930YqCg8ZwNeOx0+YKkRcT4sSnp7CAYdDhw6xdu1aAM477zwKFSpkuCIJFwoI8kbBgJxNNaw93gEWLVpEenq6yXJEgs7r9doXpcWARmbLkQA7F5gP/NdxbC/W/+fBRioybwGw0zeuAFQyWIu4WxWgpm/8xx9/kJwczit1hZbCAYclS5bY4/POO89gJRKOFBDkjoIByS7/XbajR4/aQa1ItFq/fj27d+8G4GIgn9lyJAjigDewfuaV8x3LwLpj3gpIMlSXKc4pBZ2NVSFi8XdrJScns3DhQqO1hBOFAw6LFy+2xwoH5FQUEOSMggHJCWcLrvN8LBKNtN6Ae1wNLPf97vcHVmAw20hFoeclazgw6HRPFAkRrTtwagoHHBQOSHYoIMgeBQOSUwoHxE2c4YDWG4h+5YHvgTeB/L5jR4H2QD9TRYXQIuBf37g0mS3dIqY4z7tadyCTwgEH/8VofHw8tWvXNlyNhDMFBGemYEByQ+GAuIn/73h+oJnZUiREYoBHsdYiOMdxfLjv8Q4TRYWIs2vgWmNViGQ6h8zpPrrmyKRwwOfw4cOsX78egLp16xIbG2u4Igl3CghOTcGA5FYZrIUJAZYuXUpGhjb+kuiUlpZmr6tRB4g3W46EWDOs3QzuchzbANTA2gox2mhKgYQjD3C+b7x37172799vspywoXDAx7kYYf369Q1WIpFEAUFWCgYkr/zdA4mJiVqUUKLWpk2bSE1NBaCe4VrEjKLAZ8AXQBHfsVSsbRA7A2mG6gqGFcB637gEoKtsCRfO8+/q1auN1RFOFA74ONtJzj///DM8UyQrBQQWBQMSCJpaIG6watUqe6wPSu52B7AMaO44Ng2oCKw0UVAQTHaMrzRWhcjJnOdf53nZzRQO+CxdutQeazFCySm3BwQKBiRQnOGA87wsEk0UDohTbeBX4DHHsX1AY+A1IxUFlnNKwRPGqhA5mcKBkykc8FmzZg0AMTEx1KypNVQl59waECgYkEBq4Bj7z8si0cbZvqppBQIQBwwGfsDa2QAgA3gSaAEkGqorr1YD//jGRYGmBmsROZGmFZxM4QDg9Xrtua2VKlUiLi7OcEUSqdwWECgYkECrBBTyjbXmgEQr5x2qugbrkPBzFbAc6OA4thArMPjJSEV54+wauNxYFSKnVhkrtAJ1DvgpHAB2797NkSNHAKhevbrhaiTSuSUgUDAgwRADnOsbb9y40V60TSRaeL1e+yK0OlDYbDkShsoB04G3sba6BEgCrgAeMFVULjnDgYHGqhA5NQ+Z3QObN2/m2LFjJssJCwoHyHp3qkaNGuYKkagR7QGBggEJpjq+39PS0ti8ebPJUkQCbteuXSQkJACaUiCnFwP8B/idzMAUYCRQC9hhoqgc2gj4V44pBFxisBaR0/Gfh52d5G6mcABYt26dPVbngARKtAYECgYk2Oo4xvpBLdFGixFKTjQFlgA9Hcc2YXWdfG6ioBxwdg20MVaFyJlpUcKsFA6Q9eJT4YAEUrQFBAoGJBQUDkg0UzggOVUEGA2MJXN+dBrQA+jkG4cjZzjwqLEqRM5M4UBWCgfQtAIJrmgJCBQMSKgoHJBopp0KJLe6A8uwdi/wmw5UwFrEMJxsA/7wjQsAVxqsReRMtGNBVgoHyLz4jIuLo0KFCoarkWgU6QGBggEJJYUDEs3UOSB5UQv4FXgCazE1gP1AE+AlQzWdyleOcStjVYicXW0g1jdW54DCAbxeLxs2bACgatWqxMS4/j+JBEmkBgQKBiTUSvl+Afb5WSRa+C8+SwNlzZYiESo/8BowE6trAMALPAs0A44YqsvJOaXgEVNFiGRDfjIX/Vy7di3p6ekmyzHO9Z+EDx48SHJyMgDly5c3XI1Eu0gLCBQMiCmVfb/v3LkTr9drtBaRQElJSWHHDmud+XPP8lyRs7kCazrBtY5ji7ECg5lGKrLsBub5xnFAZ4O1iGSH/3ycnJzMzp07jdZimuvDAedfgDJlyhisRNwiUgICBQNiUiXf7ykpKRw4cMBoLSKBsn//fntczmAdEj3KAtOAd7A+iAMkYf3Mvg/IMFDT11idDGB1Mrj+w4aEPef52HmediPX/3v1J/gA5crpR7WERrgHBAoGxLSKjrHbU3yJHvv27bPHmlIggeIB+gO/k3XNlo+w5lNvC3E9zikFfUP83iK54TwfO8/TbuT6cECdA2JKuAYECgYkHCgckGi0d+9ee6wrDgm0C7CmFdztOLYZqIm1FWIo7Afm+MaxQLcQva9IXjjPx87ztBspHHBcdJYtqxxfQivcAgIFAxIuKjnGzg4vkUjmvCOlcECCoQjwMfAlUMx3LA0rMLjWNw6mqYB/Obcm6IOGRAbn+VidAy7nvOhUOCAmhEtAoGBAwok6ByQaKRyQUOkKLCPrNoIzgPK+48HinFJwfxDfRySQFA5kcn04oGkFEg5MBwQKBiTcKByQaORsV9XtCAm2msBc4EmsdQkADgBNgReC8H6HydwlIQboGYT3EAkG5/lY0wpcTtMKJFyYCggUDEg4ck4rUDgg0UKdAxJq+YFXgB/JDF29wPNYIUFCAN9rGpDqGzfAWnNAJBKocyCT68MBfzpUuHBhChQoYLgacbtQBwQKBiRcOfeOcXuKL9FD4YCY0g5YDnRyHFsKVMCabhAIzikFvQP0miKhoHAgk+vDgSNHjgBWOCASDkIVECgYkHBWEMjnG/vP0yKRTuGAmFQGa8HA94A437FjWAsV3g1k5OG1j5J5DeEBHsjDa4mEWhEy/00oHHC5xMREAAoVKmS4EpFMwQ4IFAxIuPNg/bAGhQMSPfxdMPnJXEleJJQ8wEPAn0A9x/HRWGsU/JvL152BFTTge924MzxXJNx4yFx3wO3diq4OB7xerx0OqHNAwk2wAgIFAxIpivp+95+nRSKd/45UGTIXiBMxoTGwCLjHcexfoDYwKhev55xScFce6hIxxd/NtW/fPrxer9FaTHJ1OHDs2DEyMqwmKoUDEo4CHRAoGJBI4g8H1Dkg0cDr9WYJB0RMKwx8BEwAivuOpQH3Yl0fpJ7m+050HGsxQrBCr4cDWKNIqPjPy6mpqa6+7nB1OOD8H69pBRKuAhUQKBiQSOOfVpCYmOjqFF+iw9GjR0lOTga0jaGEl9uAZUBrx7GZQHlgSTa+fybg7++qDeiKWiKR87zs5nUHFA74KByQcJbXgEDBgEQif+eA1+slKSnJaC0ieXXw4EF7XNJgHSKnUgOYCzxN5pSXg0Az4NmzfK9zSkH3gFcmEhrO8/KBAweM1WGaq8MB5zxWhQMS7nIbECgYkEhVxDF2c4ufRIfU1MwmbS3WJuEoFngJmA1U8h3z+o41AQ6f4ntSsHZA8Hs0iPWJBJPzvJyWlmasDtNcHQ44Lza15oBEgpwGBAoGJJIVdYwVDkikS09Pt8f5zvA8EdPaAn8B1zmO/QVUAL4/4blzgEO+cXW0C4dELud52Xm+dhtXhwP+uX8AcXHK8SUyZDcgUDAgka6gY3z8+HFjdYgEgsIBiSRlgG+AoUC879hxoCPQE8jwHXNOKbgtRLWJBIPCAYurwwH/TgUA+fLpR7VEjtMFBId84xQUDEjkc/6A0oKEEumcbaqxBusQyS4P0A/4E6jvOP4Z1hoFG4GvHccfC1llIoHnPC9rWoFLOcOBmBhX/6eQCHSqgOBUFAxIpHKeld2c4kt0UOeARKpGwCLgPsexrcC5gH9N98poi06JbOocsLg6vF68eLE9/uCDD/jwww8NViMSHHPIbAkUiSTO3H716tVccMEFxmoRySvnxaarL74kIhUCRgJXAvdidSpmOL6+A11rSGRzXnMsXLiQK6+80lgtJrn655NzDmtGRkaWTgKRaJFiugCRAEhJ0d9kiWzOcEC9ihKpbgFKA+1OOO5F1xsSPdy8zpGrwwHnOgNxcXHExyvzlMh0upXcPR4PRYoUOeXXRMLd8ePH7e3fNPVLIp3z77BuRUikehcYcIrj8fHxWtxbIlpycrJ9IyI21r0fkd37Jwdatmxpj++55x769OljsBqR3Ln//vuZP3/+ab8+cOBAbrzxxhBWJBIYL730EhMnTgSgQYMGhqsRyRvnxaZ7Z7NKpDoOXAXMcxyLjY21F2674ooreP31102UJhIQ77//vj3FvFWrVoarMcfVt2KypPiaUiAR6GzBgNfr5bnnnuPrr78+7XNEwpUWjZVo4uxWdO862BKJfgPKkTUYaNGiRZZri19++UXTvySiOa853Nw54OqrLefFpptXpZTIdGIw0L9/f3tcvXp16te3Nh5SQCCRyrl9ocIBiXTOcEBXHBIpBgCXAP7Ji/ny5aN///58+OGH1KhRgypVqgCQmJjI77//bqpMkTzLsqOMi7e4d/XVlnONAf+8VpFIcKpg4J577rEfezwexo8fr4BAIlpycrI91powEumcd6LUOSDhbi9QH/g/x7HKlSvz2Wefcc8999gfnjp06GB/fdasWSGtUSSQsuwoo84BdypatKg9TkpKMliJSPadLRjwi4mJUUAgEc15Xnaer0UikToHJFJMAqoAqx3HOnTowKRJk2jcuHGW5/bo0cMez5kzRzfbJGI5pxWoc8ClnKu4Hz161GAlItmT3WDATwGBRDLneVnhgEQ6550ofXyScJQB3ArcRua2hAULFuTFF19kyJAhpzwPlyhRggoVKgBw+PBhFi1aFKpyRQLKv7gmqHPAtZwnOYUDEu5yGgz4KSCQSOXsHChUqJDBSkTyrkSJEvb4oLkyRE5pDVAJmOw4Vq9ePSZMmMCNN96Ix+M57fdeccUV9vjHH38MWo0iwXT48GF77Dxfu42rwwFn58CxY8cMViJyZrkNBvwUEEgk8ocDRYoU0YKEEvGKFi1K/vz5AdhnuBYRp7eB84DdjmN33HEHY8eOpWbNmmf9/t69e9vjn376SYt8S0Q6dOiQPS5Tpoy5Qgxz9dVW4cKF7SRUnQMSrvIaDPgpIJBI4z8va0qBRAOPx2NfcCockHCQhLUTwaNYUwoASpYsybBhw3j88ceJi4vL1uuUKVPG/ru9f/9+li1bFoRqRYLLHw7ExsZSvHhxs8UY5OpwwOPx2N0DCgckHAUqGPBTQCCRxH9ednZ5iUSysmXLAlY44D3zU0WC6legPPCb41jLli2ZMmUKl156aY5f7/LLL7fH2rVAItHBg9aErzJlypxxGk20c3U4AJkXndqtQMJNoIMBPwUEEgm8Xq893UudAxIt/HdXk4FEs6WIi/UH2pD5dzA2NpZHHnmEDz/80A6wcso5teDHH3/MsvK7SLjzer1ZwgE3c3044L/oVOeAhJNgBQN+Cggk3KWkpNgrB6tzQKKF86JTUwsk1PYAdYH3HMcqV67MZ599Ru/evfO0tkvlypUpWbIkALt372blypV5qlUklI4dO0ZycjKgcMD14UDp0qUBSExMJCUl5SzPFgm+YAcDfgoIJJwdOHDAHrv9B7VED4UDYsoEoCqw1nHsmmuuYdKkSTRq1Cgg7+GcjqBdCySSaDHCTK4PBypVqmSP9+3Tj2oxK1TBgJ8CAglXe/bsscfO87RIJHO2bOuKQ0IhA7gZ6Ar4b4EVLFiQl19+mcGDBwd02pbzemXWrFl4vVpZQyKDf0oBkOupNdHC9eFAxYoV7bHzYlQk1EIdDPgpIJBw5AxrnedpkUjmvCO112Ad4g6rgIrAV45j9evXZ+LEiVx//fUBX3StRo0aFCtWDIBt27axevXqgL6+SLA4wwF1Dric86JTnQNiiqlgwE8BgYSbvXszPzopHJBooWkFEipvAg2w1hnwu+uuuxgzZgw1atQI2vtefPHF9li7FkikUDiQyfXhgLNd1XkxKhIqpoMBPwUEEk4UDkg0UjggwZYEXAQMxJpSAFCqVCmGDx/OwIEDiYuLC+r7O69ftO6ARAqtOZDJ9eGA86JT4YCEWrgEA34KCCRcKByQaOScy6orDgm0X4BywALHsdatWzNlyhTatGkTkhrq1KlD4cKFAdi0aRMbNmwIyfuK5IXWHMikcEDTCsSQcAsG/BQQSDhwhgNakFCihToHJFgeAtoC/o25Y2NjGTBgAB988EHI74S2bNnSHmtqgUQCTSvI5PpwQNMKxIRwDQb8FBCIaf6wNjY21t5yViTSOf8uawlkCYRdQB1gqONY1apV+eKLL+jVqxcxMaG/1O/du7c91tQCiQQKBzK5PhwoXbo0+fPnB2D37t2GqxE3CPdgwE8BgZjk3z2mQoUKRi5uRYKhQIEClCtXDgA1W0tejQOqAescxzp16sTEiRNp0KCBoaqgUaNGFCxYEIA1a9bw77//GqtFJDv8f0djY2OpUKGC4WrMcv0Vl8fjoWbNmoC17UpGRsZZvkMk9yIlGPBTQCAmJCQkcODAAQBq1apluBqRwPKfT3cDB8/8VJFTygBuAG4HUn3HChUqxKuvvsprr71GkSJFjNXmd+GFF9pjTS2QcJaens6WLVsAOPfcc+2bxm7l+nAArMVTAI4dO2bfrRIJtEgLBvwUEEioOe8y+c/PItHCfy4Fax96kZz4G6gAfOs4dt555zFx4kQ6d+5sqKqT9erVyx5raoGEs+3bt5OSkgJkPT+7lcIBsl58+pMjkUCK1GDATwGBhNLmzZvtcd26dc0VIhIE9erVs8erDdYhkWcw0IisO1307NmTMWPGUL16dUNVnVqLFi2Ij48HYOXKlezcudNwRSKntmnTJnvsPD+7lcIBFA5IcEV6MOCngEBCxXkeVueARBt1DkhOJQGtgCewphQAlCpVig8++IBHH300bNugmzRpYo/VPSDhauPGjfZYnQMKB4CsF5/OO1YieRUtwYCfAgIJBYUDEs0UDkhOzAbKAX84jl188cVMmTKFiy++2FBV2XPnnXfaY607IOFK4UBWCgdQ54AER7QFA34KCCTY/CFtTEyMFiSUqFOlShUKFy4MaFqBnFk/oD1w1Pc4NjaW//73vwwfPjwitlu77LLL7K6GZcuWactwCUvOaQWayqhwAIBKlSpRqFAhQOGABEa0BgN+CggkWLxer30erlmzJnFxcYYrEgksj8djz2vdBBw3W46EoR3AOcBwx7Fq1aoxZswYevToEVHbu/q3VPR6vfz000+GqxHJyuv12p0DVatWDYudPkyLnLNLEHk8Hrt7YNu2baSmpp7lO0ROL9qDAT8FBBIM+/btIykpCdCUAole/vNmBrDWbCkSZsYANYANjmPXXXcdEydO5PzzzzdTVB50797dHmvdAQk3+/fv58iRI4CmFPgpHPDxX4Q697oUySm3BAN+Cggk0NavX2+PFQ5ItHJehGpqgQCkAdcBdwL+W1SFChXitdde45VXXrGnokSaq666inz58gGwaNEiDh48aLgikUzOKQUKBywKB3ycK6r+888/5gqRiOW2YMBPAYEEkvP86zwvi0QT53ZZWpRQVgIVge8cxxo0aMDkyZPp1KmToaoCIyYmxr4+SE9PZ86cOYYrEsnkXIxQ2xhaFA74XHjhhfZ41Sr9qJaccWsw4KeAQALFef51npdFool2LBC/14DGwD7HsV69evH5559TtWpVQ1UF1m233WaPZ86cabASkay0U8HJFA74OC9C1TkgOeH2YMBPAYEEgv/8W7BgQf2glqh1zjnnEBsbC2hagVslAi2AJ7HWngAoXbo0I0eOZMCAAfYq/9Hg+uuvtxdR/OOPP0hISDBckYhF0wpOpnDAp3Tp0lSvXh2w7lylp6cbrkgigYKBrBQQSF4kJCSwdetWABo3bmx/eBKJNvnz56d27doArCFzjrm4w09AeWCh49gll1zClClTuOiiiwxVFTwxMTH2GjJpaWn88ssvhisSsa5RN2ywlv4sWbIkZcuWNVxReFA44ODvHjh27JgWJZSzUjBwagoIJLc0pUDcpGnTpoC1leFSs6VICD0AXAEk+R7Hxsby2GOPMWzYMEqXLm2wsuC66aab7PGsWbMMViJi2bFjB3v27AHgggsuwOPxGK4oPCgccHBejP79998GK5Fwp2DgzBQQSG44p3QpHJBo16ZNG3s8z2AdEho7gFrASMex6tWrM3bsWO6880677T5a3XLLLfaHr99++42jR48arkjcbsmSJfb40ksvNVhJeInuM1EOad0ByQ4FA9mjgEBySuGAuInzYnSuwTok+D4HqgObHMduuOEGJk6cyHnnnWeoqtDKnz8/tWrVAiAlJYV58xSJiVmLFy+2xwoHMikccFA4IGejYCBnFBBITvjPuwUKFHDNBbO4V/369e028l/JXJROokca0Ano4RsDFC5cmMGDB/PSSy9RqFAhc8UZcP3119tjTS0Q0/zhQP78+WnZsqXhasKHwgGHMmXKULNmTcCaVpCcnGy4IgknCgZyRwGBZMe+ffv4999/AWvunxYjlGgXExPDJZdcAsABQLckostyoAIw3XGsYcOGTJo0iWuvvdZQVWZ169bNnlowb948jh8/brgicat9+/axefNmAJo1a+a6oO5MFA6c4LLLLgMgOTmZFStWGK5GwoWCgbxRQCBns2jRInvctm1bc4WIhJCzlVVN1tHjZaAJsN/32OPx0Lt3bz777DOqVq1qrjDDChQoQLVq1QBr8e/ffvvNcEXiVkuXZi4D61z/RRQOnMR5Ubpw4cLTP1FcQ8FAYCggkDNxnm8VDohbOC9Kte5A5EsAmgHPAF7fsbJly/Lhhx/yyCOPkD9/fnPFhYmOHTva4x9//NFgJeJmWm/g9BQOnEDhgDgpGAgsBQRyOv7zbWxsbFTu8y1yKhdccAGFCxcGrM4B75mfLmHsB6xpBIsdx9q0acPkyZNp1aqVoarCz1133WWPf/nlF1JSUgxWI27l36nA4/Fw8cUXG64mvCgcOEH16tXtdQf++usvrTvgYgoGgkMBgZxo3759bNpkrePdokULihQpYrgikdBwhmHbybqavUSGDOA+oANwzHcsf/78PP744wwbNoxSpUqZKy4MFS5cmMqVKwNw5MgRfv/9d8MVidscOXKE1atXA9CoUSNKlChhtqAwo3DgFPzdAykpKSxfvtxsMWKEgoHgUkAgTlpvQNxM6w5Erm1ALeAjx7EaNWowbtw47rjjDnvxPcnq6quvtseaWiChtmzZMrxeq09L6w2cTOHAKTgvTp0XreIOCgZCQwGB+Gm9AXEzZzigdQcix2igJrDFceymm25iwoQJ1KtXz1BVkaFXr172eM6cOaSlpZ3h2SKBpfUGzkzhwCn4dywArTvgNgoGQksBgYDWGxB3a9GiBXFxcYA6ByJBGnAtcLdvDFCkSBHeeOMNXnjhBW2Jlg0lSpSgfPnyABw6dEg34iSk/OsNgDoHTkXhwCk41x1YtmwZSUlJhiuSUFAwYIYCAnfbtWtXlvUG/IuzibhFgQIFaNGiBQDrgF1my5EzWAaUB2Y4jjVq1IhJkybRoUMHM0VFqCuvvNIea2qBhMrx48ftrerPPfdcKlSoYLii8KNw4DT886FSU1NZsGCB4Wok2BQMmKWAwL1++eUXe3zVVVcZrETEHOfdq1kG65DTewFoChzwPfZ4PNx77718+umnVKlSxWBlkal37972+McffyQ9Pd1gNeIWixcvtqexqGvg1BQOnEbnzp3t8c8//2yuEAk6BQPhQQGBOznPr9ddd525QkQMuvbaa+3xFIN1yMkSsEKB58ncarJcuXKMGjWKhx9+mPz58xurLZKVKVOGMmXKALB//36WLVtmtiBxhVmzMuNX53lXMikcOI127drZ88bmzp2rRDNKKRgILwoI3CUpKYk//vgDgCpVqtCkSROzBYkYctFFF9ntrT8AiWbLEZ8ZQAVgqeNY27ZtmTx5sj0VRHLv8ssvt8eaWiDBlp6ezpw5cwAoWLCgpgKdhsKB0yhQoIA9teDAgQP2/BSJHgoGwpMCAveYP38+qampgNWtpW2/xK1iYmK48cYbATgOfG+2HNfLwFpw8FrgmO9YXFwcgwYN4r333qNkyZLmiosiJ04t8G8vJxIMS5Ys4cABa2LQtddeqzWOTkPhwBloakH0UjAQ3hQQuIOmFIhkuuWWW+zxZIN1uN2/WFsUjnYcq1mzJuPGjaN79+4KMQOocuXKdtCya9cuVq5cabgiiWYzZ860xzfffLPBSsKbwoEz6Nixo/1DwLlolkQ2BQORQQFBdEtPT2fuXGtX98KFC9O2bVuzBYkYdumll1K6dGnA6hw4duanSxCMAmpjBQR+N998M+PHj6du3bqGqopuWRbjnKXlOCU4MjIy+OmnnwCrC6hjx46GKwpfCgfOoFy5crRu3RqA9evXs3XrVsMVSV4pGIgsCgii1/Llyzl48CBg7Q5ToEABwxWJmBUbG8sNN9wAwFGstQckNNKAq4F7fWOAokWL8uabb/L888/ba1BJ4DmvwWbNmqWpBRIUy5cvZ+/evYC1M1KxYsUMVxS+FA6chXNqgX8RC4lMCgYikwKC6OQ8nzrPsyJu5mx11a4FobEEKAfMdBxr0qQJkydPtteekuCpWbOm/UFt27ZtrFmzxnBFEo2cXSmaUnBmCgfO4vrrr7fH//vf/wxWInmhYCCyKSCILhkZGfb5NF++fGrvE/Fp3749xYsXB+A7INlsOVHvWaAZcND32OPxcP/99zN69GgqVapksDJ38XfpgqYWSOB5vV57N4zY2FitcXQWCgfOon79+vb2WitWrGDLli1mC5IcUzAQHRQQRI+lS5eyc+dOAK688krKli1ruCKR8BAXF2d30hwGfjJbTtRKAJoALwH+JvZy5crx8ccf8+CDDxIbG2usNjdyXpNpS0MJtH/++YcdO3YA1lb1pUqVMlxReFM4kA233367PZ4+fbrBSiSnFAxEFwUE0cF5Hr3jjjsMViISfjS1ILi+B8oDfzmOXX755UyZMoXmzZsbqsrd6tWrZ28rt3HjRjZs2GC4IokmmlKQMwoHsqFbt272rgXTp0/XYikRQsFAdFJAENlSU1P54QdrqbVChQplmbolItYCnf4PSt+SuUCe5E0G0BPoCBz3HYuLi+Opp57i3XffpUSJEqZKE6Bly5b2WFMLJFCcUwpiYmLsRV/l9BQOZEPlypW5/PLLAfj3339ZsWKF4YrkbBQMRDcFBJFr3rx5JCQkAHDjjTdSpEgRwxWJhJeCBQva63DsB7SRct5tAWoAnzmO1a5dmy+//JKuXbvaN4DEnLvvvtsea2qBBMq6devsKeGXXnop5cqVM1xR+FM4kE3O1tdp06YZrETORsGAOyggiEzO86emFIicmrP1dYLBOqLBSKA24NyM+tZbb+XLL7+kTp06hqqSEzVu3JiCBQsCsGbNGm0fLgHhXExeUwqyR+FANt10003Ex8cD8MMPP5Cammq4IjkVBQPuooAgshw5coRffrHug5YrV44rrrjCcEUi4enaa6+1pxaMw1pAT3ImBbgSeABI9x0rWrQob7/9Ns8++6z9QVTCR9OmTe2xphZIXqWmpvLNN98A1s5IN910k9mCIoTCgWwqXry4vfXFgQMHWLBggeGK5EQKBtxJAUHk+PHHH0lJSQGga9euWhFc5DSKFCnCnXfeCcBR4Auz5UScRViLDjqb05s2bcqUKVO48sorDVUlZ9OzZ097rKkFkldz5sxh7969gLU1vbYnzR6FAzngbIH1J1ESHhQMuJsCgsjg/P/h3AVGRE7Wp08fezyczC335MyeBloAh3yPY2Ji6NOnDx9//DEVK1Y0V5icVatWrewu3RUrVthb3orkxoQJmZOy+vbta7CSyKJwIAeuueYaKlSoAGRNo8QsBQMCCgjC3bp161i6dCkADRo00JZhImfRqFEjLrnkEgD+AeaaLSfsHQIaAa+QGaSUL1+ejz/+mL59+6pTKUI0btzYHqt7QHJr48aN/PnnnwDUrVuXdu3aGa4ocigcyIH8+fPbq6mmpaXpQ0cYUDAgTgoIwtekSZPs8f3336/VwUWywdk9MMJgHeFuKlARcO4l1b59e6ZMmUKzZs0MVSW54Z9OAwoHJPecXQMPPPCArjlywOP1etWplgObN2+mVq1aeL1eKlWqxPfff0++fPlMl+VKCgZO1rBhQwBq1KjBd999Z7gaczIyMujatSurVq0CwOPx8MILL3DjjTcarsydkpKSaN++PYmJiRQqVIgdO3ZQvHhx02WJhL3k5GSqVq3K3r17icVacb+C6aLCSAbQk6xrMsTHx/PYY49x66236gNBhGratCmpqal4PB5mz55NmTJlTJckEcR5zVGwYEG2b99OyZIlTZcVMdQ5kEM1atSgQ4cOAOzYsYN58+YZrsidFAzImaiDILx8//33JCYmAtZChAoGRLInPj7e/tmWBowyW05Y2QRUI2swcM455/Dll19y2223KRiIYOeffz5g/ez+6aefDFcjkWb69On2NUf37t0VDOSQwoFccC5q8eWXXxqsxJ0UDEh2KCAID16vN8t50tkmLSJn55yGMxIrJHC7EcC5wHbHsS5duvDll19y7rnnGqpKAqV79+72WFsaSk54vV7Gjx9vP9ZChDmncCAXrrnmGmrVqgXA/Pnz2bRpk+GK3EPBgOSEAgLzFi9ezNq1awFrJWrN/xXJmerVq9OpUycAtgHTzJZjVArQHugLpPuOFStWjHfeeYenn36aAgUKmCtOAubqq6+2p+wuWrSIgwcPGq5IIsVff/1lX3O0bNmSpk2bGq4o8igcyIV8+fJlSaKcCZUEj4IByQ0FBGY5uwYeeughg5WIRC7nNcdwg3WYtBAoB8x2HGvatClTpkyhffv2hqqSYIiJibF/ZqenpzNnzhzDFUmkUNdA3ikcyKW7776bggULAvDNN99w+PBhwxVFNwUDkhcKCMzYvn27PV+0fPny3HLLLYYrEolMV111ld2xOAtYa7ackBsEtAT8V1oxMTH07duXTz75xN5iWqLLrbfeao81tUCy48CBA8ycOROAUqVKcdtttxmuKDIpHMilkiVL0rNnT8BaFVNrDwSPggEJBAUEoTd69GjS063m3759+xIXF2e4IpHIFBMTk2W9jg8M1hJKB4CGwOuAf2utChUqMHr0aPr06aPdoqLYddddR0yM9THl999/JyEhwXBFEu6+/vprUlNTAejdu7emGeWSwoE8GDhwoP2DaezYsSQlJRmuKPooGJBAUkAQOvv27bP/uxYpUoQHH3zQcEUika1Xr17Ex8cDMBqI9o9K3wKVgJWOY1deeSWTJ0/WPGIXiI2NtReXTEtL45dffjFckYSz1NRUJkyYAFjbV99///2GK4pcCgfyoGbNmnTt2hWAQ4cO8dVXXxmuKLooGJBgUEAQGl988QUpKSmA9W+5VKlShisSiWylS5e2V3E/BLxrtJrgyQBuB24Akn3H4uPjefbZZ3nrrbe0FaqL3HTTTfb4xx9/NFiJhLtvvvmGnTt3AtbC8bVr1zZcUeTyeL1e79mfJqezcuVKGjZsCFhzamfMmEH+/PkNVxX5FAzkjv/vYo0aNfjuu+8MVxPeMjIy6Nq1K6tWrQKspPmFF17gxhtvNFxZ5EtISOCqq67i6NGjxMXFsWnTJipVqmS6LJGIt27dOurXr096ejrFgc1ACbMlBdQG4FJgh+PYueeey5AhQzjnnHMMVSWmpKSk0KxZM7xeL/Hx8cydO5dChQqZLkvCTEpKCh07dmTXrl2ANQ2lZcuWhquKXOocyKMGDRrQuXNnAHbv3s20aW7eZCgwFAxIKKiDIHgmTJjA0aNHAejRo4eCAZEAOffcc7nrrrsAa3G+/zNbTkANBeqSNRjo2rUr48aNUzDgUnFxcfZCnMnJycydO9dwRRKOvvrqKzsY6Nixo4KBPFI4EACDBg2yx5988om9AJfknIIBCSUFBIF37NgxxowZA1j/fR977DHDFYlEl2eeeYbY2FjACgcOmC0nz44DlwMPAf6rp+LFi/Puu+/y1FNPaVExl/PfgANNLZCTHT9+nI8++