Additional Programming Concepts in Python.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Additional Programming Concepts in Python\n",
    "\n",
    "In this notebook, you will learn about additional programming concepts in Python. They are not part of the learning objectives of the course, but you may run into them at some point, or wonder what they are, or find them fun and useful if you already have some programming experience. \n",
    "\n",
    "*(Much of this material we wrote up in an earlier version of the notebooks, but then moved here when we tweaked the course to fit in the time we have available.)*\n",
    "\n",
    "## Tuples\n",
    "\n",
    "### What is a tuple?\n",
    "\n",
    "The first more complicated data structure we will discuss is a `tuple`. A tuple is a collection of values inside curved brackets. Here is the basic syntax for making a tuple: \n",
    "\n",
    "```\n",
    "my_tuple = (a, b, c, ...etc...)\n",
    "```\n",
    "\n",
    "As a concrete example, this will create a tuple of three integers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(5, 6, 7)\n"
     ]
    }
   ],
   "source": [
    "tup1 = (5,6,7)\n",
    "print(tup1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Like the other data types we've see, we can see the tuples we create using `%whos`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variable   Type     Data/Info\n",
      "-----------------------------\n",
      "tup1       tuple    n=3\n"
     ]
    }
   ],
   "source": [
    "%whos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tuples are like lists, but behave a bit differently than python lists. In fact, we've already seen tuples before in the previous notebook when we were looking at `for` loops!\n",
    "\n",
    "If you are given a tuple, you can check how long it by using the `len()` function built into python:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "len(tup1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that tuples do not have to contain integers, they can contain any type of data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# A tuple of strings\n",
    "str_tup = ('foo', 'bar')\n",
    "print(str_tup)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Different than how numpy arrays are typically used, tuples can even be mixed, with each element of a different type:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mixed_tup = (1, 1.05, 7+3j, 'foo')\n",
    "print(mixed_tup)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And you can even have tuples of tuples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "tup_of_tup = (str_tup, mixed_tup)\n",
    "print(tup_of_tup)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tuples support all the same indexing and slicing as arrays. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tuples can also contain other tuples! If your tuple contains another tuple, like the example `tup_of_tup`, you can use the square brackets a second time to index into the tuple you extract by the first set of square brackets:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(tup_of_tup)\n",
    "print(tup_of_tup[0])\n",
    "print(tup_of_tup[0][0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Looping over tuples without using indices\n",
    "\n",
    "As mentioned briefly in Notebook 2, python `for` loops can also directly iterate over \"iteratable\" objects. \n",
    "\n",
    "The `tuple` (along with lists, which we will see in a bit, and numpy arrays, which we will see in the next notebook), is one such iteratable objecte. \n",
    "\n",
    "For example, to print all of entries of a `tuple` out in order, we can use directly following:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for n in tup1:\n",
    "    print(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "During each subsequent iteartion of the loop, the variable `n` will be assigned to the next item that is stored in the tuple. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lists\n",
    "\n",
    "In this section, we will introduce a very commonly used data structure in python: the `list`. \n",
    "\n",
    "A list is a list of values, like a `tuple`, but that is made using square brackets:\n",
    "\n",
    "```\n",
    "my_list = [a, b, c, ...etc...]\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "l1 = list(range(10))\n",
    "l1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Like tuples, you can extract single elements of the list using indexing, and extract portions of the list using slicing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(l1[0])\n",
    "print(l1[0:5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, so far so good. But if I have `tuple`, why would I ever want a list?\n",
    "\n",
    "### Lists vs tupples: Tupples are \"immutable\", lists are \"mutable\"\n",
    "\n",
    "This is a bit of python-speak for saying that you cannot change the values of a tupple, but you can change the values of a list.\n",
    "\n",
    "What does this mean? It means if I have a list `[5,6,7]` and I want to change the last number in my list to an 8, I can just directly do this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "l1 = [5,6,7]\n",
    "l1[2] = 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If I try this with a tuple, I will find that I can't do it!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "'tuple' object does not support item assignment",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-3-9c8e47fcf882>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[0mt1\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;36m5\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m6\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m7\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mt1\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m8\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m: 'tuple' object does not support item assignment"
     ]
    }
   ],
   "source": [
    "t1 = (5,6,7)\n",
    "t1[2] = 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because of this functionality, **lists are much more powerful as we can change them once we've made them!**\n",
    "\n",
    "In addition to changing lists by individual indexing, we can also change whole parts of the list using slicing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "l2 = list(range(10))\n",
    "print(l2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Replace three entries by zeros\n",
    "l2 = list(range(10))\n",
    "l2[4:7] = [0,0,0]\n",
