nbs/00-Studying-Mathematics-with-Python.ipynb

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    "# Studying Mathematics with Python\n",
    "\n",
    "## Preface\n",
    "\n",
    "This handbook is an introduction to doing Mathematics with Open Source Software. \n",
    "\n",
    "One alternative to matlab, mathematica and maple is to use [sagemath](http://www.sagemath.org/). Sagemath is a fantastic collection of resources build on top of the open source programming language [Python](https://www.python.org/). However it is quite a large piece of software that can be difficult to install on Windows. There is a cloud based system [cloud.sagemath.com](https://cloud.sagemath.com/) but this requires a connection to the internet. Instead this notebook will describe the use of 3 lightweight Python libraries:\n",
    "\n",
    "1. [Sympy](01 - Symbolic mathematics with Sympy.ipynb) for carrying out symbolic mathematical calculations;\n",
    "2. [Numpy](02 - Linear algebra with Numpy.ipynb) for dealing with large numeric arrays: perfect for linear algebra;\n",
    "3. [Pandas](03 - Data analysis with Pandas.ipynb) a brilliant tool for data analysis.\n",
    "\n",
    "**Python and the above libraries are all free to use (and potentially change if you needed to!).** The recommended distribution of Python that includes a number of libraries for scientific work is Anaconda. If you have not yet installed Python on your machine go to the following link: https://www.continuum.io/downloads and follow the download instructions.\n",
    "\n",
    "**Note** that there is a choice between python 3 and python 2. **Python 3 is recommended and assumed in this handbook.**\n",
    "\n",
    "## Interacting with Python\n",
    "\n",
    "Once you have installed Anaconda, you will now have Python on your machine. You can interact with Python in multiple ways:\n",
    "\n",
    "### The Python shell\n",
    "\n",
    "- On Windows open a \"Command Prompt\":\n",
    "\n",
    "  ![Image of a command prompt Window](./img/01-00_command_prompt_on_windows.png)\n",
    "  \n",
    "- On Mac OS or Linux open a \"Terminal\":\n",
    "\n",
    "  ![Image of a terminal Window](./img/01-01_terminal_on_linux.png)\n",
    "  \n",
    "This is a simple utility that allows you to give commands to your computer. In there type:\n",
    "\n",
    "```bash\n",
    "python\n",
    "```\n",
    "\n",
    "This should then look something like:\n",
    "\n",
    "```python\n",
    "Python 3.5.2 |Anaconda 4.2.0 (64-bit)| (default, Jul  2 2016, 17:53:06) \n",
    "[GCC 4.4.7 20120313 (Red Hat 4.4.7-1)] on linux\n",
    "Type \"help\", \"copyright\", \"credits\" or \"license\" for more information.\n",
    ">>> \n",
    "```\n",
    "\n",
    "The `>>>` is a prompt for you to type a command. Let us carry out a straightforward addition:\n",
    "\n",
    "\n",
    "```python\n",
    ">>> 2 + 2\n",
    "```\n",
    "\n",
    "and press `ENTER`:\n",
    "\n",
    "\n",
    "```python\n",
    ">>> 2 + 2\n",
    "4\n",
    "```\n",
    "\n",
    "**This is not a very efficient and practical way of using Python.** We will instead learn to use a [Jupyter notebook](http://jupyter.org/).\n",
    "\n",
    "### Jupyter notebooks\n",
    "\n",
    "If you are still in the Python shell (with a `>>>` waiting for you to give a command) then type:\n",
    "\n",
    "```python\n",
    ">>> exit()\n",
    "```\n",
    "\n",
    "This will leave the prompt and you will now be back at the Command Prompt or the Terminal window. Here simply type:\n",
    "\n",
    "```bash\n",
    "jupyter notebook\n",
    "```\n",
    "\n",
    "After a little moment this will open a Jupyter notebook page in a browser. Note that this is all just running on your computer and you do not need to be connected to the internet to use a Jupyter notebook.\n",
    "\n",
    "In your browser window you should have a tab that looks like the following:\n",
    "\n",
    "![The Jupyter browser window](img/01-02_jupyter_home.png)\n",
    "\n",
    "You can use that to navigate your directories/folders. When you are in a location that you want to use, **create a new notebook** by clicking on `New` (on the top right) and `Python ...`.\n",
    "\n",
    "These notebook are very versatil and powerful. You can write blocks of code in each cell and see the output. To execute code in a cell:\n",
    "\n",
    "- Click on \"play\" symbol;\n",
    "- Press `SHIFT + ENTER`.\n",
    "\n",
    "#### Writing code in a Jupyter cell\n",
    "\n",
    "Here is an example:\n",
    "\n",
    "![A simple Python command in a Jupyter cell](img/01-03_command_in_jupyter_cell.png)\n",