sh+/OKLLxqsJjooHAiA1q1bc9lllwGwefNmZs+efZbvkFNRMCAmKCAIrK+//poDB6yPK7feeqvu+IkEWM2aNbn77rsBOAK8ZbacPPkDKA/87DjWrFkzJk+eTLt27cwUJWGlW7dueDweAObOncvx48cNVyThZPLkyezZsweAG264QYuVBoDCgQBxdg+MGDFC3QM5pGBATFJAEBjHjx/n448/th8/8cQTBqsRiV5PPfWUvb7Ru8Bes+XkyuNAazJ3XciXLx8PPvggo0aNokKFCgYrk3BSqFAhqlWrBlidac5rRXG3Y8eOMWrUKPvxCy+8YLCa6KFwIECuuuoqWrRoAVgLBn3//feGK4ocCgYkHCggyLsvv/zSTvA7d+5MkyZNzBYkEqWqVavGvffeC8BR4A2z5eTIAeB8YAjgXxG7YsWKjB49mvvvv9/eIlrE79prr7XHs2bNMliJhJMJEyawf/9+wOpUbNSokeGKooN2Kwig2bNn0759ewAqVarEd999R1xcnOGqwpuCgcDSbgV5p10McichIYFrrrmGhIQEPB4Pf/31l/33UUQCb/v27dSuXZvk5GQKApuwWvTD2ddANzK3KATr5spzzz1HsWLFDFUl4S4xMZHWrVsDULRoUX755RftDOZySUlJdOjQgYMHD+LxeFixYgXnn3++6bKigjoHAqhdu3ZcddVVAOzYsYOJEycarii8KRiQcKQOgtz55JNPSEiwGoTvvPNOBQMiQVa5cmUeeOABAI4Bg82Wc0YZWKHATWQGAwUKFOD555/nzTffVDAgZ1SkSBF715sjR47w+++/G65ITBs3bhwHDx4ErF1NFAwEjsKBAHv99dft8YcffkhiYqLBasKXggEJZwoIcmbPnj2MHTsWsLae0mrBIqHxxBNPULBgQQBGkHUbwHCxAagCjHccq1OnDhMmTODmm2+2F5sTOZMOHTrYY+1a4G6JiYl8+umngHW99txzz5ktKMooHAiwCy64gK5duwJw8OBBPvvsM8MVhR8FAxIJFBBk34gRI+wVpPv27Uv16tUNVyTiDhUqVKBv376AtSXga2bLOcl7QF1gp+NY9+7dGTdunL1/vUh29OzZ0x7Pnj2btLQ0c8WIUWPGjOHw4cMA3HHHHdStW9dwRdFFaw4Ewfr166lfvz5paWkULFiQ77//njJlypguKywoGAgurTkQeFqD4Mw2bdrEjTfeSHp6OkWLFmXjxo0634mE0N69e6lZsyZHjx4lDlgPVDVc03HgamCu41iJEiV46aWXaNu2rZmiJOJdccUV7N69G4BRo0bRsmVLwxVJqCUkJNChQweOHDlCvnz5WL16tbZMDjB1DgTBOeecw3333QdY22yMHDnScEXhQcGARCJ1EJzZ+++/b2/dOnDgQAUDIiFWtmxZHnroIQBSsLYINOk3oBxZg4EWLVowefJkBQOSJ1dccYU91q4F7jRixAiOHDkCQI8ePRQMBIE6B4Jk165d1K5dm6SkJPLly8ekSZM499xzTZdljIKB0FDnQPCog+BkCxcu5O677wagfPnyrF+/niJFihiuSsR9Dhw4QJ06dextvWYBV5z5W4Liv8DbZG5RmC9fPvr168fdd9+tLQolz/bs2WPvClamTBl++uknYmJ0n9MtVq1aRdeuXcnIyKBgwYKsWrVK0xiDQP+igqRChQo88cQTAKSnp/PKK6/g1hxGwYBEA3UQZJWamsorr7xiP37ppZcUDIgYUqpUKQYPztyvoB9ZtwsMtn1AfeAtMoOBypUr8+mnn3LvvfcqGJCAKFeuHKVLlwZg3759LFu2zGxBEjIZGRm8/PLLZGRkAPDss88qGAgShQNBNHDgQGrXrg3A4sWL+f777w1XFHoKBiSaKCDING7cODZs2ABYLcO9e/c2XJGIu/Xq1YuLLroIgLXAkBC972SgMrDacaxDhw5MmjSJJk2ahKgKcYvLL7/cHmtqgXtMnjyZ5cuXA1C/fn0GDBhguKLopXAgiAoUKMC7775rP37rrbdctbWhggGJRgoIrNbO4cOHA9b0imHDhqm1U8SwmJgYRowYYd+lfwVrG8FgyQBuA27FWusAoGDBgrz44osMGTKEokWLBvHdxa38U9nA2tLQrV25brJ///4sn6eGDx9OXFycwYqim67mgqxjx45cd911gLWi8IgRIwxXFBoKBiSauT0geOutt0hKSgLgvvvuo1mzZoYrEhGARo0a0b9/f8CaVvAQmW3+gbQOq1tgkuNYvXr1GD9+PDfeeCMejycI7yoCVatWpUSJEoC1vtfKlSvNFiRB9/bbb5OQkADAnXfeqYVNg0zhQAi88847FChQAICxY8eyfv16wxUFl4IBcQO3BgQLFy60p0iVKlUqy7oDImLe888/T+XKlQGYAXwV4Nd/B6gH7HIcu+OOOxg7diy1atUK8LuJnKxNmzb2WFMLotuiRYuYOnUqYG2H+sYbbxiuKPopHAiBmjVrMmjQIMBanPDVV1+N2jYoBQPiJm4LCFJTU3n11Vftx6+//rq9OJSIhIeiRYtmacHtDxwJwOseB9oA/8GaUgBQsmRJhg4dyuOPP642XwkZ53WlphZEr9TUVF5++WX78auvvkr58uUNVuQOCgdC5LHHHrMT9YULFzJt2jTDFQWeggFxIzcFBF988YXd+dS8eXMtQigSpm666SY6dOgAwHbghTy+3q9AOd/vfi1btmTy5MlcdtlleXx1kZypVauWvabF1q1bWbt2reGKJBi++OILe+Hj5s2bc9999xmuyB0UDoRIgQIFeO+99+zHr7/+Onv37jVYUWApGBA3c0NAsHHjRoYNGwZoEUKRcOfxeBg6dKg9pfEdYHkuX+s/WB0D/u6DfPny0b9/fz788EPKlSuX51pFcqN169b2WFMLos+OHTv44IMPgJMXW5Xg0pVdCHXs2JFu3boBkJCQwIsvvhgVrVAKBkSiOyBIT0/nmWeeISXFWpP8P//5D82bNzdclYicSe3atXnyyScBSAf6kDkdIDv2YK0t8I7jWOXKlfn888+55557FA6KUc7OtR9//NFgJRIMr7/+OseOHQOgb9++XHjhhYYrcg+d2UPs/ffft+fL/PzzzxE/vUDBgEimaA0IPvvsM3t/4Tp16mSZAygi4euxxx7j3HPPBWA+MDqb3zcRqAqscRy75pprmDRpEo0aNQpskSK5cN5551G4cGEANmzYwMaNGw1XJIHy888/M2fOHADKly+va44QUzgQYqVLl7bbZABee+019uzZY7Ci3FMwIHKyaAsITpxOMHr0aAoWLGi4KhHJjvj4eIYPH24//i+w9QzPzwBuAboAKb5jBQsW5KWXXmLw4MH2PG+RcNCiRQt7rKkF0eHw4cO89NJL9uO3336b4sWLG6zIfRQOGHDDDTdw++23A3DkyBFeeOGFiJteoGBA5PSiJSBIS0vj6aeftqcTPProo1x00UWGqxKRnLjiiivsa45DwF1Y0wxOtAaoBExxHKtfvz4TJ07khhtuwOPxBLtUkRzp1auXPdbUgsjn9Xp54YUX7JumV155pT0dW0JH4YAh7733HhUqVABg7ty59h6ekUDBgMjZRUNA8Nlnn7FixQoA6taty4svvmi4IhHJjaFDh1K1alUAfgbeOuHrbwHnAbsdx+666y7GjBlDjRo1QlGiSI5dcMEF9qKbq1evZuvWM/XFSLj79ttv7Q6QUqVKMXr0aIWSBigcMKRUqVKMHDnSfjx48GB27dplsKLsUTAgkn2RHBCsX7/enk4QExPDp59+qukEIhGqRIkSfPHFF/aF9tPAUiAJuBhruoF/scJSpUoxbNgwBg4cSFxcnJF6RbLLuVCdugci19atW3nttdfsxx9++CGVK1c2WJF7KRww6LrrruOOO+4ArOkFjz/+OGlpaYarOj0FAyI5F4kBwbFjxxg4cCCpqamANZ2gVatWhqsSkby47LLLeOyxxwBIxVpboDzWQoV+rVq1YvLkyVx66aUGKhTJuR49ethjrTsQmdLS0hg0aBBJSUmANV3k5ptvNlyVeykcMOy9996jWrVqACxZsoQRI0YYrujUFAyI5F6kBQSDBw9m/fr1ADRq1EjTCUSixIsvvsgFF1wAwEYg0Xc8NjaWAQMGMHLkSMqWLWusPpGcat26td3hsmLFiojowpWsPvroI/766y8AatWqxbvvvmu4IndTOGBYyZIlGT9+PPny5QOsfyALFiwwXFVWCgZE8i5SAoLp06czZYq1JFmhQoWYMGGCPadTRCJbXFwcb775ZpZj5cuX5/PPP6dXr17ExOiyUCJP48aN7bGmFkSWhQsX2ru45cuXjzFjxmhXFMP0UyAMtG7dmldeeQWwPjAMGjSIffv2Ga7KomBAJHDCPSDYsmVLli6BESNGUK9ePYMViUigtWvXji5dutiPk5KSKFmypMGKRPLGP0UXNLUgkhw4cIAnnniCjAxrxZNnn32W1q1bG65KFA6EiYEDB3L11VcDsH//fp544gnS00+12VDoKBgQCbxwDQiSk5P573//a8/569GjB3fddZfRmkQkOMaNG2evK3DkyJEsa4yIRJq2bdsSGxsLwNKlS8PmBpucXkZGBk8++aS9bWG7du146qmnDFcloHAgbMTExPD5559TsWJFAP744w8+/vhjY/UoGBAJnnAMCN566y1Wr14NWNsWDh061FgtIhJcMTExfPfdd9SuXRuAlStX8vbbbxuuSiR3YmJiaNCgAWD9PJ09e7bhiuRsPvnkE3777TfAmto0duxYe4q1mKVwIIyUK1eOcePG2XP+hg0bxsKFC0Neh4IBkeALp4Bg1qxZfPnllwDEx8czceJEihQpEvI6RCR0ihUrxsSJE+3F3MaMGaMPVRKxunXrZo9nzpxpsBI5myVLltg3IDweD2PHjqVChQqGqxI/hQNhpm3btjz77LOA1XLz3//+lx07doTs/RUMiIROOAQEa9euzdLK9+6779KoUaOQvb+ImNO0adMsHQNPP/00W7ZsMViRSO506NDBvvO8aNEiDh06ZLYgOaV9+/YxcOBAe+r0M888Q/v27Q1XJU4KB8LQ008/zRVXXAFYi3U8/PDD9jzgYFIwIBJ6JgOCgwcP8vDDD3Ps2DHAuvNy3333Bf19RSR89O3bl1tuuQWw1h948MEHOXz4sOGqRHImJibGXkA3PT2dOXPmGK5ITnTs2DEeeughe50B5w1RCR8KB8JQvnz5mDBhgj0XcM2aNTz99NP2ap7BoGBAxBwTAUFqaioDBgxg+/btAFx44YWMGjUKj8cTtPcUkfDj8XgYNWoU559/PgCbN2/mkUce0QKFEnFuvfVWe6xdC8KLfwHClStXAlCtWjW+/PJLrTMQhhQOhKlSpUoxdepUe6/PWbNmMXLkyKC8l4IBEfNCHRAMHjyYRYsWAdZiQN988w2FChUKynuJSHgrXrw406ZNo1y5coDVlv3888/j9XoNVyaSfddff729bteCBQs4cuSI4YrE75133uHHH38EoGjRokyfPl3rDIQphQNh7LzzzmPcuHH2nbzhw4fb/7ACRcGASPgIVUAwYcIEJkyYAEBcXBxff/01VapUCeh7iEhkqVGjBlOnTqVAgQIATJ06lQ8//NBwVSLZFxsby7nnngtAWloaP//8s9mCBIDJkyczevRowOqOnjRpkr27hIQfhQNhrlOnTrz22mv24yeffJI1a9YE5LUVDIiEn2AHBAsXLuT111+3H3/44Ye0bt06IK8tIpGtZcuWfPHFF/bjoUOHMmPGDIMVieTMjTfeaI8DfUNNcm7BggW8/PLL9uP333+fq6++2mBFcjYKByLAY489Rvfu3QFrMY+HH36Yffv25ek1FQyIhK9gBQT//vsvAwYMIC0tDYABAwbQo0ePPNcrItHjlltuyXJT4umnn2bZsmXmChLJgVtvvdXuuP3tt99CsqC3nNqGDRt49NFH7Z0J/vOf/9CnTx/DVcnZKByIAP7Fgpo1awbAjh07eOCBB3I9l0rBgEj4C3RAsG/fPu677z57e6err76awYMHB6pcEYkijz/+OL179wYgJSWFhx9+mK1btxquSuTs4uLiqFmzJgDJycnMmzfPcEXutG/fPvr162d/Vrnuuut44403DFcl2aFwIEIULFiQb7/9lmrVqgHWDgb9+/cnOTk5R6+jYEAkcgQqIDhy5AgPPPCAvTNBgwYNGD9+PLGxsQGvWUQin8fjYcSIEbRr1w6wtj3t16+ftjiUiNC5c2d7rKkFoXf8+HH69+9vX3M0bdqUcePGaWeCCKFwIIJUqlSJH374gdKlSwPW3OFBgwbZ7Tpno2BAJPLkNSBITk6mf//+9lol1apV43//+x8lSpQIVskiEgXy58/P5MmT7b3jN23axKOPPqotDiXsde/e3Z5aMHfuXI4fP264IvfIyMjgqaeeYvny5QBUqVKF7777jsKFCxuuTLJL4UCEqVevHtOnT7e3HJs1axavvfbaWbcbUjAgErlyGxCkp6czaNAgFi5cCEDp0qX54YcfqFy5ctBrFpHIV7JkSaZPn06ZMmUA+OOPP3j55Ze1xaGEtUKFClG1alUAkpKSslz/SnANHTqUmTNnAlCkSBGmTZtGpUqVDFclOaFwIAK1bNmSKVOm2C3BEyZMYOTIkad9voIBkciX04DA6/Xy2muvMWvWLMC6WJo+fbp9F1BEJDtq1arFt99+S3x8PABfffUV7733ngICCWsdO3a0x5paEBpjxozho48+AjKvWRo3bmy4KskphQMRqkOHDnzyySf242HDhjFp0qSTnqdgQCR65CQgGDlyJBMmTACsvZ+nTJlCy5b/396dR0VdL/4ff7FoiBu4lZmJqamVVq75LQzSm7iklhrIoqltQNltX+yGluu1TmXuknoVF7xqqWmaBoimoeFWKlBqSkopagpBCMz8/uDn5zaB5QJ8YD7Pxzlz+PCeIV6n42FmXvNeOpdrXgDO4f/+7/80f/