{
"cells": [
{
"cell_type": "markdown",
"id": "468e25aa-24a4-4eaf-b4e0-c47792f38b1c",
"metadata": {
"tags": []
},
"source": [
"##### Python for High School (Summer 2022)\n",
"\n",
"* [Table of Contents](PY4HS.ipynb)\n",
"* "
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# pow curve\n",
"pd.DataFrame({'x':table.values(),'y':table.keys()}).plot(x='x',y='y',lw=2, grid=True);"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "4c64a485-66d6-4b51-ac6e-95cb97db34fd",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# reversing x with y gives the inverse function\n",
"# log curve\n",
"pd.DataFrame({'x':table.keys(),'y':table.values()}).plot(x='x',y='y',lw=2, grid=True);"
]
},
{
"cell_type": "markdown",
"id": "163232ca-d171-4d8f-9a0e-ba2e37ee3923",
"metadata": {},
"source": [
"Lets play the game of \"hitting the target\" by dialing in an exponenet for base b. To what must I raise b, to hit my target? Say b is 2:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "4150d8c6-7e51-4274-a5fe-3507ad63637d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8.0"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"b = 2 # base (change base)\n",
"target = 256 # raise b to what power to get target? (change target)\n",
"\n",
"answer = log(target, b) # in the old days, maybe look it up in a big table \n",
"answer"
]
},
{
"cell_type": "markdown",
"id": "429a4a1d-4b99-4110-83e3-aeee533c5c8e",
"metadata": {},
"source": [
"**Exercise**\n",
"\n",
"Think of \"hit my target\" as a looping game with an input prompt, and a \"too high\" \"too low\" pair of hints. Once the guesser gets it right to three significant digits, the game ends. Q to quit any time. What would the code look like?\n",
"\n",
"To get you started:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "8c18a2bc-10f4-4c21-91e2-dc62a314de5b",
"metadata": {},
"outputs": [],
"source": [
"from random import randint\n",
"from math import log10, isclose\n",
"\n",
"def game():\n",
" target = randint(0, 100)\n",
" exponent = log10(target) # what to guess\n",
"\n",
" playing = True\n",
" while playing:\n",
" prompt = f\"Raise 10 to what power to get {target}? > \"\n",
" answer = input(prompt)\n",
" if answer.upper() == \"Q\":\n",
" playing = False\n",
" continue\n",
" try:\n",
" answer = float(answer)\n",
" except:\n",
" print(\"Floating point number please\")\n",
" continue\n",
" if isclose(answer, exponent, rel_tol=1e-03):\n",
" print(\"You win! You're a genius!\")\n",
" playing = False\n",
" else:\n",
" print(answer, exponent)\n",
" if answer > exponent:\n",
" print(\"Too high\")\n",
" else:\n",
" print(\"Too low\")\n",
" \n",
" print(\"Thanks for playing\")"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "8f11aefb-e44a-4dc2-9cac-8d51a6884d36",
"metadata": {},
"outputs": [
{
"name": "stdin",
"output_type": "stream",
"text": [
"Raise 10 to what power to get 20? > 1.1\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"1.1 1.3010299956639813\n",
"Too low\n"
]
},
{
"name": "stdin",
"output_type": "stream",
"text": [
"Raise 10 to what power to get 20? > 1.3\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"You win! You're a genius!\n",
"Thanks for playing\n"
]
}
],
"source": [
"game()"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "0490fc5d-f555-47fb-9b41-662b92b43717",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\"1.1\".isdigit()"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "6a2c1a5b-93ae-4678-bb4f-205ed64ae2c8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"isclose(1.342, log10(22), rel_tol=1e-03)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "99a6c7ae-31c7-496d-a230-0404d12b3493",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[0;31mSignature:\u001b[0m \u001b[0misclose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrel_tol\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m1e-09\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mabs_tol\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m0.0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mDocstring:\u001b[0m\n",
"Determine whether two floating point numbers are close in value.\n",
"\n",
" rel_tol\n",
" maximum difference for being considered \"close\", relative to the\n",
" magnitude of the input values\n",
" abs_tol\n",
" maximum difference for being considered \"close\", regardless of the\n",
" magnitude of the input values\n",
"\n",
"Return True if a is close in value to b, and False otherwise.\n",
"\n",
"For the values to be considered close, the difference between them\n",
"must be smaller than at least one of the tolerances.\n",
"\n",
"-inf, inf and NaN behave similarly to the IEEE 754 Standard. That\n",
"is, NaN is not close to anything, even itself. inf and -inf are\n",
"only close to themselves.\n",
"\u001b[0;31mType:\u001b[0m builtin_function_or_method\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"?isclose"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "e98833fd-fc80-4dcc-a72a-1c8dd6d9ca57",
"metadata": {},
"outputs": [
{
"name": "stdin",
"output_type": "stream",
"text": [
"Raise 10 to what power to get 22? > 1.1234\n",
"Raise 10 to what power to get 22? > q\n"
]
}
],
"source": [
"game()"
]
},
{
"cell_type": "markdown",
"id": "5137f1b5-8982-4ec7-acc9-27e066f672f2",
"metadata": {},
"source": [
"Why not a long irrational-looking floating point value for an answer, in the case of $\\log_{2}(256)$? Because 256 is a whole number power of 2. Numbers that are not whole number powers get these \"many significant digit\" answers."