    "print(l2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Remove entries from a list by replacing them with an empty list []\n",
    "l2 = list(range(10))\n",
    "l2[4:7] = [] \n",
    "print(l2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions for manipulating lists\n",
    "\n",
    "In fact, our list object itself has functions built in that allow you to change it! Some examples of things we can do:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This will add an element to the end of the list\n",
    "l1.append(10)\n",
    "l1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This will remove an element from the end of the list\n",
    "l1.pop()\n",
    "l1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are many more functions built into lists, some of which you can find here:\n",
    "\n",
    "https://docs.python.org/3/tutorial/datastructures.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The problem with lists for scientific computing\n",
    "\n",
    "Lists look an awful lot like numpy arrays: why don't we just use lists? \n",
    "\n",
    "In scientific computing, it is very common to want to perform numerical operations on <a href=https://en.wikipedia.org/wiki/Row_and_column_vectors>vectors and matrices</a> of numbers. And also, many times in experiments, the data you will take will be represented by a vector of numbers: for example, the position of a particle as a function of time.\n",
    "\n",
    "A vector is a collection of numbers in a one-dimentional array:\n",
    "\n",
    "$$\n",
    "x = [1, 2, 3, 4, 5]\n",
    "$$\n",
    "\n",
    "In Notebook 3, we already introduced python `list`s. A list is also a vector, right? It certainly looks the same! Why do we need something new? \n",
    "\n",
    "The reason we need something new is that python `list`s are not designed to work in the same way as we expect vectors to from our mathematics classes. For example, in math:\n",
    "\n",
    "$$\n",
    "2x = [2,4,6,8,10]\n",
    "$$\n",
    "\n",
    "Let's check if this works with lists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2, 3, 4, 5, 1, 2, 3, 4, 5]\n"
     ]
    }
   ],
   "source": [
    "l = [1, 2, 3, 4, 5]\n",
    "print(2*l)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This certainly does something, but it does not do what we want! It has made the list twice as long by appending two of them together!\n",
    "\n",
    "Also addition and subtraction doesn't work like we would expect:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(l+l)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Addition makes the list twice as long? And:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "print(l-l)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And subtraction doesn't work at all...clearly, although they look a lot like vectors, in terms of mathematics, lists do not act much like vectors. This is one of the reasons numpy arrays were created."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dictionaries\n",
    "\n",
    "Another useful data type in python is a \"dictionary\":\n",
    "\n",
    "https://docs.python.org/3/tutorial/datastructures.html#dictionaries\n",
    "\n",
    "At a basic level, a dictionary is a bit like a list that supports non-numeric indices. Dictionaries can be created using the curly brackets in python `{` and `}`. \n",
    "\n",
    "Here we will create an empty dictionary and start filling it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "delft_lunch_rating = {}\n",
    "delft_lunch_rating['greek olive'] = 10\n",
    "delft_lunch_rating['brandmeester'] = 7\n",
    "delft_lunch_rating['aula'] = \"expensive\"\n",
    "delft_lunch_rating['citg'] = \"bad\"\n",
    "delft_lunch_rating['doner'] = \"good but a bit salty\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is what our dictionary looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'greek olive': 10, 'brandmeester': 7, 'aula': 'expensive', 'citg': 'bad', 'doner': 'good but a bit salty'}\n"
     ]
    }
   ],
   "source": [
    "print(delft_lunch_rating)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I can then look up values in my dictionary using the \"keys\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'expensive'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delft_lunch_rating['aula']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are also functions for getting all the keys:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "delft_lunch_rating.keys()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "My dictionaries can also hold lists if I want:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "delft_lunch_rating[\"greek olive\"] = ['good', 10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dictionaries are actually a way to implement a basic database in python (I use them in my grading scripts to look up the email addresses and student numbers of a given netid...)\n",
    "\n",
    "And the Jupyter notebook files actually store a list cells, and each cell consist of a dictionary that contains the text of the cell (and other fun things like metadata). You can see this in action using the <a href=https://nbformat.readthedocs.io/en/latest/api.html>nbformat</a> library, you can actually load a notebook file into your python kernel and poke around it to see what it looks like. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Strings\n",
    "\n",
    "In notebook 1, we already saw strings as variable types. \n",
    "\n",
    "It turns out that strings are not just simple (immutable) variables like `int`s and `float`s: `str`s are actually data structures that are indexable (like `tuple`s and `list`s). \n",
    "\n",
    "Strings are immutable, which means they cannot be changed. But they do have lots of built-in functions that can return a new string (or lots of other things!). \n",
    "\n",
    "Let's look at an example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s1 = \"This is a string\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Indexing a string returns the characters of the string:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(s1[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also slice:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(s1[0:6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Strings do not allow you to directly change "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Built-in string functions\n",
    "\n",
    "Strings have a bunch of useful built-in functions: \n",
    "\n",
    "https://docs.python.org/3/library/stdtypes.html#string-methods\n",
