SM286D · Introduction to Applied Mathematics with Python · Spring 2020 · Uhan
\n", "\n", "Lesson 2.
\n", "\n", "Loops, slicing, and plotting
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## This lesson..." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Loops\n", "- Slicing\n", "- Building lists of numbers\n", "- Plotting with `matplotlib`" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Loops" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- A __for loop__ allows you to perform the same task or set of tasks on every item in a list.\n", "\n", "- For example, suppose we have a list of band member names. We can use a `for` loop to print each name in the list:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Robbie Robertson\n", "Levon Helm\n", "Richard Manuel\n", "Rick Danko\n", "Garth Hudson\n", "And those are the members of The Band!\n" ] } ], "source": [ "# A list containing the names of the members of The Band\n", "the_band = ['Robbie Robertson', 'Levon Helm', 'Richard Manuel', 'Rick Danko', 'Garth Hudson']\n", "\n", "# Print each member's name\n", "for name in the_band:\n", " print(name)\n", " \n", "# How do we write code outside the for loop?\n", "print('And those are the members of The Band!')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Note the __indentation__ in the above code. In particular, the above code is not the same as\n", "\n", "```python\n", "for name in the_band:\n", "print(name)\n", "```\n", "- Always use Tab to indent – this will keep your indentation consistent." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Often we will want to write a `for` loop over consecutive integers. \n", "\n", "- We can do this using the `range()` function, like this:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0\n", "1\n", "2\n", "3\n", "4\n" ] } ], "source": [ "# Print the first five integers, starting at 0\n", "for i in range(5):\n", " print(i)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- `range(n)` starts at `0`, increases by `1` until `n - 1`.\n", "- `range(start, stop)` starts at `start`, increases by `1` until `stop - 1`.\n", "- `range(start, stop, step)` starts at `start`, increases by `step` until `stop - 1` is reached or passed." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__Example.__ Use a `for` loop to print the even numbers starting at 10 and ending at 20." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "10\n", "12\n", "14\n", "16\n", "18\n", "20\n" ] } ], "source": [ "# Write your code here\n", "for i in range(10, 21, 2):\n", " print(i)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- We can use `for` loops to repeatedly do something to a variable.\n", "\n", "- For example, we can add the integers from 1 to 10:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "55\n" ] } ], "source": [ "# Create a variable to stored the total value\n", "total = 0\n", "\n", "# Add 1 to 10 to the total value, one at a time\n", "for i in range(1, 11):\n", " # total += i is the same as total = total + i\n", " total += i\n", "\n", "# Print the total\n", "print(total)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "- Let's take a look at what this code is doing step-by-step. Run the cell below, and a [Python Tutor](http://pythontutor.com) visualization should appear:\n", " - You can use Python Tutor with your own code — just use the link above to go to the Python Tutor website." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n",
"There is a story about Carl Friedrich Gauss, pictured below. This problem was assigned to his class when he was a child; his teacher anticipated being able to relax while the children worked. But Gauss summed the list immediately.
\n", "\n", "Let `L` be the list of numbers 1, 2, ..., 1000. Let `M = sorted(L, reverse=True)`. Add `L` and `M`. What do you notice? Is it easy to add the numbers in `L + M`? Can you do it in your head? How are the sums of `L`, `M` and `L + M` related? Can you see what Gauss did? If so, you should be able to give an algebraic expression for the sum of the numbers from 1 to 1000, without using Python. \n", "\n", "![](\"img/Gauss_banknote.png\")
\n",
"\n",
"C. F. Gauss is featued on the German 10DM bill, alongside the graph of the Gaussian distribution from Statistics, buildings at the University of Göttingen, and (on the reverse side), an outline of the German state of Hannover, which Gauss surveyed. Some of the greatest mathematicians were professors at Göttingen, including Gauss, Riemann and Hilbert.\n",
"
"
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"The sum of the numbers 1, 2, ..., 1000 is 500500\n",
"The sum of the numbers 1, 2, ..., 1000 is 500500\n"
]
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"# Write your code here\n",
"# Using a for loop\n",
"sum_of_numbers = 0\n",
"for i in range(0, 1001):\n",
" sum_of_numbers += i\n",
"print(f'The sum of the numbers 1, 2, ..., 1000 is {sum_of_numbers}')\n",
"\n",
"# Using sum()\n",
"list_of_numbers = list(range(0, 1001))\n",
"sum_of_numbers2 = sum(list_of_numbers)\n",
"print(f'The sum of the numbers 1, 2, ..., 1000 is {sum_of_numbers2}')"
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"__Problem 8.__ The Fibonacci numbers are defined in the following way. The first and second Fibonacci numbers $F_1$ and $F_2$ are both equal to 1. The following Fibonacci numbers are defined according to the rule \n",
"\n",
"\\begin{equation*}\n",
"F_{n} = F_{n-1} + F_{n-2},\n",
"\\end{equation*}\n",
"\n",
"for $n \\geq 3$. Create a list of the first 100 Fibonacci numbers. Print the list.\n",
"\n",
"_Hint._ Read about __negative indexing__ on page 35 of PCC.\n",
"\n",
"\n", "\n", "Let `L` be the list of numbers 1, 2, ..., 1000. Let `M = sorted(L, reverse=True)`. Add `L` and `M`. What do you notice? Is it easy to add the numbers in `L + M`? Can you do it in your head? How are the sums of `L`, `M` and `L + M` related? Can you see what Gauss did? If so, you should be able to give an algebraic expression for the sum of the numbers from 1 to 1000, without using Python. \n", "\n", "
\n",
"\n",
"C. F. Gauss is featued on the German 10DM bill, alongside the graph of the Gaussian distribution from Statistics, buildings at the University of Göttingen, and (on the reverse side), an outline of the German state of Hannover, which Gauss surveyed. Some of the greatest mathematicians were professors at Göttingen, including Gauss, Riemann and Hilbert.\n",
"\n",
"Note. It is a fascinating fact that \n",
"\\begin{equation*}\n",
"F_n = \\left \\lfloor{\\frac{\\phi^n}{\\sqrt{5}} + \\frac{1}{2}} \\right \\rfloor,\n",
"\\end{equation*}\n",
"where \n",
"\\begin{equation*}\n",
"\\phi = \\frac{1+\\sqrt{5}}{2}\n",
"\\end{equation*} \n",
" is the golden ratio. This is a situation where an approximate computation leads to a closed form formula. The Fibonacci numbers are so fascinating that a whole journal, the Fibonacci Quarterly, is devoted to their study. Consecutive Fibonacci numbers arise in many biological applications, including: the arrangements of leaves on a plant stem, the arrangement of pinecone bracts, and the fruit sprouts of a pineapple. See the image below for the appearance in Yellow Chamomile. (source)\n",
"![](\"img/yc.png\")
\n",
"
"
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"[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, 1779979416004714189, 2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738, 19740274219868223167, 31940434634990099905, 51680708854858323072, 83621143489848422977, 135301852344706746049, 218922995834555169026, 354224848179261915075]\n"
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"# Write your code here\n",
"fib = [1, 1]\n",
"for i in range(0, 98):\n",
" fib.append(fib[-1] + fib[-2])\n",
" \n",
"print(fib) "
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