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "1b75b918-8507-44f6-ac81-9bd2d6392c46",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"256.0"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"checking = pow(b, answer) # show that we hit the target, up above\n",
"checking"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "6fbdb0f8-4416-45ea-9410-40a5ad28bc8a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.266786540694902"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"answer = log(77, 2) # not a whole number power of 2\n",
"answer"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "c5310d3f-a41a-48aa-87e4-d6426c3da337",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"77.00000000000003"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"checking = pow(2, answer)\n",
"checking"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "033ab0a2-7b4a-4df7-9519-e2ebdf857911",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4.0"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"b = 10\n",
"target = 10000\n",
"log(target, b)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "98607540-e16e-4c01-8601-10005caafdd8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.8864907251724818"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(77, 10) "
]
},
{
"cell_type": "markdown",
"id": "214c5de8-20a0-49da-81c4-96314f74e6fb",
"metadata": {},
"source": [
"The log of a huge target number need not itself be be huge, as a rule of thumb, because raising a base to a power is a fast way to shrink and grow i.e. exponentially.\n",
"\n",
"With X in the role of exponenet, it covers a huge breadth of verticality, from insect height to mountain height.\n",
"\n",
"The idea is to fit our desired range, from very tiny to very large, within the domain of a reasonably-wide X, say from -10 to 10."
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "9467e133-11a0-49ad-8990-2105f650b58f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"16.172963232704753"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log10(14892349939984940) # log base 10 of a large number"
]
},
{
"cell_type": "markdown",
"id": "843d5503-4dc0-4d93-9e67-b55670e22005",
"metadata": {},
"source": [
"**What we're also exercising and talking about in this Notebook:**\n",
"\n",
"* your ability to embed standard math notations in Jupyter Markdown cells, by means of $\\LaTeX$, a way of encoding math typography, and ultimately, entire books using $\\TeX$.\n",
"\n",
"* your ability to plot a function, as you might on a graphing calculator, starting with data, using `pandas` and `matplotlib`, both 3rd party Python packages, which in turn depend on `numpy`."
]
},
{
"cell_type": "markdown",
"id": "f7f7425a-7d59-420c-b3cc-0ec249bc466a",
"metadata": {},
"source": [
"### Getting the Log of N\n",
"\n",
"The the idea of getting the logarithm of a number is the inverse of raising a number to a power.\n",
"\n",
"Ask yourself: \"what number would I have to raise a number **b** to, to get another number **N**?\" That first number, **b** is your base, and with that you want to find the \"log\" (to that base) of your target number **N**.\n",
"\n",
"$$\n",
"\\log_{b}(N)\n",
"$$\n",
"\n",
"For example, I would have to raise 10 to the 2nd power to get 100. That means the base 10 log of 100 is 2.\n",
"\n",
"$$\n",
"\\log_{10}(100) = 2\n",
"$$\n",
"\n",
"To what power would I need to raise the number 2 to get 100? That's the same as asking \"what is the base 2 log of 10?\""
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "9d895b90-e65e-4ac0-b53f-0ce38526066e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"32"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pow(2, 5) ## too low"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "d9b9165a-71cb-46cf-9bc5-8c7989002456",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"128"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pow(2, 7) ## too high, but closer"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "fee59090-4c81-4c6a-9470-cd90b3a9838b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"103.96830673359815"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pow(2, 6.7) # very close"
]
},
{
"cell_type": "markdown",
"id": "abe27d9b-1208-414e-be80-cb16be53f94b",
"metadata": {},
"source": [
"Lets use Python to find a more precise answer:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "aa6beb9e-e8b4-414b-9b26-5a18ea03df77",