584/vo6Gh99NFHFASosIYMGWJcJyQksBymjMXExDhscvzBBx84FDSoPCgHKrGwsDCHnT/feecdffbZZ8b3FAOA87mcgmDhwoWaNm2a8f3cuXMVEBBQ7lkBOI+goCBNnTrV+H7OnDkUBKiwatSoYUxnz8rKUlJSksmJnNefi4GoqCg9++yzJibCtaAcqOReeuklvfjii5KK3iSMGjVK69evpxgAnNhfFQSLFy/Wv//9b+Ox7777rsLCwsyKCsCJREZGUhCg0ujRo4dxzdKCslFSMTB69GjzAuGaudj5i17p2e12RUZGasaMGSXeTzFgHW3atJEk+fj4aM2aNSanQVmz2WwKCgrSwYMHS7x/zJgxeuutt8o5FQBnN23aND3zzDPG90888YSeffZZY4d4oCI4e/asunbtKkny8vJSfHw8R/iWIooB58TMASfg4uKiqVOn6oknnih2H8UA4Lz+PIPgj958802KAQBlIjIy0mHp0pw5c9ikEBWOt7e3GjRoIEn69ddflZycbHIi57Fw4UKHYmD06NEUA06CcsBJuLq6asaMGWrWrJkx5uLioptuusnEVADKmqurq/r27esw1qlTJ7399tsmJQJgBREREQ4FQXR0NAUBKpzu3bsb1xdP78G1WbhwocPyxdGjRysqKsrERChNlANOxM3NTQcOHDAKArvdrldffdVhk0IAzmXBggUO7X2nTp20fft2pvcCKHMUBKjoRowYYVzHxcXJZrOZmKbyoxhwfpQDTqZq1apKTU01lhjYbDa98cYbJR5zCKDystvtmjVrlsOJJaNGjdL27dvl6sqfdgDlIyIiQtOnTze+pyBARdKgQQPVrVtXknTq1Cnt3bvX5ESV15+LgTFjxlAMOCFeQTohNzc3zZw5U+Hh4ZKK3kS8/fbbmjFjBk/WgBMoLCzU+PHjHXYNHz16tMaOHUsxAKDchYeHUxCgwvLz8zOuWVpwdUoqBtjXyDnxKtJJubq6atq0aXrppZeMsenTp2vcuHEqLCw0MRmAa3HhwgW98sorWrp0qTE2adIk2nsApiqpIBg7dqwKCgpMTAU4Li3YtGkTpdUVsNls+uijjygGLIRywIm5uLho8uTJDtOOY2Nj9fLLLysvL8/EZACuRnZ2tiIiIvTFF19IKpolNH/+fL3yyismJwOA4gXBsmXLFBkZqaysLBNTweoaN24sLy8vSVJGRob2799vbqBKIi8vT6+++qpmz55tjFEMOD/KAQt46aWXtGDBAuNs140bNyo8PJwna6ASyczM1PDhw5WUlCRJqlatmlatWqWhQ4eanAwA/ic8PFwLFy5U1apVJUnbtm3TkCFDdPz4cZOTwcruu+8+45qlBX/v9OnTGjFihNavXy+paEbyhx9+SDFgAZQDFhEWFqbVq1fL09NTkrRz504NHz5cmZmZJicD8HfS09MVFhamgwcPSpLq1KmjuLg49e7d2+RkAFBcaGioNm3aZGwE98MPPyg4OFj79u0zORms6uJG3RJLC/7OoUOHFBISYmzeWL16da1atUojR440ORnKA+WAhfTs2VNxcXHGk3VKSopCQkKUmppqcjIAl7Jr1y6Fhobqp59+klQ0PXLr1q265557TE4GAJfm6+urr7/+Wrfeeqsk6cyZMxo+fLixLAooT7fccotq1qwpSTp27JjS0tJMTlQxbd++XWFhYcZMn0aNGmnr1q3q06ePyclQXigHLKZz587aunWrbr75ZknSiRMnFBYWpk2bNpmcDMCfrVixQiNGjNCZM2ckSbfffru2bdum1q1bm5wMAP5e8+bNtX37dmO3+Ly8PL344ouKjo7mk1uUuy5duhjXLC0obvny5Q7Ljtu1a6cdO3borrvuMjcYyhXlgAW1atVK27dvV4cOHSRJubm5ev755zVjxgzZbDaT0wEoKCjQhAkTNHr0aGOn7+7du2vLli266aabTE4HAJevTp062rBhg8P+KB9++KGioqKUn59vYjJYzZ9PLUARm82m9957T2PGjDFONOvbt68SExN14403mpwO5Y1ywKJuvPFGJSYmKjg42BibPn26XnrpJeXk5JiYDLC2c+fO6emnn9bixYuNseeee06ff/65vL29TUwGAFenatWqmjdvnsaNG2eMffLJJ3r66ad17tw5E5PBSm677TZVr15dUtG6+sOHD5ucyHw5OTl6/vnnNX/+fGPshRde0MqVK43/V7AWygELq1atmmJiYjRx4kS5uLhIKppmNWTIEJ04ccLkdID1HDp0SIMHDzZOJKhSpYqio6P1wQcfGKeNAEBl5OLiojfeeENLly7VddddJ0nasWOHQkNDlZ6ebnI6WEXHjh2Na6vPHjh58qSGDRumuLg4SUXHI0+fPl3vvfee3NzcTE4Hs1AOWJyLi4teffVVrV692tioJTU1VYMHD9Y333xjcjrAOuLj4xUSEmK8SG7QoIHi4+MdpkECQGUXGBio+Ph41a9fX5L0448/avDgwYqPjzc5GazgscceM66tXA7s3LlTgwcP1oEDByRJNWvW1Nq1axUeHm5yMpiNcgCSpD59+igpKUnNmzeXVLSr8IgRIzRr1ixj/RGA0pefn6/Jkydr5MiR+u233yRJd999t3bu3Kl7773X5HQAUPq6dOmipKQkY3PVc+fOaeTIkRo/frzy8vJMTgdn1r59e3l4eEiSDh48aLlZKwUFBZo6dapGjBihkydPSpJuvvlmbdu2TT169DA5HSoCygEYWrduraSkJHXv3l1S0QYlU6dO1VNPPaXMzEyT0wHOJz09XUOGDNGCBQuMsUGDBmnLli3GiSIA4IyaNm2qbdu2qX///sbYkiVLFBwczFpwlKl27doZ11aaPXDixAkNGzZMs2bNMk4L8ff3V1JSku644w6T06GioByAgzp16mj9+vUaPXq0XF2L/nkkJSVpwIAB2r59u8npAOfxxRdf6NFHH9V3330nqWjDrilTpig2NpZNgABYgpeXl1auXKnp06cb+xCkpaUpMDBQy5cv57hDlIk/npxhlXJgw4YNGjhwoPbs2SOpaH+B8ePHa+PGjbrhhhvMDYcKxcXOX15cQkJCgoKDg5WRkSGpaH+Cxx9/XBEREWyOVkG1adNGkuTj46M1a9aYnAYlycvL0+TJkxUbG2uMNW/eXLGxsQ6fZgCAlXz77bcKCgoy1kBL0j/+8Q9FRUWpdu3aJiaDM2rfvr0uXLggSU79Bjk3N1eTJk3SihUrjDEfHx8tWbJE99xzj4nJUFExcwCX5Ofnpz179iggIECSZLfbNWfOHI0YMUI///yzyemAyufIkSMKCQlxKAaCgoKUnJxMMQDA0tq0aaOdO3fq6aefNsY2btyoQYMGGZ92AqXlzjvvNK6//PJLE5OUndTUVAUFBTkUA4GBgdqzZw/FAC6JcgB/qUGDBlq7dq0mTZpkHGuya9cuPfLII1q1ahVT/oDLYLPZFBMTo0cffVSpqamSJA8PD82ZM0eLFy9WrVq1TE4IAObz9PTUjBkztGLFCnl5eUmSMjIy9Nhjj2nmzJlskIxSExwcbFxv3LjRxCSlz263a/HixQ77d3h6emru3LlasmQJM3Hwl1hWgMu2fft2BQUF6dixY8ZY165dFRUVpQYNGpiYDBexrKDiOXbsmP71r39p165dxljr1q21bNkyNgACgEs4duyYQkJCtHXrVmOsQ4cOmjBhgtNOAUf5sdlsat++vQoKCuTi4qK4uDjVq1fP7FjX7OzZs3rrrbeUkJBgjN11111aunSpWrZsaV4wVBrMHMBl69Kli/bs2aOwsDBjLDExUf3799fq1auZRQD8gc1m06JFizRgwACHYuCZZ57Rzp07KQYA4C/cfPPNio+Pd9gg+ZtvvtHAgQO1Zs0aXnPgmri6uhrPw3a7XXFxcSYnunYJCQkaOHCgQzHwz3/+U19//TXFAC4b5QCuiLe3txYsWKBVq1YZzX1WVpZGjRqlZ5991jgzFbCy9PR0DR8+XBMnTtTvv/8uqejYrvj4eH300UecRgAAl8Hd3V1RUVFKSEhQ48aNJUnnzp3TG2+8occff5wjD3FNAgMDjevKvLQgIyNDI0eOdHgdXq9ePX322Wd6//33jZNAgMtBOYCr0rdvX+3fv18hISHG2ObNm5lFAEsrLCw0ZgskJycb45GRkdq3b5/8/PzMCwcAlZSvr6/27NmjQYMGGWM7duzQgAEDNGXKFKOEBa5Er169jP20du7cqV9//dXcQFcoPz9fc+fOVb9+/RQfH2+MP/jgg9q3b5969+5tYjpUVpQDuGp16tRRTEyMPv30U11//fWS/jeLYPjw4frhhx9MTgiUn3379ik4OFgTJ05Ubm6upKK9H+Li4jR16lTVqFHD5IQAUHnVqVNHy5Yt05o1a+Tj4yNJKigo0Jw5c9S/f38lJiaaGxCVjqurqzHdvrCw0OENdkX3zTffaNCgQXr//feN1xw33HCDFi9erPXr16thw4YmJ0RlRTmAa9avXz/t37/fYefXi+sCJ0+erOzsbBPTAWXrzJkzioqKUkhIiMP53BEREfr222/l7+9vYjoAcC59+vTR/v379cYbb6hKlSqSpOPHjysyMlLPP/88Ry3jivxxNsqmTZtMTHJ5zpw5o1GjRmnYsGE6dOiQpKKS45lnnlFKSooGDx4sFxcXk1OiMuO0ApSqtWvXauTIkQ7rAOvVq6cXX3xRvXv35g9WGeO0gvJTWFio5cuXa8qUKTp//rwx3qZNG02dOlVdu3Y1MR0AOL+DBw8qIiLCYQO2atWqKTIyUsHBwUZ5AFxKQUGB2rdvL5vNJnd3dyUmJqpmzZpmxyrGZrNpxYoV+uCDDxxec3To0EEzZ85U+/btTUwHZ8LMAZSq3r17a//+/RozZow8PDwkSZmZmXr99dc1bNgwpaWlmZwQuHZ79+7V4MGDNXbsWONJulatWvrggw+0a9cuigEAKAetW7dWXFycFi5caBypnJubq3fffVeBgYHavXu3yQlR0bm7u6t58+aSioqCzZs3m5youJSUFIWFhentt982XnPUrl1b06ZN09dff00xgFJFOYBS5+HhobfeeksHDhxQv379jPHk5GQ9+uijGjdunDIzM01MCFydEydOaNSoUQoNDdXBgweN8bCwMKWmpuq5556Tu7u7iQkBwFpcXFwUGhqqlJQUhYeHGzMUv//+ew0ZMkRRUVG85sBfevjhh43rirS04Ny5c5o0aZICAwO1b98+Y/ziv/eIiAhjQ0WgtLCsAGVu3bp1GjlypLE2Siqa9jd06FANHTqUjdpKEcsKysbZs2c1e/ZsxcbGKj8/3xhv06aNpk2bJl9fXxPTAQAu2rFjh8LDw7Vr1y5jrFq1agoKCtKwYcPk7e1tYjpURBcuXFCHDh1kt9vl4eGhzZs3y9PT07Q8WVlZiomJ0cKFC5WVlWWMt2zZUtOnT9cDDzxgWjY4P2YOoMz16tVL3333nd555x3jfPfc3FzNnDlTvXr1UkxMjC5cuGBySqC4nJwczZw5Uz179lRMTIxRDHh5eRlLCCgGAKDi6NSpk3bs2KEpU6aoVq1akopec8ybN08BAQGaMmWKzp07Z3JKVCRVq1Y1TsD4/ffftXXrVlNy5OTkKDo6WgEBAZo+fbpRDHh4eGjcuHHau3cvxQDKHDMHUK5++eUXvfPOO5o1a5YKCgqM8UaNGikyMtLhzFlcOWYOlI78/HwtX75cs2bN0unTp43xatWq6bnnntMrr7zCp08AUMH98ssvmjBhgmbOnKm8vDxjvEaNGhoyZIhCQ0Mr5OZzKH/R0dH68MMPJUkBAQGaPHlyuf3u3NxcxcbGau7cuTp79qwx7ubmpmHDhunNN99UkyZNyi0PrI1yAKY4dOiQ/vWvf2nJkiUO4y1atNDTTz+tbt26URJcBcqBa5Ofn6+1a9dq9uzZSk9PN8bd3Nw0YsQIRUVF6cYbbzQxIQDgSv30008aP368oqOjHZaG1apVS4899phCQkJMnUYO8+Xk5Khz586SJE9PTyUmJuq6664r09+Zl5en5cuXKzo62mFfDFdXV4WGhuqtt95Ss2bNyjQD8GeUAzDV7t279frrr2vDhg0O4z4+Pho+fLj69OnDUURXgHLg6uTm5mrlypWaP39+sTOyBw4cqLFjx6ply5YmpQMAlIajR49q7NixmjdvngoLC41xb29vDRs2TEFBQapWrZqJCWGmXr16GR8MTJkyRf7+/mXyey5cuKBPPvlEs2fP1smTJ41xFxcXBQUFKSoqitccMA3lACqE+Ph4vfbaa9qxY4fD+PXXX6+hQ4dqwIABtPqXgXLgypw/f15Lly5VTEyMw1Q+SXrggQc0ceJEdezY0aR0AICycOjQIb3zzjtauHChbDabMV63bl09/vjjGjRoUJl/aoyKZ+rUqZo1a5YkqW/fvho3blyp/vfz8/O1evVqzZo1SxkZGQ73DRw4UFFRUbrjjjtK9XcCV4pyABWG3W7Xxo0bNWHCBCUkJDjc5+XlpeDgYAUHB6t27drmBKwEKAcuT2ZmphYsWKBly5bpt99+c7ivT58+eu2113TvvfealA4AUB7S0tI0ZswYLVmyRH98OVy3bl0NHDhQAwcO1A033GBiQpSn7OxsdenSRZJUs2ZNbd68uVRmr54+fVqffPKJli1bVqwU6Nevn0aPHq277rrrmn8PUBooB1Ahff3115owYYJWr17tMO7p6al+/fpp8ODBatq0qUnpKi7Kgb+WkpKixYsXa+3atQ4nZLi6uiowMFCvvfaa2rZta2JCAEB5279/v8aMGaP//ve/DuNubm7y8/NTUFCQOnfuLBcXF5MSorw8+OCDxhv4mTNnXvUHBXa7XXv37tXSpUv1xRdfOOx1IRUtYRgzZow6dOhwzZmB0kQ5gArtu+++06RJk7RkyRKH9YGS1KVLFwUHB8vX15fNC/8/yoHi8vPz9eWXX2rx4sXavXu3w31Vq1bVsGHD9PLLL7PpDwBY3N69ezVu3DitXLmy2GsOHx8fBQYGqm/fvsYRiXA+7733nubPny9JGjBggEaPHn1FP5+Tk6O1a9cqNjZWqampDve5uLioZ8+eevPNN40ZCkBFQzmASuHIkSN69913NX/+fOXk5Djc17BhQw0YMECPPPKI6tevb1LCioFy4H+OHz+uFStWaOXKlQ7HEUpS7dq19eSTT+r5559Xw4YNTUoIAKiIjh8/rjlz5mj27NnFpoF7eHiod+/eCgwMVOvWrU1KiLJy9uxZde3aVVLRRpVxcXFyd3f/2587fPiwYmNjtXr1amVnZzvcV7duXY0YMUJPPfWUbrnlljLJDZQWygFUKmfPntW8efM0bdo0HT582OE+d3d3+fn56eGHH1aXLl0secqB1cuBvLw8JSYmauXKlfrqq6/05z9vt99+u5555hmFhoaqRo0aJqUEAFQG+fn5WrVqlaZPn674+Phi97dt21ZBQUF68MEH2cDQiXTr1s04ReDjjz9Wp06dSnxcfn6+EhISFBsbq6SkpGL3d+7cWZGRkRo0aJA8PDzKNDNQWigHUCkVFhbq888/14wZM/T5558XexPo7e2tgIAA9e7dW23btrXMOkErlgM2m03Jycn67LPPtHHjRmVlZTnc7+7urv79+ys8PFz+/v6W+bcAACg9Bw4c0IwZM/Sf//yn2POMt7e3+vXrp169eqlVq1Y8z1Ry48eP15IlSyRJQUFBGjVqlMP9hw8f1rp16/TJJ584HEUoSdWqVVNwcLDCw8PVvn37cssMlBbKAVR6P/74o6Kjo/Xxxx8XO6Nekho3bqw+ffqod+/eatKkiQkJy4+VyoG0tDR99tlnWrdunX755Zdi9zdp0kRPPvmkhg8fzm7TAIBSkZ2drUWLFmnatGn69ttvi93fpEkT9ejRQwEBAWrRooUJCXGtfvnlF3Xv3l2SVL9+fW3atEnp6elav369NmzYoO+//77Yz7Ro0UIREREaOnSovL29yzsyUGooB+A08vPz9fnnn2vRokVavXq1fv/992KPadOmjQICAuTv76/GjRubkLJsOXM5YLfbdfjwYcXHx2vdunUlPjnXqFFDjzzyiEJCQtStWzc2qgQAlAm73a5t27Zp2rRpWr58ebHd6CWpWbNm6tGjh3r06KGmTZsyo6AS8fPzM/Yr8vHx0Y8//ljsMa6ururbt68iIiLUrVs3ubq6lnNKoPRRDsApnT9/XitXrlRMTIzi4uKKLTuQip6077//fvn7+6tNmzZO8UbS2cqB/Px87d69W/Hx8dq8ebPS09OLPcbd3V0BAQEKCQlR37595enpaUJSAIBVnTp1SitWrFBsbKw2b95c4msOHx8f+fv7y9/fX23btnWK1xzOxm6368CBA4qLi9OyZcv066+/lvi4e+65R4GBgRo0aJAaNWpUviGBMkY5AKd3/PhxLV26VIsWLSp2lN1FderUka+vr/z9/dWlS5dK+wbTGcqB8+fPa+vWrUpISNDWrVuLre28qEuXLgoNDdWjjz6qevXqlXNKAACKy8jI0PLlyxUbG6uvvvqqxMfUqVNHfn5+8vf3V6dOnSrtaw5nkJeXp+TkZMXFxSkhIaHEZYqS1KFDB6MQcPYlqrA2ygFYSkpKilatWqU1a9Zo27ZtJbb7VapU0Z133qmOHTuqY8eOatu2baXZhbgylgM5OTnas2ePdu7cqZ07d2r//v0qKCgo9jg3Nzfdf//9euihh9S3b1+OAwIAVGjp6elasWKFPv30U23ZskU2m63YY9zc3HTbbbepXbt2xs3Ly6v8w1pEVlaW9uzZo127dik5OVnfffddiUtCJOmGG27QnXfeqRdeeEEPPvhgOScFzEE5AMs6efKk1q1bpzVr1mjDhg367bffSnxc1apV1bZtW3Xq1EkdOnSo0GVBZSgHLrcMkKTatWurV69eeuihhxQQEMAmPwCASikzM1Nr167VqlWrtGHDBuXk5Fzysc2bN1e7du3Uvn17tWvXjk11r0FmZqZ27dpllAFpaWklljRS0eu97t27q1+/fnrooYfUsGHDck4LmI9yAJD0+++/Kz4+XqtXr9YXX3yhw4cPX/KxVatW1R133KHbbrvNuPn4+FSI9YMVrRzIz8/X4cOHdeDAAe3fv18HDx7UgQMHLlkGSFLLli3Vs2dPPfTQQ/L19VWVKlXKMTEAAGUrNzdXmzZt0tq1a7VlyxYdOHDgLx/fqFEjoyho166dfHx82NywBHa7XcePH1dycrJRCJS0keAfNW/eXL6+vurVq5d69OihmjVrlk9YoIKiHABKcOzYMW3evFkJCQlKSEj4y7JAKjrXtlWrVrrtttvUunVrozAo7ze2ZpYDFy5cMIqAi7fU1FRduHDhL3+uVatW8vPzk5+fn+6//34+IQEAWEpmZqa2bt2qxMREbdmyRbt371ZhYeElH1+nTh21aNFCTZs2VdOmTXXLLbeoadOmatCggSVKA7vdrtOnT+vIkSM6cuSIDh8+rCNHjuiHH37QyZMnL/lzLi4uatu2rXx9fdW1a1fdd999zA4A/oRyALgMR48edSgLjhw58rc/4+bmpkaNGqlJkyZq0qSJfHx8jK8NGjQokyNvyrocKCws1M8//6yjR4/qxx9/1NGjR43rjIyMS07V+6OWLVs6lAE8MQMA8D9ZWVnavn27tmzZosTERCUlJSkvL+9vf6569eoOZcHFW+PGjSvlLLyCggIdP37coQC4eH2pzYr/qEqVKurQoYO6du0qX19f3XvvveznAPwNygHgKlxcw5acnGzc/m7q2h95eHiocePGatCggerXr6/69eurXr16xvXF7690b4NrKQdyc3OVmZmpU6dO6dSpUw7Xp06d0smTJ5Wenv63MwH+qEWLFmrfvr1xu/vuu3liBgDgCuTl5embb74xZhYkJSXpzJkzl/3z7u7uaty4sZo0aaJ69erJ29tbXl5eJX719PQs09kHdrtdubm5+vXXX3X27NkSv54+fdr48OFSmwWWpHbt2urYsaN8fX3l6+urzp07cxIEcIUoB4BScvr0aaMw2LNnj9LS0pSWlnbJjQ4vR40aNeTp6SlPT09Vr15d1atXd/je09NTVatWlaurq1xdXTVjxgxJkpeXlwYPHiybzSabzaa8vDz99ttvysnJMW5//D47O/svN0f6O7Vq1dKtt96qli1b6u677zaKgNq1a1/1fxMAAJTs1KlTSklJ0cGDB5WSkmJcHz16tMSTmC5XlSpVHMoCb29v1apV66r2VbLZbDp//nyxN/+XMwvirzRu3FitWrVS69at1apVK+P6+uuvt8SyCqAsUQ4AZchutysjI8MoCv54O3LkyBV9Cm82Dw8PNWvWTLfeeqtxa9GihW699VbLrHMEAKAiy8nJUVpamkNhkJKSotTU1Gt+U16eqlSpohYtWhgFwMWvLVu2VI0aNcyOBzgtygHAJHa7XWfOnNGJEyeUkZGhjIyMYteZmZnKzs5Wdna2srKyrunTgD9zdXVVzZo1VaNGDdWsWVP169dXw4YN1bBhQ914440OXxs2bCgvLy8KAAAAKqHCwkLjdcXl3krzA4wqVaqoXr16l3WrW7euGjVqJHd391L7/QAuD+UAUEnY7XZjCUBWVpbxNS8vz1g+sGPHDhUUFKhatWq6++675ebmJhcXF3l4eDgUATVr1pSHhwdv9gEAQDF2u13Z2dnKzMzUmTNnrurDCRcXF3l5ealevXqqVasWrzmASoByAAAAAAAAiyv9s9QAAAAAAEClQjkAAAAAAIDFUQ4AAAAAAGBxlAMAAAAAAFgc5QAAAAAAABZHOQAAAAAAgMVRDgAAAAAAYHGUAwAAAAAAWBzlAAAAAAAAFkc5AAAAAACAxVEOAAAAAABgcZQDAAAAAABYHOUAAAAAAAAWRzkAAAAAAIDFUQ4AAAAAAGBxlAMAAAAAAFgc5QAAAAAAABZHOQAAAAAAgMVRDgAAAAAAYHGUAwAAAAAAWBzlAAAAAAAAFkc5AAAAAACAxVEOAAAAAABgcZQDAAAAAABYHOUAAAAAAAAWRzkAAAAAAIDFUQ4AAAAAAGBxlAMAAAAAAFgc5QAAAAAAABZHOQAAAAAAgMVRDgAAAAAAYHGUAwAAAAAAWBzlAAAAAAAAFkc5AAAAAACAxVEOAAAAAABgcZQDAAAAAABYHOUAAAAAAAAWRzkAAAAAAIDFUQ4AAAAAAGBxlAMAAAAAAFgc5QAAAAAAABZHOQAAAAAAgMVRDgAAAAAAYHGUAwAAAAAAWBzlAAAAAAAAFkc5AAAAAACAxVEOAAAAAABgcZQDAAAAAABYHOUAAAAAAAAWRzkAAAAAAIDFUQ4AAAAAAGBxlAMAAAAAAFgc5QAAAAAAABZHOQAAAAAAgMVRDgAAAAAAYHGUAwAAAAAAWBzlAAAAAAAAFkc5AAAAAACAxVEOAAAAAABgcZQDAAAAAABYHOUAAAAAAAAWRzkAAAAAAIDF/T+tU673XHQsMwAAAABJRU5ErkJggg==",
"text/plain": [
"<Figure size 640x480 with 2 Axes>"
]
},
"metadata": {
"image/png": {
"height": 266,
"width": 515
}
},
"output_type": "display_data"
}
],
"source": [
"fig, ax = pyplot.subplots(1, 2)\n",
"ax[0].pie(\n",
" [1, 1, 1, 1, 1, 1, 1, 1],\n",
" colors=['red', 'red', 'red', 'lightgray', 'lightgray', 'lightgray', 'lightgray', 'lightgray'],\n",
" wedgeprops={\"edgecolor\":\"k\",'linewidth': 1}\n",
")\n",
"ax[0].set_title('Figure 1.6.1.3')\n",
"\n",
"ax[1].pie(\n",
" [1, 1, 1, 1, 1],\n",
" colors=['red', 'red', 'red', 'lightgray', 'lightgray'],\n",
" wedgeprops={\"edgecolor\":\"k\",'linewidth': 1}\n",
")\n",
"pyplot.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is clear that $3$ parts of $8$ must be smaller than $3$ parts of $5$. In essence, keeping the numerator constant, the final values increases when the denominator get smaller and decreases when the demominator gets bigger."
]
},
{
"cell_type": "markdown",
"id": "3443d8d0",
"metadata": {},
"source": [
"Understanding the mechanics, leads us to comparing fractions and ratios. When is one smaller or larger than the other?"
]
},
{
"cell_type": "markdown",
"id": "b172e1a0",
"metadata": {},
"source": [
"To answer this question, we must have equal $b$ parts (equal denominators). In __Equation 1.6.1.3__ we made such a comparison. We can make a direct comparison by _equalling_ $b=8$ and $b=5$. The easiest way to do this is to start by considering multiplying both by $1$. The number $1$ can be written as a fraction, in many ways, as we see in __Equation 1.6.1.4__, as long as $b \\ne 0$."
]
},
{
"cell_type": "markdown",
"id": "33ee71ec",
"metadata": {},
"source": [
"$$\n",
"1 = \\frac{1}{1} = \\frac{2}{2} = \\frac{1206}{1206} = \\frac{b}{b}\n",
"\\tag{Equation 1.6.1.4}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "64563544",
"metadata": {},
"source": [
"Multiplying by $1$ does not change a number. All we have to do is write $1$ as a fraction. We do so my choosing the _opposite_ value of $b$, shown in __Equation 1.6.1.5__, remembering that we multiply numerator by numerator and denominator by denominator, we now have equal parts $b=40$."
]
},
{
"cell_type": "markdown",
"id": "db6a340b",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\frac{3}{8} \\text{ vs. } \\frac{3}{5} \\\\ \\\\\n",
"&\\frac{3}{8} \\times \\frac{5}{5} \\text{ vs. } \\frac{3}{5} \\times \\frac{8}{8} \\\\ \\\\\n",
"&\\frac{3 \\times 5}{8 \\times 5} \\text{ vs. } \\frac{3 \\times 8}{5 \\times 8} \\text{ (see section 1.6.2 below)} \\\\ \\\\\n",