    "\n",
    "some of which we will look at here:\n",
    "\n",
    "### Splitting a string\n",
    "\n",
    "Strings have a built-in function `split()`. By default, it will return a list of \"words\" by using whitespaces as the separator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s1.split()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Passing `.` as an argument, `split()` will use that as a separator, which is useful for working with filenames:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "input_file = 'myfile.dat'\n",
    "output_file = input_file.split('.')[0]\n",
    "output_file += \"_processed.dat\"\n",
    "print(output_file)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Replacing parts of strings\n",
    "\n",
    "The function `replace()` replace substrings in your string for you:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "s2 = \"This is a long sentence. It is a good idea to end it.\"\n",
    "print(s2)\n",
    "print(s2.replace(\" is\", \" was\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that without the space, it will also replace the \"is\" in \"This\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(s2)\n",
    "print(s2.replace(\"is\", \"was\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Testing for the contents of a string\n",
    "\n",
    "You can check if a substring is found inside a string using the `in` logical operator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "if \"this\" in \"somewhere in this sentence\":\n",
    "    print(\"We found a 'this'\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# It is case sensitive:\n",
    "if \"this\" in \"This is a sentence\":\n",
    "    print(\"This will not get printed\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# But you can use the .lower() function of a string to do case insensitive checks\n",
    "s3 = \"This is a sentence\"\n",
    "if \"this\" in s3.lower():\n",
    "    print(\"Using .lower(), s3 is converted to all lower-case:\\n\")\n",
    "    print(\"s3.lower = '%s'\\n\" % s3.lower())\n",
    "    print(\"And now we do find the substring 'this'\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we can also see an example of special characters in strings: a `\\n` in a string specifies a \"new line\":\n",
    "\n",
    "https://docs.python.org/3/reference/lexical_analysis.html#strings\n",
    "\n",
    "Note that if you want to print a backslash `\\`, you need to put a double backslash in your string: `\\\\`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### String formatting\n",
    "\n",
    "Until now, we have been printing values of our variables using the standard `str()` conversion of numbers to strings:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 11/300\n",
    "print(\"The value of a is\", a)\n",
    "\n",
    "# The above print() statement is equivalent to:\n",
    "\n",
    "output = \"The value of a is\"\n",
    "output += \" \"\n",
    "output += str(a)\n",
    "print(output)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But maybe we don't want so many digits in our text output. Say I want only two digits. How can I do this? \n",
    "\n",
    "For this, python supports \"string formatting\". My personal preference is to work with traditional \"printf-style\" formatting, inherited from the C programming language:\n",
    "\n",
    "https://docs.python.org/3/library/stdtypes.html#printf-style-bytes-formatting\n",
    "\n",
    "It sounds a bit scary at first, but it's actually pretty easy to use. It works by using a special operator `%` that works with strings. \n",
    "\n",
    "To use it, you include a special text in your string that starts with `%`, followed by a sequence of numbers and letters that you use to tell python how you want the string to be formatted. \n",
    "\n",
    "Here are some examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Floating point format with 4 digits\n",
    "print(\"The value of a is %.4f\" % a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Exponential notation with two digits\n",
    "print(\"The value of a is %.2e\" % a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# A string with the formatting not at the end\n",
    "print(\"The value of a is %.2e seconds\" % a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Some additional matrix creation routines\n",
    "\n",
    "There are several functions for making matrices which you may find useful someday: \n",
    "\n",
    "https://docs.scipy.org/doc/numpy/reference/routines.array-creation.html\n",
    "\n",
    "including this one which I use often:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The identity matrix\n",
    "print(np.eye(10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "# A band diagonal matrix\n",
    "print(np.eye(10,k=-1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mutable objects and \"call by reference\"\n",
    "\n",
    "### The `=` operator\n",
    "\n",
    "Now that we have introduced some more advanced data types, it is time to go back and revisit one of our first topics: the `=` opeartor.\n",
    "\n",
    "At the start of the first notebook, we introduced the **assignment operator** `=`, and saw that it could be used to give new values to a variable based on the value of another variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 5\n",
    "b = a\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happens if we change the value of `a` after the statment `b = a`? For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 5\n",
    "b = a\n",
    "a = 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What value does `b` have? Does it have the value of `5` that `a` had when we performed the assignment operation, or does it have `6` (the new values of `a`)?  \n",
    "\n",
    "The obvious answer would be that `b` should have the answer `5`, right? Let's check it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5\n"
     ]
    }
   ],
   "source": [
    "a = 5\n",
    "b = a\n",
    "a = 6\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK, that seems to make sense. \n",
    "\n",
    "Now let's take a look at and examples with lists. We will create a list `a`, using the assignment operator to make a list variable `b = a`, and then change the values in the list `a`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = [2,1]\n",
    "b = a\n",
    "a[0] = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now question: did `a[0] = 1` change the value of of `b`? Let's check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",