"metadata": {},
"outputs": [],
"source": [
"from math import log"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "e7372d0a-f6cd-45b9-afca-2b9e2f2c959e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[0;31mDocstring:\u001b[0m\n",
"log(x, [base=math.e])\n",
"Return the logarithm of x to the given base.\n",
"\n",
"If the base not specified, returns the natural logarithm (base e) of x.\n",
"\u001b[0;31mType:\u001b[0m builtin_function_or_method\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"? log"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "db0b4f78-34d2-4a05-9511-336e77237fd6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.643856189774725"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(100, 2)"
]
},
{
"cell_type": "markdown",
"id": "29784f37-89eb-46d2-8fcf-e33c8f7a1b92",
"metadata": {},
"source": [
"That's it!\n",
"\n",
"$$\n",
"\\log_{2}(100) \\approx 6.643856189774725\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "dc983574-27fd-4219-92c3-d0e44d269de9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"100.00000000000004"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ** 6.643856189774725 # same as pow(2, 6.643856189774725)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "209aa855-2099-4109-8693-3f78d95c499f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.0"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(100, 10)"
]
},
{
"cell_type": "markdown",
"id": "8c1d9d6c-451a-465d-87ea-14f53a353884",
"metadata": {},
"source": [
"Back in the days before electronic calculators, engineers used slide rules. You could multiply two large numbers by adding their logarithms.\n",
"\n",
"\n",
"\n",
"
\n",
" \n",
" \n",
" Domain \n",
" Log 10 \n",
" \n",
" \n",
" \n",
" \n",
" 195 \n",
" 979.901508 \n",
" 2.991182 \n",
" \n",
" \n",
" 196 \n",
" 984.926131 \n",
" 2.993404 \n",
" \n",
" \n",
" 197 \n",
" 989.950754 \n",
" 2.995614 \n",
" \n",
" \n",
" 198 \n",
" 994.975377 \n",
" 2.997812 \n",
" \n",
" \n",
" 199 \n",
" 1000.000000 \n",
" 3.000000 \n",
" \n",
" \n",
"
\n",
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"df1.plot(x=\"Domain\", y=\"Log 10\", grid=True, lw=4);"
]
},
{
"cell_type": "code",
"execution_count": 42,
"id": "287e351e-57ec-4e2a-97d2-53b2d0d92053",
"metadata": {},
"outputs": [],
"source": [
"pow_range = np.power(10, log10_range)"
]
},
{
"cell_type": "code",
"execution_count": 43,
"id": "1189da17-5e72-4e38-b6bd-97ba9443f714",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" Domain \n",
" 10 ** x \n",
" \n",
" \n",
" \n",
" \n",
" 0 \n",
" -1.000000 \n",
" 0.100000 \n",
" \n",
" \n",
" 1 \n",
" 0.709662 \n",
" 5.124623 \n",
" \n",
" \n",
" 2 \n",
" 1.006434 \n",
" 10.149246 \n",
" \n",
" \n",
" 3 \n",
" 1.181096 \n",
" 15.173869 \n",
" \n",
" \n",
" 4 \n",
" 1.305319 \n",
" 20.198492 \n",
" \n",
" \n",
"
\n",
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"df2.plot(x=\"Domain\", y=\"10 ** x\", grid=True, lw=4);"
]
},
{
"cell_type": "markdown",
"id": "57806b93-3f71-4579-8ece-f1190bc0764e",
"metadata": {},
"source": [
"Getting a little fancier, we can put two plots into one figure, by means of subplots. \n",
"\n",
"To do this, we need to import another library that pandas has been using all along. Now we would like to gain more control of the nitty gritty details, so we import it and talk to it directly. About our subplots."
]
},
{
"cell_type": "code",
"execution_count": 45,
"id": "4bf77f49-bb40-4962-9513-4b0057e5b502",
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"import matplotlib.pyplot as plt\n",
"\n",
"# makes two subplots, each with its own axis object in axes\n",
"fig, axes = plt.subplots(nrows=2, ncols=1, sharey=False, sharex=False)\n",
"\n",
"axes[0].set_ylim(-2, 4) # set the vertical limits on the first axis object\n",
"\n",
"# df1, df2 are pandas DataFrames, which will plot on their own, but\n",
"# here we direct them to use the axis objects associated with each \n",
"# subplot\n",
"\n",
"df1.plot(x=\"Domain\", y=\"Log 10\", ax=axes[0], lw=4)\n",
"df2.plot(x=\"Domain\", y=\"10 ** x\", ax=axes[1], lw=4);"
]
},
{
"cell_type": "markdown",
"id": "101c2f8a-8edf-411a-bb76-4c30f1e9e4cd",
"metadata": {},
"source": [
"### Natural Logarithms\n",
"\n",
"We're not done with the story of logarithms. Logarithms of many bases have suggested themselves. We might even use an irrational base."