"&\\frac{15}{40} \\text{ vs. } \\frac{21}{40}\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.1.5}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "3aadcb59",
"metadata": {},
"source": [
"Clearly $15 < 21$ and three eighths is smaller than three fifths."
]
},
{
"cell_type": "markdown",
"id": "e6266b69",
"metadata": {},
"source": [
"The `sympy` package makes such a comparison easy. We use the `Rational` function to create the fractions and the `>` and `<` symbols (which are comparison operators) to compare the fractions. We print the two fractions to the screen to see the use of the `Rational` function."
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{3}{8}$"
],
"text/plain": [
"3/8"
]
},
"execution_count": 105,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Print the rational three-eighths\n",
"sympy.Rational(3, 8)"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{3}{8}$"
],
"text/plain": [
"3/8"
]
},
"execution_count": 106,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Print the rational three-eighths\n",
"sympy.Rational(3, 8)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" In the code cell below, we ask the question: _Is three eighths smaller than three fifths?_ Note the use of the less than `<` comparison operation."
]
},
{
"cell_type": "code",
"execution_count": 107,
"id": "226df007",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\text{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(3, 8) < sympy.Rational(3, 5)"
]
},
{
"cell_type": "markdown",
"id": "37d397ad",
"metadata": {},
"source": [
"The `sympy` package returns `True` to indicate that three eighths is smaller than three fifths."
]
},
{
"cell_type": "markdown",
"id": "62a0537f",
"metadata": {},
"source": [
"Depending on the propblem, we can multiply both sides by $1$ expressed in smaller values for the numerator and denominator. The next problem shows the intuition."
]
},
{
"cell_type": "markdown",
"id": "2f6298d2",
"metadata": {},
"source": [
"__Problem 1.6.1.1__ Compare the two fractions in __Equation 1.6.1.6__ and determine which is larger."
]
},
{
"cell_type": "markdown",
"id": "9cf0b973",
"metadata": {},
"source": [
"$$\n",
"\\frac{3}{8} \\text{ vs. } \\frac{1}{4}\n",
"\\tag{Equation 1.6.1.6}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "7ca21e4d",
"metadata": {},
"source": [
"The results is shown in __Equation 1.6.1.7__."
]
},
{
"cell_type": "markdown",
"id": "644b22dd",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\frac{3}{8} \\text{ vs. } \\frac{1}{4} \\\\ \\\\\n",
"&\\frac{3}{8} \\times \\frac{1}{1} \\text{ vs. } \\frac{1}{4} \\times \\frac{2}{2} \\\\ \\\\\n",
"&\\frac{3}{8} \\text{ vs. } \\frac{2}{8} \\\\ \\\\\n",
"&\\frac{3}{8} > \\frac{2}{8} \\\\ \\\\\n",
"&\\frac{3}{8} > \\frac{1}{4} \\text{ (original problem)}\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.1.7}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "0ee463ac",
"metadata": {},
"source": [
"Clearly, three eights is larger than two eights and therefor three-eights is larger than a quarter."
]
},
{
"cell_type": "markdown",
"id": "c76170fe",
"metadata": {},
"source": [
"We use `sympy` again to verify the result. Note that in this case we use the larger than `>` comparison operator."
]
},
{
"cell_type": "code",
"execution_count": 108,
"id": "efecda1d",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\text{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 108,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(3, 8) > sympy.Rational(1, 4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we use the less than `<` comparison operator (keeping the order of the fractions the same), we get `False`."
]
},
{
"cell_type": "code",
"execution_count": 109,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\text{False}$"
],
"text/plain": [
"False"
]
},
"execution_count": 109,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(3, 8) < sympy.Rational(1, 4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"__Problem 1.6.1.1__ Is $\\frac{21}{43}$ larger than $\\frac{51}{100}$?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We multiply both sides with $1$ as represented in __Equation 1.6.1.8__."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\frac{21}{43} \\times \\frac{100}{100} \\text{ vs. } \\frac{51}{100} \\times \\frac{43}{43} \\\\ \\\\\n",
"&\\frac{2100}{4300} \\text{ vs. } \\frac{2193}{4300} \\\\ \\\\\n",
"&\\frac{2100}{4300} < \\frac{2193}{4300}\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.1.8}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We note that $\\frac{21}{43}$ is not larger than $\\frac{51}{100}$. In fact, it is smaller. We confirm this with code below."
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\text{False}$"
],
"text/plain": [
"False"
]
},
"execution_count": 110,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(21, 43) > sympy.Rational(51, 100)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We do not require the `Rational` function to create the fractions. We can simply use the `/` symbol to create the fractions and the great than `>` or less than `<` comparison operators as required."
]
},
{
"cell_type": "code",
"execution_count": 111,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 111,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(21 / 43) > (51 / 100)"
]
},
{
"cell_type": "markdown",
"id": "691c6408",
"metadata": {},
"source": [
"__Homework 1.6.1.1__ Is $\\frac{3}{13}$ smaller than $\\frac{7}{30}$?"
]
},
{
"cell_type": "markdown",
"id": "185c817c",
"metadata": {},
"source": [
"### 1.6.2 Multiplication of fractions"
]
},
{
"cell_type": "markdown",
"id": "cf30df60",
"metadata": {},
"source": [
"The multiplication of fractions is quite simple. In __Equation 1.6.2.1__, we see two fractions, where each of $a,b,c$, and $d$ are integers, and neither $b$ nor $d$ are equal to $0$."
]
},
{
"cell_type": "markdown",
"id": "9c676807",
"metadata": {},
"source": [
"$$\n",
"\\frac{a}{b} \\times \\frac{c}{d} = \\frac{a \\times c}{b \\times d}\n",
"\\tag{Equation 1.6.2.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "5ca90ec2",
"metadata": {},
"source": [
"__Equation 1.6.2.2__ shows an example of multiplying two fractions."
]
},
{
"cell_type": "markdown",
"id": "5b4ee4bf",
"metadata": {},
"source": [
"$$\n",
"\\frac{3}{11} \\times \\frac{7}{20}\n",
"\\tag{Equation 1.6.2.2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "4477987b",
"metadata": {},
"source": [
"We use `sympy` to solve the problem."
]
},
{
"cell_type": "code",
"execution_count": 112,
"id": "2e8811ef",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{21}{220}$"
],
"text/plain": [
" 21\n",
"───\n",
"220"
]
},
"execution_count": 112,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(3, 11) * sympy.Rational(7, 20)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Below, we calculate the same product using simple Python mathematical notation."