]
},
{
"cell_type": "code",
"execution_count": 46,
"id": "0d8a9e4d-053b-4619-b2a1-c9a6f1b2a4bc",
"metadata": {},
"outputs": [],
"source": [
"from math import log\n",
"import math"
]
},
{
"cell_type": "code",
"execution_count": 47,
"id": "7df53500-6556-4705-873a-9341e01405b0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.0"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(math.pi ** 3, math.pi)"
]
},
{
"cell_type": "markdown",
"id": "491630bf-1efe-464e-ad65-942d127e8c67",
"metadata": {},
"source": [
"Suprisingly, there is a \"best base\" or at least one that stands out as the most natural. The number has many interesting properties. One of them is, the exponential curve for this number has a y coordinate equal to its slope.\n",
"\n",
"$$\n",
"e^{x} \\approx \\frac{e^{x+h} - e^{x}}{h}\n",
"$$\n",
"\n",
"High school topics include some calculus. Here is one of those times, as we're interested in the slope of a curve at a specific point. The slope of a curve is \"rise over run\" meaning how much it changes vertically versus how much it (the curve) changes horizontally. We set our horizontal change to a tiny h (h for horizontal) and see how much wiggling x, wiggles y.\n",
"\n",
"What's the change in the y coordinate, per a tiny (to infinitessimal) change in x?\n",
"\n",
"As h goes to 0, the result gets more precise:\n",
"\n",
"$$\n",
"\\lim_{h \\to 0} \\frac{e^{x+h} - e^{x}}{h} = e^{x}\n",
"$$\n",
"\n",
"From another point of view, the skill in focus here, possibly getting some exercise, is maybe not calculus so much, as that of typesetting. How does one get standard math notations to appear in a Notebook short of taking screen shots? \n",
"\n",
"Here is where $\\LaTeX$ fits in, as a puzzle piece and as a workout opportunity that pays off in the long run. \n",
"\n",
"**Exercise**\n",
"\n",
"On your local device, pop open a Markdown cell and practice entering some mathy expressions, you're more than welcome to use any of those here. In Markdown, we use single or double dollar signs to signal \"LaTeX Within\". \n",
"\n",
"* Single dollar signs when embedded in a sentence\n",
"* Double dollar when centering math expressions in their own \"div boxes\" (HTML-speak)."
]
},
{
"cell_type": "markdown",
"id": "0ab74568-699d-45cb-a9c9-f4acef10fcd3",
"metadata": {},
"source": [
"### How Steep?\n",
"\n",
"We frequently encounter the phenomenon of \"gradient\" in life, meaning an uphill or downhill path, most simply. \n",
"\n",
"Steps count, as providing gradients and they're manageably steep we hope, except to babies, others with different abilities.\n",
"\n",
"We may speak of a \"learning curve\" again using steps as a metaphor. When the steps get too steep, we feel frustrated by the inaccessibility of this path, unless we're rock climbers of some kind.\n",
"\n",
"Thinking back to our work with graphs, one might add \"steepness\" as a characteristic of an edge. In electric circuits, edges stand for wires and may come with voltage drops associated with resistence and loads.\n",
"\n",
"A roller coaster is a great example of a curving track of varying slope. In riding a roller coaster, we experience the changing acceleration, perhaps as exhileration.\n",
"\n",
"When looking at a curve, we see varying steepness, which we define as vertical change per horizontal change. When a curve is rising steeply to my right, that means if I step right just a little, the y-position over my head is skyrocketing. It may double in altitude with every step to my right. \n",
"\n",
"Real projectiles would not behave in this way, doubling in speed indefinitely. A mathematical curve is the construct of our imaginations.\n",
"\n",
"The solution Newton and others came up with, for measuring the slope of a curve at any given point, was to wiggle the X-axis position a tiny bit, by adding some h (a step to the right), and seeing how that changed Y. \n",
"\n",
"The ratio of the change in y -- a function of x, so f(x) -- to the change in h, could be expressed as:\n",
"\n",
"$$\n",
"\\frac{f(x+h) - f(x)}{h}\n",
"$$\n",
"\n",
"where x is allowed to vary, giving us changing slopes (potentially) throughout the curve. x is where we make our measurement of \"steepness\" or \"slope\". A curve may slope up (positive) or slope down (negative). A curve with 0 slope at a point, is momentarily tangent to a perfectly horizontal straight line."
]
},
{
"cell_type": "code",
"execution_count": 48,
"id": "4a56ce3b-a72e-43c4-8415-d27264ee087c",
"metadata": {},
"outputs": [],
"source": [
"def slope_at(x, h=1e-8):\n",
" \"\"\"\n",
" Our measuring technique: change x (horizontal) a very tiny bit \n",
" and ratio that with the resulting change in y (vertical)\n",
" \"\"\"\n",
" return (math.exp(x+h) - math.exp(x))/h"
]
},
{
"cell_type": "code",
"execution_count": 49,
"id": "bcaf95d2-b4b7-40b9-ab02-e3caeb148788",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"54.59815000108392"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"slope_at(4) # pretty steep already"
]
},
{
"cell_type": "markdown",
"id": "a917f00f-cae6-4423-9f8a-bd7064d7ea9c",
"metadata": {},
"source": [
"Looking at the curve below, we see by x = 4 we have already left the ground and are climbing steeply past 50."
]
},
{
"cell_type": "code",
"execution_count": 50,
"id": "88263afd-b2d0-4f08-8e51-bca5d77d76e9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" Domain \n",
" e ** x \n",
" \n",
" \n",
" \n",
" \n",
" 0 \n",
" 0.000000 \n",
" 1.000000 \n",
" \n",
" \n",
" 1 \n",
" 0.060606 \n",
" 1.062480 \n",
" \n",
" \n",
" 2 \n",
" 0.121212 \n",
" 1.128864 \n",
" \n",
" \n",
" 3 \n",
" 0.181818 \n",
" 1.199396 \n",
" \n",
" \n",
" 4 \n",
" 0.242424 \n",
" 1.274335 \n",
" \n",
" \n",
"
\n",
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"df3.plot(x=\"Domain\", y=\"e ** x\", grid=True, lw=4);"
]
},
{
"cell_type": "markdown",
"id": "cfff9a72-8af8-4a8a-b197-28e858838eb0",
"metadata": {},
"source": [
"Now let's zoom out by computing over a wider domain. Note that the y-axis is in units of many thousands, don't let those single digits fool you."
]
},
{
"cell_type": "code",
"execution_count": 52,
"id": "551a0193-2b54-405e-bc5d-45c4d2271b45",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
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\n",
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\n",
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\n",
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" 6 \n",
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" 3.065840e-02 \n",
" \n",
" \n",
" 35 \n",
" 3.838384 \n",
" 4.645034e+01 \n",
" \n",
" \n",
" 83 \n",
" 15.959596 \n",
" 8.534232e+06 \n",
" \n",
" \n",
" 86 \n",
" 16.717172 \n",
" 1.820436e+07 \n",
" \n",
" \n",
" 70 \n",
" 12.676768 \n",
" 3.202213e+05 \n",
" \n",
" \n",
" 20 \n",
" 0.050505 \n",
" 1.051802e+00 \n",
" \n",
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" \n",
" \n",
" 19 \n",
" -0.202020 \n",
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" \n",
" \n",
" 69 \n",
" 12.424242 \n",
" 2.487596e+05 \n",
" \n",
" \n",
" 84 \n",
" 16.212121 \n",
" 1.098588e+07 \n",
" \n",
" \n",
"
\n",
"
"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"df4.plot(x=\"Domain\", y=\"e ** x\", grid=True, lw=4);"
]
},
{
"cell_type": "markdown",
"id": "77d2ab7e-5588-4d56-ae4f-dbe05aa842e9",
"metadata": {},
"source": [
"In exercising with the cells below, changing arguments, you will find that the slope as computed by our measuring technique, with tiny h, is very close to the y-axis value of the curve at that same point. The curve is registering as own slope as it goes, starting from extremely flat (slope almost 0) and transitioning to extremely vertical."
]
},
{
"cell_type": "code",
"execution_count": 55,
"id": "c0ad05ae-03fd-454b-a263-24596814ddc2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.018315638936061696"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"slope = slope_at(-4) # close to 0 slope\n",
"slope"
]
},
{
"cell_type": "code",
"execution_count": 56,
"id": "40c9ecbf-e3ae-46c4-ba7f-fb92991b13fb",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.018315638888734186"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pow(math.e, -4) # almost the same (off because h > 0)"
]
},
{
"cell_type": "code",
"execution_count": 57,
"id": "c37f17e1-cb65-499d-bc4e-8821396d5ac8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.206031351510319e-08"
]
},
"execution_count": 57,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"math.exp(4) - slope_at(4) # almost the same"
]
},
{
"cell_type": "code",
"execution_count": 58,
"id": "4bd4ab6e-33b6-4368-a0b2-f0e91f7e6eb6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.7182818218562943"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"slope_at(1) # reveals our base (approximately)"
]
},
{
"cell_type": "code",
"execution_count": 59,
"id": "d46476b2-b4b3-495d-ab64-9730b5f8ff5f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.718281828459045"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"math.exp(1)"
]
},
{
"cell_type": "markdown",
"id": "6dc30184-03af-4168-b9ef-2e1f92d4bef3",
"metadata": {},
"source": [
"That's the number. We call it `e`."
]
},
{
"cell_type": "code",
"execution_count": 60,
"id": "b527c239-0f14-47ed-8bb4-1be236f20508",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.718281828459045"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"math.e"
]
},
{
"cell_type": "markdown",
"id": "d9dfed77-10d3-463e-91b3-c90723f55b5e",
"metadata": {},
"source": [
"A log to this base has it's own symbol, LN for Natural Log, although more often it's written in lowercase. $\\ln(x)$\n",
"\n",
"In Python, the `log` function we've been using all along, uses base e as its default."
]
},
{
"cell_type": "code",
"execution_count": 61,
"id": "7ce4adac-3a55-4d44-87c5-5643094cddc7",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.907755278982137"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(1000)"
]
},
{
"cell_type": "code",
"execution_count": 62,
"id": "c4b09aaa-105c-4a7e-bf0e-fd3fad07a526",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6.907755278982137"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log(1000, math.e)"
]
},
{
"cell_type": "markdown",
"id": "651e7ceb-2b94-46bf-98c4-dbd8e783e0b4",
"metadata": {},
"source": [
"### Exploring e with Sympy\n",
"\n",
"If you have sympy installed, then you're equipped to explore the wonderful world of a computer-assisted algebra system (CAS). Sympy will help you check your concepts, including in calculus."