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 0.0954545454545454$"
],
"text/plain": [
"0.09545454545454544"
]
},
"execution_count": 113,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(3 / 11) * (7 / 20)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The result is a numerical approximation."
]
},
{
"cell_type": "markdown",
"id": "3e53a4a0",
"metadata": {},
"source": [
"Note that `sympy` will automatically simplify any fraction. In __Equation 1.6.2.3__ we see such a simplification where five tenths is eqal to a half."
]
},
{
"cell_type": "markdown",
"id": "30e7e877",
"metadata": {},
"source": [
"$$\n",
"\\frac{5}{10} = \\frac{1}{2}\n",
"\\tag{Equation 1.6.2.3}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "4ae48c61",
"metadata": {},
"source": [
"Below, we see that `sympy` simplifies the fraction."
]
},
{
"cell_type": "code",
"execution_count": 114,
"id": "3ad3dbfc",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{1}{2}$"
],
"text/plain": [
"1/2"
]
},
"execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(5, 10)"
]
},
{
"cell_type": "markdown",
"id": "4c363e39",
"metadata": {},
"source": [
"### 1.6.3 The least common multiple"
]
},
{
"cell_type": "markdown",
"id": "6ccf343b",
"metadata": {},
"source": [
"We have seen that comparisons of fractions are easier when the denominator is the same. Arithmetic with fraction is in general easier when the denominators are the same. In fact, in the case of addition and subtraction the denominators have to be the same. To do this, we need to find the least common multiple."
]
},
{
"cell_type": "markdown",
"id": "9fe84ec5",
"metadata": {},
"source": [
"In __Equation 1.6.3.1__, we see the addition of two fractions. We rewrite each fraction by multiplying it by $1$, written as a fraction where the numerator and denominator are the denominator of the _other fraction_."
]
},
{
"cell_type": "markdown",
"id": "c8a111e7",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"\\frac{3}{4} + \\frac{5}{6} &= \\left( \\frac{3}{4} \\times \\frac{6}{6} \\right) + \\left( \\frac{5}{6} \\times \\frac{4}{4} \\right) \\\\ \\\\ \n",
"&= \\frac{18}{24} + \\frac{20}{24} \n",
"\\end{align*}\n",
"\\tag{Equation 1.6.3.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "48a3e6fb",
"metadata": {},
"source": [
"In __Equation 1.6.3.2__, we see an alternative way to solve __Equation 1.6.3.1__."
]
},
{
"cell_type": "markdown",
"id": "95160f6f",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"\\frac{3}{4} + \\frac{5}{6} &= \\left( \\frac{3}{4} \\times \\frac{3}{3} \\right) + \\left( \\frac{5}{6} \\times \\frac{2}{2} \\right) \\\\ \\\\ \n",
"&= \\frac{9}{12} + \\frac{10}{12} \n",
"\\end{align*}\n",
"\\tag{Equation 1.6.3.2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "9bad06fd",
"metadata": {},
"source": [
"We have recognized that $12$ is the least common multiple between $4$ and $6$. Both $4$ and $6$ divide $12$. We can use the `lcm` function from the `sympy` package to find the least common denominator."
]
},
{
"cell_type": "code",
"execution_count": 115,
"id": "defda522",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOCAYAAADNGCeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQ4EaWT7Q2CMBCG0TgA0Q10Az82wA10BR3Bv/yVDdAJjG4gK8gGOoJxA3zepm0Q1IBeUt7rfbzt3dFOURSBkziOx+hH1gT94ewOsQ3RN3Y/Be/aY89l66GE4J4lhwKUUBNLlIJz50Tfol9kY2VdPg/WkrXGcXCBb1CJivFCjm6pClRN0NWnoUTEXSFQJWXJ2ITYh23IlHQjqdZLyxz2ykd80yFZfvBraAH+vM3NalwQiEgDMxP+iwwSNf4EaQK2GoDivUCQslEPffk/3QyCFUR90P9zOqXxABQsgWABjEB/I3T1rV2ZJKnhM9A9KXFIdMC9erOBcVEC+PI/2dPV8Axd/SpLhC0xZCjmOeCNbMQR2w39DO6s7QyqHPWrKuahPwGbS2D0goaxHgAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 12$"
],
"text/plain": [
"12"
]
},
"execution_count": 115,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.lcm(4, 6)"
]
},
{
"cell_type": "markdown",
"id": "6bc67d07",
"metadata": {},
"source": [
"We still have to determine if the solutions in __Equation 1.6.3.1__ and in __Equation 1.6.3.2__ are the same. To do this, we see the rule of addition of fractions in __Equation 1.6.3.3__. Fractions can only be added (or subtracted) if they have the same denominator. The addition of two fractions with the same denominator is shown in __Equation 1.6.3.3__ where $c \\ne 0$ and $a,b,$ and $c$ are integers."
]
},
{
"cell_type": "markdown",
"id": "64ceac90",
"metadata": {},
"source": [
"$$\n",
"\\frac{a}{c} + \\frac{b}{c} = \\frac{a+b}{c}\n",
"\\tag{Equation 1.6.3.3}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "8d828d86",
"metadata": {},
"source": [
"Our two solutions are shown in __Equation 1.6.3.4__."
]
},
{
"cell_type": "markdown",
"id": "2f919c8b",
"metadata": {},
"source": [
"$$\n",
"\\frac{38}{24} \\text{ and } \\frac{19}{12}\n",
"\\tag{Equation 1.6.3.4}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "63274710",
"metadata": {},
"source": [
"These two fractions are indeed the same. In __Equation 1.6.3.5__ we simplify thirty-eight twenty-fourths by multiplying it by $1$ in the form of a half over a half."
]
},
{
"cell_type": "markdown",
"id": "96caa66f",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\frac{38}{24} \\times \\frac{\\frac{1}{2}}{\\frac{1}{2}} = \\frac{\\frac{38}{2}}{\\frac{24}{2}} = \\frac{19}{12}\n",
"\\end{align*}\n",
"\\tag{1.6.3.5}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "97240b55",
"metadata": {},
"source": [
"We can use `sympy` to verify this."
]
},
{
"cell_type": "code",
"execution_count": 116,
"id": "032d14f6",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{19}{12}$"
],
"text/plain": [
"19\n",
"──\n",
"12"
]
},
"execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(38, 24)"
]
},
{
"cell_type": "markdown",
"id": "613de897",
"metadata": {},
"source": [
"This brings us to an important topic for the least common multiple. We can _break down_ or rewrite any integer into the multiplication of other interegers. In more advanced algebra, we learn that these _other integers_ (if the _break down_ is complete) are all prime numbers (numbers that are only divisible by $1$ and themselves). We see an example in __Equation 1.6.3.6__, where we consider the integers that multiply to get $18$ and $9$, with $18 = 2 \\times 3 \\times 3$ and $9 = 3 \\times 3$."
]
},
{
"cell_type": "markdown",
"id": "f16cade3",
"metadata": {},
"source": [
"$$\n",
"\\begin{array}{cccc}\n",
"18: & 2 & 3 & 3 \\\\ \n",
"9: & & 3 & 3\n",
"\\end{array}\n",
"\\tag{Equation 1.6.3.6}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "b6f70ee7",
"metadata": {},
"source": [
"We have taken care to write similar numbers in the same columns. In __Equation 1.6.3.7__, we add a new line by _bringing down_ or representing all the numbers."
]
},
{
"cell_type": "markdown",
"id": "bb492f9b",
"metadata": {},
"source": [
"$$\n",
"\\begin{array}{cccc}\n",
"18: & 2 & 3 & 3 \\\\ \n",
"9: & & 3 & 3 \\\\ \n",
"- & - & - & - \\\\ \n",
" & 2 & 3 & 3\n",
"\\end{array}\n",
"\\tag{1.6.3.7}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "3fe0d3e5",
"metadata": {},
"source": [
"If we multiply the numbers in the new row in __Equation 1.6.3.7__, we find the least common multiple of $18$ and $9$, which is $2 \\times 3 \\times 3 = 18$. So, instead of __Equation 1.6.3.8__, we can do __Equation 1.6.3.9__."
]
},
{
"cell_type": "markdown",
"id": "d098ce5d",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"\\text{Longer method: } \\\\ \\\\\n",
"\\frac{4}{9} + \\frac{11}{18} &= \\left( \\frac{4}{9} \\times \\frac{18}{18} \\right) + \\left( \\frac{11}{18} \\times \\frac{9}{9} \\right) \\\\ \\\\ \n",
"&= \\frac{72}{162} + \\frac{99}{162} \\\\ \\\\ \n",
"&= \\frac{171}{192}\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.3.8}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "1938146e",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"\\text{Shorther method: } \\\\ \\\\\n",
"\\frac{4}{9} + \\frac{11}{18} &= \\left( \\frac{4}{9} \\times \\frac{2}{2} \\right) + \\left( \\frac{11}{18} \\right) \\\\ \\\\ \n",
"&= \\frac{8}{18} + \\frac{11}{18} \\\\ \\\\ \n",
"&= \\frac{19}{18}\n",
"\\end{align*}\n",
"\\tag{1.6.3.9}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "a7c6b3f4",
"metadata": {},
"source": [
"We recognized that we do not have to perform the multiplication by rewriting $1$ as a fraction using the _other_ fraction's denominator. Since $18$ was the least common multiple, we only have to consider how to change the $9$ into an $18$ and that was mutiplying by $2$."
]
},
{
"cell_type": "markdown",
"id": "3e93d2a0",
"metadata": {},
"source": [
"__Problem 1.6.3.1__ Find the least common multiple of $12$ and $15$."
]
},
{
"cell_type": "markdown",
"id": "03ef918a",
"metadata": {},
"source": [
"The process of _breaking down_ or factoring an integer can be tedious. For the relatively small integers that we deal with in the life sciences, we can start by dividing repeatedly by $2$, then $3$, then $5$, and so on, using the primes in an iterative process. Remember, that primes are only divisible by $1$ and themselves. We do so for $12$ and $15$ in __Equation 1.6.3.10__"
]
},
{
"cell_type": "markdown",
"id": "872ab028",
"metadata": {},
"source": [
"$$\n",
"\\begin{array}{ccccc}\n",
"12: & 2 & 2 & 3 & \\\\ \n",
"15: & & & 3 & 5 \\\\ \n",
"- & - & - & - & - \\\\ \n",
" & 2 & 2 & 3 & 5\n",
"\\end{array}\n",
"\\tag{Equation 1.6.3.10}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "dca7edfd",
"metadata": {},
"source": [
"We have that $2 \\times 2 \\times 3 \\times 5 = 60$. This is smaller than $12 \\times 15 = 180$. We can use the `lcm` function from the `sympy` package to verify our result."
]
},
{
"cell_type": "code",
"execution_count": 117,
"id": "62672103",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 60$"
],
"text/plain": [
"60"
]
},
"execution_count": 117,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.lcm(12, 15)"
]
},
{
"cell_type": "markdown",
"id": "aff14797",
"metadata": {},
"source": [
"__Homework 1.6.3.1__ Find the least common multiple of $13$ and $17$."
]
},
{
"cell_type": "markdown",
"id": "999a3c21",
"metadata": {},
"source": [
"__Problem 1.6.3.2__ Add the fractions in __Eqiation 1.6.3.11__."
]
},
{
"cell_type": "markdown",
"id": "aea13fe1",
"metadata": {},
"source": [
"$$\n",
"\\frac{5}{12} + \\frac{7}{15}\n",
"\\tag{Equation 1.6.3.11}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "48704e59",
"metadata": {},
"source": [
"We have to _turn_ $12$ into $60$ and $15$ into $60$. This is simple and is shown in __Equation 1.6.3.12__."
]
},
{
"cell_type": "markdown",
"id": "0da244a2",
"metadata": {},
"source": [
"$$\n",
"\\frac{60}{12} = 5, \\; \\; \\frac{60}{15} = 4\n",
"\\tag{Equation 1.6.3.12}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "66f4adfd",
"metadata": {},
"source": [
"We have found the way to rewrite $1$ as fractions for this problem. The solution is shown in __Equation 1.6.3.13__."
]
},
{
"cell_type": "markdown",
"id": "3bd33781",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"\\frac{5}{12} + \\frac{7}{15} &= \\left( \\frac{5}{12} \\times \\frac{5}{5} \\right) + \\left( \\frac{7}{15} \\times \\frac{4}{4} \\right) \\\\ \\\\ \n",
"&= \\frac{25}{60} + \\frac{28}{60} \\\\ \\\\ \n",
"&= \\frac{25 + 28}{60} \\\\ \\\\ \n",
"&= \\frac{53}{60}\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.3.13}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "cb6bfbed",
"metadata": {},
"source": [
"We use `sympy` to confirm the result."
]
},
{
"cell_type": "code",
"execution_count": 118,
"id": "0b1a57fd",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{53}{60}$"
],
"text/plain": [
"53\n",
"──\n",
"60"
]
},
"execution_count": 118,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Rational(5, 12) + sympy.Rational(7, 15)"
]
},
{
"cell_type": "markdown",
"id": "f5ae86cf",
"metadata": {},
"source": [
"__Homework 1.6.3.2__ Add the fractions in __Equation 1.6.3.14__."
]
},
{
"cell_type": "markdown",
"id": "763126af",
"metadata": {},
"source": [
"$$\n",
"\\frac{7}{12} + \\frac{5}{6}\n",
"\\tag{Equation 1.6.3.14}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "f759b6b2",
"metadata": {},
"source": [
"### 1.6.4 Remainders"
]
},
{
"cell_type": "markdown",
"id": "c46afacb",
"metadata": {},
"source": [
"A fraction does not change in value if we divide the numerator and the denominator by the same value. This is shown in __Equation 1.6.4.1__."