]
},
{
"cell_type": "code",
"execution_count": 63,
"id": "60196e4d-7f99-4cc9-b36d-905e06388cfd",
"metadata": {},
"outputs": [],
"source": [
"import sympy as sym"
]
},
{
"cell_type": "code",
"execution_count": 64,
"id": "cf5067ca-f7a8-4ed6-9362-b70559ec398f",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e$"
],
"text/plain": [
"E"
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sym.exp(1)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"id": "a0897af7-9751-4d95-8473-b2700cbbd55f",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n"
]
}
],
"source": [
"print(sym.exp(1).evalf(100))"
]
},
{
"cell_type": "markdown",
"id": "4a127998-7600-4714-8023-08a6ddbfe25c",
"metadata": {},
"source": [
"Here we're differentiating, meaning asking for a function that will give the rate at which $e^{x}$ is changing (the \"slope\") at any x. \n",
"\n",
"As we have seen, this function is $e^{x}$ itself."
]
},
{
"cell_type": "code",
"execution_count": 66,
"id": "148b576a-9ab3-4487-a323-9e8bb2925616",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e^{x}$"
],
"text/plain": [
"exp(x)"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = sym.Symbol('x')\n",
"sym.diff(sym.exp(x))"
]
},
{
"cell_type": "markdown",
"id": "f3116399-e5bf-4a9b-ba9d-5ab4c44bc790",
"metadata": {},
"source": [
"The way we write that in calculus is:\n",
"\n",
"$$\n",
"\\dfrac{d e^{x}}{dx} = e^{x} \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "5f2b6e3d-48f2-48c2-9d4e-1a4b32d8d248",
"metadata": {},
"source": [
"Here's a means for approaching e to whatever degree of precision you wish."
]
},
{
"cell_type": "code",
"execution_count": 67,
"id": "008254d5-e164-463e-9649-1f6a5fbd02da",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left(1 + \\frac{1}{n}\\right)^{n}$"
],
"text/plain": [
"(1 + 1/n)**n"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"n = sym.Symbol('n')\n",
"expr = (1 + 1/n)**n\n",
"expr"
]
},
{
"cell_type": "markdown",
"id": "5d4fbe24-8a01-46d1-8ea8-babd07fa990d",
"metadata": {},
"source": [
"What happens as n grows larger and larger? \n",
"\n",
"The exponent tends to make the number bigger, since it's greater than 1. But 1/n gets smaller, meaning the number being raised to the nth power is getting ever closer to exactly 1.\n",
"\n",
"The tension between these two tendencies, to blow up because of exponentiation vs to settle on 1 because of the vanishing difference from 1, actually resolves to a specific number, the number e.\n",
"\n",
"$$\n",
"\\lim_{n \\to \\infty} (1 + 1/n)^{n} = e\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "7521cb4b-5ff6-4df1-8d8a-2c443cc5e11e",
"metadata": {},
"source": [
"Sympy lets us compute that limit directly, for a symbolic answer."
]
},
{
"cell_type": "code",
"execution_count": 68,
"id": "51390251-05f5-4f24-ba75-5b23f48961e0",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e$"
],
"text/plain": [
"E"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sym.limit(expr, n, sym.oo)"
]
},
{
"cell_type": "markdown",
"id": "747b1736-02d2-4126-ae89-f8710546103a",
"metadata": {},
"source": [
"How about the area under the $e^{x}$ curve? The operation of \"accumulating\" (integration) is the opposite of \"slicing thinly\" (differentiation)."
]
},
{
"cell_type": "code",
"execution_count": 69,
"id": "5f313a4d-6810-4eba-be0b-7db9a0575962",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e^{x}$"
],
"text/plain": [
"exp(x)"
]
},
"execution_count": 69,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sym.integrate(sym.exp(x), x)"
]
},
{
"cell_type": "code",
"execution_count": 70,
"id": "b1d3000e-bd7d-4a92-96f0-31e49028a523",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle e$"
],
"text/plain": [
"E"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sym.integrate(sym.exp(x), (x, -sym.oo, 1))"
]
},
{
"cell_type": "code",
"execution_count": 71,
"id": "c330f191-5039-4105-9eb2-9bcc29aaf5f3",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 15.1542622414793$"
],
"text/plain": [
"15.1542622414793"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sym.integrate(sym.exp(x), (x, -sym.oo, math.e))"
]
},
{
"cell_type": "code",
"execution_count": 72,
"id": "118dd919-9bde-4e96-8b05-2e5f956be4a2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"15.154262241479262"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"math.exp(math.e)"
]
},
{
"cell_type": "markdown",
"id": "643d01fb-e711-4b4b-be17-f9cf4e5be57f",
"metadata": {},
"source": [
"We get to read off the area under the curve by reading the y-coodinate, just as we're able to read the slope the same way. The slope at exp(x) matches the area under the curve up to that point x.\n",
"\n",
"So we get a sense that e is indeed a \"special number\" in the Mathematics Hall of Fame."
]
},
{
"cell_type": "markdown",
"id": "e62f98a8-5db1-4b36-ae6f-7456b33761a2",
"metadata": {},
"source": [
"### Segue to Probability Studies\n",
"\n",
"Data Science. Statistics. Casino Math... \n",
"\n",
"The field goes by many names and nicknames, and has to do with likelihood and expectations, and whether a sample size is \"good enough\" for predicting the characteristics of a larger population.\n",
"\n",
"At this juncture, showing $e$ working in tandem with $\\pi$ to give us a Gaussian, along with the typesetting, reinforces the $e$ deserves its Hall of Fame status."
]
},
{
"cell_type": "code",
"execution_count": 73,
"id": "3af7699e-08bd-47fc-8bae-246b2c94ce27",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{\\sqrt{2} e^{- \\frac{x^{2}}{2}}}{2 \\sqrt{\\pi}}$"
],
"text/plain": [
"sqrt(2)*exp(-x**2/2)/(2*sqrt(pi))"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x, σ, μ, f = sym.symbols(['x','σ', 'μ', 'f'])\n",
"expr1 = (1/(σ * sym.sqrt(2 * sym.pi)))\n",
"expr2 = sym.exp(sym.Rational(-1,2) * ((x - μ)/σ)**2)\n",
"gaussian = expr2 * expr1\n",
"standard = gaussian.copy()\n",
"standard = standard.subs(μ, 0)\n",
"standard = standard.subs(σ, 1)\n",
"standard"
]
},
{
"cell_type": "code",
"execution_count": 74,
"id": "57c69e48-6acb-4f77-8b3f-18f9714097aa",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"domain = np.linspace(-6, 6, 500)\n",
"f = sym.lambdify(x, standard, 'numpy') \n",
"bell_curve = f(domain)\n",
"df5 = pd.DataFrame({'x':domain, 'y':bell_curve})\n",
"ax = df5.plot(x='x', y='y', lw=4)\n",
"ax.set_title(\"Probability Density Function\");\n",
"ax.legend(['PDF']);"
]
},
{
"cell_type": "markdown",
"id": "5d02dd9f-3a1b-432f-939d-4ee66ecf3bf7",
"metadata": {},
"source": [
"### YouTube Gallery\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "0fa101ef-23ba-4c2e-9c0c-33dd16b7a6b4",
"metadata": {},
"outputs": [],
"source": [
"from IPython.display import YouTubeVideo"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "1ebf0992-bb7c-4a5c-a735-e7bf44920a51",
"metadata": {},
"outputs": [
{
"data": {
"image/jpeg": 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\n",
"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"YouTubeVideo(\"VRzH4xB0GdM\")"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "d7f92503-692a-4953-948c-155c94c4c35a",
"metadata": {},
"outputs": [
{
"data": {
"image/jpeg": 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\n",
"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"YouTubeVideo(\"WZLQoP6CM0k\") "
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "a2d9f164-4cbc-4e01-b203-073ac0f69b6b",
"metadata": {},
"outputs": [
{
"data": {
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"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"YouTubeVideo(\"1qbg7orb1lc\") # Bill Nye"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "d0a9d574-c69b-4d58-b517-8ce6bc8dae38",
"metadata": {},
"outputs": [
{
"data": {
"image/jpeg": 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"text/html": [
"\n",
" \n",
" "
],
"text/plain": [
""
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"YouTubeVideo(\"UaGifkgfmWk\") # logs, powers, atomic physics"
]
}
],
"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": "3.9.12"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\" "\\"Open"){kind=icon}
![\"Vintage](\"https://live.staticflickr.com/838/41646298500_2a359d5fcc.jpg\")
\n",
"\n",
"[Attribution](https://flic.kr/p/26s9aBj)\n",
"\n",
"For example, suppose I wanted to multiply 185 by 987. The slide rule showed numbers as powers of some base."
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "adb604b6-8851-4376-aa87-b978009fc623",
"metadata": {},
"outputs": [],
"source": [
"import math\n",
"from math import log10 # Python as a special function for base 10"
]
},
{
"cell_type": "markdown",
"id": "34e5917c-c944-4b72-ae95-a9014a97ffe8",
"metadata": {},
"source": [
"What power of 10 gives me 185? \n",
"\n",
"`10**2` means 10 times 10 in Python and equals 100. \n",
"\n",
"`10**3` means 10 times 10 times 10 and equals 1000. \n",
"\n",
"However we're allowed to have fractional exponents, so an exponent of 10 just a tad more than 2 should do it."
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "5918b935-c11c-4abc-8b87-e3c282b63b51",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.2671717284030137"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log10(185)"
]
},
{
"cell_type": "markdown",
"id": "f7b788e6-1b1c-4f54-92c7-f527f5773af5",
"metadata": {},
"source": [
"Likewise 987 is close to 1000, so an exponent of almost 3 is not that surprising. The log10 function figures out a precise answer."
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "9f6e259a-f2b5-467e-bb2d-79af9e25f83d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.9943171526696366"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log10(987)"
]
},
{
"cell_type": "markdown",
"id": "281585d2-2c45-4e89-b293-09a8e927f483",
"metadata": {},
"source": [
"Now we add these two exponents..."