]
},
{
"cell_type": "markdown",
"id": "939cc634",
"metadata": {},
"source": [
"$$\n",
"\\frac{p}{q} = \\frac{\\frac{p}{k}}{\\frac{q}{k}}, \\text{ where } p,k \\ne 0\n",
"\\tag{Equation 1.6.4.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "6dd869c1",
"metadata": {},
"source": [
"In __Equation 1.6.4.2__, we note the equality between six-eights and a three-quarters."
]
},
{
"cell_type": "markdown",
"id": "122b67b7",
"metadata": {},
"source": [
"$$\n",
"\\frac{6}{8} = \\frac{\\frac{6}{2}}{\\frac{8}{2}} = \\frac{3}{4}\n",
"\\tag{Equation 1.6.4.2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "8b52a4e7",
"metadata": {},
"source": [
"In the case of six-eigths, we can divide the numberator and the denominator by the same number (and hence not change the value of the fraction). In this case, we can divide both by $2$ to get theee quarters. To do the simplification, we made use of the fact that $2$ divides both $6$ and $8$."
]
},
{
"cell_type": "markdown",
"id": "d419d85a",
"metadata": {},
"source": [
"We could not divide $3$ and $4$ by another integer that divides both $3$ and $4$. By divides in this case, we mean, division without a remainder, or then, a remainder of $0$."
]
},
{
"cell_type": "markdown",
"id": "679ac386",
"metadata": {},
"source": [
"Consider, though, dividing $11$ by $4$. We know that $4$ does not divide $10$. If we divide $11$ by $4$ and still require this division to produce a whole number, we can at best get $2$ with a remainder of $3$. We see this in __Equation 1.6.4.3__."
]
},
{
"cell_type": "markdown",
"id": "13ede2c8",
"metadata": {},
"source": [
"$$\n",
"\\frac{11}{4} = 2 \\text{ with a remainder of } 3 \n",
"\\tag{Equation 1.6.4.3}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "aa1d4264",
"metadata": {},
"source": [
"We have that $4$ divides into $11$ twice, with a remainder of $3$ since $2 \\times 4 = 8$ and $8 + 3 = 12$. We can write this in more egneral terms as shown in __Equation 1.6.4.4__."
]
},
{
"cell_type": "markdown",
"id": "6843eb40",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&11 = 2 \\times 4 + 3 \\\\ \\\\\n",
"&b = c a + r\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.4.4}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "fac266fe",
"metadata": {},
"source": [
"If $r=0$, we say that $a$ divides $b$ and $b=ca$, or then, $b$ is a constant multiple of $a$."
]
},
{
"cell_type": "markdown",
"id": "0609ee99",
"metadata": {},
"source": [
"__Problem 1.6.4.1__ Caclulate the remainder when $13$ is divided by $2$."
]
},
{
"cell_type": "markdown",
"id": "05ef6ba6",
"metadata": {},
"source": [
"We note that $2$ goes into $13$ a total of $6$ times. Since $6 \\times 2 = 12$, we have a remainder of $1$. In __Equation 1.6.4.5__ we see the calculation. In the last line we also see a common notation for the remainder, where we state that $13$ divided by $2$ is $6$ and a half."
]
},
{
"cell_type": "markdown",
"id": "27f8a583",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\frac{13}{2} = 6 \\text{ with a remainder of } 1 \\\\ \\\\\n",
"&\\frac{13}{2} = 6 \\, \\frac{1}{2}\n",
"\\end{align*}\n",
"\\tag{Equation 1.6.4.5}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "58b6d4ab",
"metadata": {},
"source": [
"The `Mod` function in `sympy` returns the remainder and we use it below to verify the result."
]
},
{
"cell_type": "code",
"execution_count": 119,
"id": "ef98c374",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZElEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUACBj0chKHhJQQLN0ZQZAGDGBRBIOACxKC4OQfE94B4NwDm+hiAOyllRAAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle 1$"
],
"text/plain": [
"1"
]
},
"execution_count": 119,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.Mod(13, 2)"
]
},
{
"cell_type": "markdown",
"id": "a779a9c2",
"metadata": {},
"source": [
"__Homework 1.6.4.1__ What is the remainder when we divide $125$ by $7$."
]
},
{
"cell_type": "markdown",
"id": "58405e97-adb2-4226-96db-266f49950c7e",
"metadata": {
"jp-MarkdownHeadingCollapsed": true,
"tags": []
},
"source": [
"## 1.7 Trigonometric functions (optional)"
]
},
{
"cell_type": "markdown",
"id": "688c56f7",
"metadata": {},
"source": [
"In this section, we take a brief look at basic trigonometry. The trigonomtery that we explore here concenrs the right triangled triangle. A right triangle is a triangle with one angle of $90^{\\circ}$."
]
},
{
"cell_type": "markdown",
"id": "15779e9a",
"metadata": {},
"source": [
"__Figure 1.7.1__ shows a right triangle with the angle $\\theta$. The sides are named in relation to the angle $\\theta$."
]
},
{
"cell_type": "code",
"execution_count": 120,
"id": "f5efe4d3",
"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {
"image/png": {
"height": 410,
"width": 515
}
},
"output_type": "display_data"
}
],
"source": [
"# Use matplotlib to plot a triangle\n",
"fig, ax = pyplot.subplots()\n",
"ax.plot([0, 1, 0, 0], [0, 0, 1, 0], 'k')\n",
"ax.set_title('Figure 1.7.1')\n",
"pyplot.xlim(-1, 1.5)\n",
"pyplot.ylim(-0.5, 1.5)\n",
"pyplot.vlines(0, 0, 1, color='black')\n",
"pyplot.gca().spines['top'].set_visible(False)\n",
"pyplot.gca().spines['right'].set_visible(False)\n",
"pyplot.gca().spines['bottom'].set_visible(False)\n",
"pyplot.gca().spines['left'].set_visible(False)\n",
"pyplot.text(0.25, -0.125, 'adjacent', fontsize=12, color='black')\n",
"pyplot.text(-0.5, 0.5, 'opposite', fontsize=12, color='black')\n",
"pyplot.text(0.55, 0.5, 'hypoteneuse', fontsize=12, color='black')\n",
"pyplot.text(0.75, 0.047, r'$\\theta$', fontsize=12, color='black')\n",
"pyplot.xticks([])\n",
"pyplot.yticks([])\n",
"pyplot.show()"
]
},
{
"cell_type": "markdown",
"id": "6bbf694e",
"metadata": {},
"source": [
"The trigonometric functions sine, cosine, tangent, cosine, secant, and cotangent are defined in __Equation 1.7.1__."
]
},
{
"cell_type": "markdown",
"id": "106d5f22",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\text{Sine: } \\sin{\\theta} = \\frac{\\text{opposite}}{\\text{hypotenuse}} \\\\ \\\\\n",
"&\\text{Cosine: } \\cos{\\theta} = \\frac{\\text{adjacent}}{\\text{hypotenuse}} \\\\ \\\\\n",
"&\\text{Tangent: } \\tan{\\theta} = \\frac{\\text{opposite}}{\\text{adjacent}} \\\\ \\\\\n",
"&\\text{Cosecant: } \\csc{\\theta} = \\frac{\\text{hypotenuse}}{\\text{opposite}} \\\\ \\\\\n",
"&\\text{Secant: } \\sec{\\theta} = \\frac{\\text{hypotenuse}}{\\text{adjacent}} \\\\ \\\\\n",
"&\\text{Cotangent: } \\cot{\\theta} = \\frac{\\text{adjacent}}{\\text{opposite}}\n",
"\\end{align*}\n",
"\\tag{Equation 1.7.1}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "27264c1d",
"metadata": {},
"source": [
"It is common to use $\\pi$ when dealing with trigonometric functions. We can express $pi$ as a numerical approximation using the `numpy` package and as an exact value using the `sympy` package."
]
},
{
"cell_type": "code",
"execution_count": 121,
"id": "8c28f6cb-2469-4d89-ae96-eb1a437c7995",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.14159265358979$"
],
"text/plain": [
"3.141592653589793"
]
},
"execution_count": 121,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"numpy.pi"
]
},
{
"cell_type": "code",
"execution_count": 122,
"id": "44f1c93e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAJCAYAAAAGuM1UAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtUlEQVQYGXWQsQ3CMBREE2AAZmCECHpLkI45YARat1BSpqUCsQEpPAElZVpaOkrzTvKPTKR86XT/7t+345QxxkLlvT9COzCXHqmm1EIKK/MAWliBK1gm3cLKPadwBb6Is3OuCyFs0Rf0i35N3yrI7E1fzCRgwarGOyWhmxobiCe5IKjbPplX4eX6f4HgHugdVgtrjPsbOEmP1V+62VCMv8l1v4CpcDf8BLx6bEGDQz6kvw908QNYwThArcyywAAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle \\pi$"
],
"text/plain": [
"π"
]
},
"execution_count": 122,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.pi"
]
},
{
"cell_type": "markdown",
"id": "4ea5bd1b",
"metadata": {},
"source": [
"There are two common measures of any angle $\\theta$. We can use degrees or radians. Degrees span a circle from $0^{\\circ}$ to $360^{\\circ}$. Radians span a circle from $0$ to $2\\pi$. We can convert between degrees and radians using __Equation 1.7.2__."
]
},
{
"cell_type": "markdown",
"id": "b146abce",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"&\\text{Degrees to radians: } \\theta \\times \\frac{\\pi}{180} \\\\ \\\\\n",
"&\\text{Radians to degrees: } \\theta \\times \\frac{180}{\\pi}\n",
"\\end{align*}\n",
"\\tag{Equation 1.7.2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "5a4a8be1",
"metadata": {},
"source": [
"We can also use code to convert between degrees and radians. We use the `radians` function from the `numpy` package to convert degrees to radians. We use the `degrees` function from the `numpy` package to convert radians to degrees."
]
},
{
"cell_type": "code",
"execution_count": 123,
"id": "a3626e15-9f8d-485b-adbf-ebbd6dcf70ef",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 3.14159265358979$"
],
"text/plain": [
"3.141592653589793"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"numpy.radians(180)"
]
},
{
"cell_type": "code",
"execution_count": 124,
"id": "faab1700-c8d9-4844-beca-e22d7169ad0c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle 180.0$"
],
"text/plain": [
"180.0"
]
},
"execution_count": 124,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"numpy.degrees(numpy.pi)"
]
},
{
"cell_type": "markdown",
"id": "c2424563",
"metadata": {},
"source": [
"__Problem 1.7.1__ Calculate the sine of $\\pi$ radians."
]
},
{
"cell_type": "markdown",
"id": "49c84b96",
"metadata": {},
"source": [
"The `sympy` package contains functions for all the trigonometric functions. These functions use angles is radians. We use the `sin` function to calculate the sine of $\\pi$."
]
},
{
"cell_type": "code",
"execution_count": 125,
"id": "561043e3",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFXWS4Q2CMBCFhQkIbqAjqBvgBsoGOkf/GUbQFXQENjAyAhtA2AC/V3tGG2hyeXdfH71LSzKO48KWc64KeYeuiQrWiiVmBLyoL+hDG2iGiO3J2zTAM5qZKbAB1UdX1d6IHolGIFpP6kKnm7EA9JFJpZ8PLdIwy4TnD+U6MQ9IM82tb+s5g/GlTpyazQzWrdOM1lL3Fi9jn3tktyZWsYvaTqzteu7A7YRxA2vU1RtJboAePZiZXG1L4iT2+9ba0E8xEPopdoTe3r/YGx/SQ0OZAIYmAAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 0$"
],
"text/plain": [
"0"
]
},
"execution_count": 125,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Calculate the sime of pi radians\n",
"sympy.sin(sympy.pi)"
]
},
{
"cell_type": "markdown",
"id": "e5f886b0",
"metadata": {},
"source": [
"__Problem 1.7.2__ Calculate the cosine of $\\frac{\\pi}{2}$ radians."
]
},
{
"cell_type": "markdown",
"id": "f3732f40",
"metadata": {},
"source": [
"We use the `cos` function from the `sympy` package to calculate the cosine."
]
},
{
"cell_type": "code",
"execution_count": 126,
"id": "c2de27a1",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFXWS4Q2CMBCFhQkIbqAjqBvgBsoGOkf/GUbQFXQENjAyAhtA2AC/V3tGG2hyeXdfH71LSzKO48KWc64KeYeuiQrWiiVmBLyoL+hDG2iGiO3J2zTAM5qZKbAB1UdX1d6IHolGIFpP6kKnm7EA9JFJpZ8PLdIwy4TnD+U6MQ9IM82tb+s5g/GlTpyazQzWrdOM1lL3Fi9jn3tktyZWsYvaTqzteu7A7YRxA2vU1RtJboAePZiZXG1L4iT2+9ba0E8xEPopdoTe3r/YGx/SQ0OZAIYmAAAAAElFTkSuQmCC",
"text/latex": [
"$\\displaystyle 0$"
],
"text/plain": [
"0"
]
},
"execution_count": 126,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# calculate the cosine of a half pi radians\n",
"sympy.cos(sympy.pi / 2)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "52980b50",
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"id": "b378e350",
"metadata": {},
"source": [
"__Problem 1.7.3__ Calculate the tangent of $\\frac{\\pi}{6}$."
]
},
{
"cell_type": "markdown",
"id": "f45464d6",
"metadata": {},
"source": [
"We use the `tan` function from the `sympy` package to calculate the tangent."
]
},
{
"cell_type": "code",
"execution_count": 128,
"id": "a17e0b8b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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",
"text/latex": [
"$\\displaystyle \\frac{\\sqrt{3}}{3}$"
],
"text/plain": [
"√3\n",
"──\n",
"3 "
]
},
"execution_count": 128,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# calculate the tangent of pi over 6\n",
"sympy.tan(sympy.pi / 6)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.9"
}
},
"nbformat": 4,
"nbformat_minor": 5
}