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "b8dc5ac9-9398-4c20-a757-7fedd251369e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.26148888107265"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log10(185) + log10(987)"
]
},
{
"cell_type": "markdown",
"id": "06c34371-7127-4f2d-abb2-2a4a82c4aa35",
"metadata": {},
"source": [
"... and now raise 10 to that power, to get our final answer.\n",
"\n",
"On a slide rule, we could read these values (to fewer decimal places) off the sliding sticks. \n",
"\n",
"We had the results of a multiplication, but thanks to logs, we got there by addition. Someone good with a slide rule could get precise enough answers quite quickly."
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "cdedf774-e229-4879-a4f1-77e3556e6169",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"182594.9999999999"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"log_sum = log10(185) + log10(987)\n",
"pow(10, log_sum)"
]
},
{
"cell_type": "code",
"execution_count": 33,
"id": "fb1eb2b7-2d61-4c74-a558-9ba7b68842d3",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"182595"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"185 * 987 # checking"
]
},
{
"cell_type": "markdown",
"id": "6777e475-2598-4025-b226-26ce4575d1c4",
"metadata": {},
"source": [
"In general: `a * b == pow(10, log10(a) + log10(b))`\n",
"\n",
"or, using our friend $\\LaTeX$, the typesetting language understood by Jupyter:\n",
"\n",
"$$\n",
"(A)(B) = b^{\\log_b(A) + \\log_{b}(B)}\n",
"$$ \n",
"\n",
"What this shows is a way of turning a multiplication problem into an addition problem. Adding is simpler than multiplying, if working by hand. When logs were first invented, this was their most valued application: to provide a short cut to multiplication. Lookup tables played a role. You could look up log to the base 10 of A and B, add them, and then look up 10 to the answer power, to get A times B. You have reached your product by adding and consulting tables.\n",
"\n",
"With floating point numbers, as with lookup tables, precise equality with the desired outcome is not assured. The answer for A times B will only have so many significant digits. \n",
"\n",
"The math module has an `isclose` we can use, which means \"almost equal\". This function comes in handy when working with floating point numbers, whereas asking for precise equality can be dangerous, as one may never get it, even when one would, were these numbers entirely precise."
]
},
{
"cell_type": "code",
"execution_count": 34,
"id": "26f115a2-373c-4718-8f3d-a2fbc54b9896",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[0;31mSignature:\u001b[0m \u001b[0mmath\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0misclose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrel_tol\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m1e-09\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mabs_tol\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m0.0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mDocstring:\u001b[0m\n",
"Determine whether two floating point numbers are close in value.\n",
"\n",
" rel_tol\n",
" maximum difference for being considered \"close\", relative to the\n",
" magnitude of the input values\n",
" abs_tol\n",
" maximum difference for being considered \"close\", regardless of the\n",
" magnitude of the input values\n",
"\n",
"Return True if a is close in value to b, and False otherwise.\n",
"\n",
"For the values to be considered close, the difference between them\n",
"must be smaller than at least one of the tolerances.\n",
"\n",
"-inf, inf and NaN behave similarly to the IEEE 754 Standard. That\n",
"is, NaN is not close to anything, even itself. inf and -inf are\n",
"only close to themselves.\n",
"\u001b[0;31mType:\u001b[0m builtin_function_or_method\n"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"? math.isclose"
]
},
{
"cell_type": "code",
"execution_count": 35,
"id": "5386c945-bba5-4fc5-94ba-c24cee6a019a",
"metadata": {},
"outputs": [],
"source": [
"def test_log(a, b):\n",
" return math.isclose(a * b, pow(10, log10(a) + log10(b)))"
]
},
{
"cell_type": "code",
"execution_count": 36,
"id": "fba13bb2-e3bf-4178-a467-24bf28a18fc1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"test_log(185, 987)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"id": "ceda64ad-ddca-4968-b7bc-dbd520515b76",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"test_log(17, 8947)"
]
},
{
"cell_type": "markdown",
"id": "ba58d6e8-9fba-4714-b976-999a5859aecb",
"metadata": {},
"source": [
"Before electronic calculators, the \"computers\" (people who computed) would use giant lookup tables, same for trig functions.\n",
"\n",
"\n",
"\n",
"[Attribution](https://commons.wikimedia.org/wiki/File:Abramowitz%26Stegun.page97.agr.jpg)"
]
},
{
"cell_type": "markdown",
"id": "ee7a8afb-ff60-4d36-9248-20d2bd4da05a",
"metadata": {},
"source": [
"### Plotting Logarithmic and Exponential Functions\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 38,
"id": "429c9e90-bb55-4708-8ec9-212a0ddf9d87",
"metadata": {},
"outputs": [],
"source": [
"import pandas as pd\n",
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": 39,
"id": "440e0de6-9031-4332-8287-e73dab6e4914",
"metadata": {},
"outputs": [],
"source": [
"domain = np.linspace(0.1, 1000, 200) # give me 200 points between .1 and a thousand\n",
"log10_range = np.log10(domain)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"id": "95e0fdc2-9be7-4bd4-9202-69c7a9760c84",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"