    "\n",
    "---\n",
    "\n",
    "**EXERCISE**: Create a cell and run the following:\n",
    "\n",
    "```python\n",
    "2 + 2\n",
    "```\n",
    "---\n",
    "\n",
    "#### Writing text/mathematics in a Jupyter notebook\n",
    "\n",
    "It is also possible to write text with a markup language called [markdown](https://daringfireball.net/projects/markdown/basics) which can also include LaTeX. To do this you need to change the mode of a cell. There is a dropdown menu that lets you choose between:\n",
    "\n",
    "1. Markdown\n",
    "2. Code\n",
    "3. Raw NBConvert (not covered in this handbook)\n",
    "4. Heading (not covering in this handbook)\n",
    "\n",
    "---\n",
    "\n",
    "**EXERCISE**: Create a **markdown** cell and type the following:\n",
    "\n",
    "```markdown\n",
    "Here is a simple list of topics we will cover in this handbook:\n",
    "\n",
    "- Symbolic mathematics\n",
    "- Linear algebra\n",
    "- Basic data analysis\n",
    "\n",
    "Note that I can also use LaTeX:\n",
    "\n",
    "$$a x ^ 2 + b x + c$$\n",
    "```\n",
    "\n",
    "---"
   ]
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    "We have gone through an overview of using Jupyter, we will now move on to basic Python syntax.\n",
    "\n",
    "## Basic Python\n",
    "\n",
    "Python is a fully fledged professional programming language. Used across a wide number of sectors:\n",
    "\n",
    "- Commercial products like Dropbox, Instagram and others;\n",
    "- Research areas like machine learning, astrophysics and others.\n",
    "\n",
    "It is not the intention of this handbook to give a strong grounding in Python as a programming language. There are a number of great resources for this which you can find here http://docs.python-guide.org/en/latest/intro/learning/\n",
    "\n",
    "We will go over some of the very basics here which if you have never used Python before will be enough to get you started.\n",
    "\n",
    "### Variables\n",
    "\n",
    "We can assign values to variables for future use:"
   ]
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    "# Anything after a `#` will be ignored. This is what we call a 'comment'\n",
    "a = 20  # Assigning a value to a\n",
    "b = 21  # Assigning a value to b\n",
    "a * b  # Calculating the product of a and b"
   ]
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    "a / (a + b)  # Calculating a / (a + b)"
   ]
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   "cell_type": "code",
   "execution_count": 10,
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       "2097152000000000000000000000"
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    "a ** b  # Calculating a raised to the power of b"
   ]
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   "source": [
    "---\n",
    "**EXERCISE** For $a=15$ and $b=16$, use Python to find the values of the following:\n",
    "\n",
    "- $a\\times b$\n",
    "- $a + b$\n",
    "- $a - b$\n",
    "- $\\frac{a}{a + b}$\n",
    "- $a ^ b$\n",
    "\n",
    "---\n",
    "\n",
    "### Functions\n",
    "\n",
    "Similar to variables we can create functions that can be easily reused."
   ]
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    "def proportion(a, b):\n",
    "    \"\"\"\n",
    "    We can use triple \" to describe what our function does.\n",
    "    \n",
    "    Here for example: we're creating a function to calculate \n",
    "    the proportion of a of a + b\n",
    "    \"\"\"\n",
    "    return a / (a + b)"
   ]
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   "source": [
    "We can read our description by typing:"
   ]
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    "proportion?"
   ]
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   "source": [
    "We can **use** our function by passing it variables:"
   ]
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    "proportion(20, 21)"
   ]
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       "0.75"
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    "proportion(3, 1)"
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   "source": [
    "---\n",
    "\n",
    "**EXERCISE** Create a function exponent that raises $a$ to the power of $a + b$ and use it calculate:\n",
    "\n",
    "- $3 ^ {(3 + 2)}$\n",
    "- $2 ^ 6$\n",
    "\n",
    "---"
   ]
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   "source": [
    "### Repeating things with for loops\n",
    "\n",
    "We can use a `for` loop to repeat bits of code with Python. For example the following will calculate:\n",
    "\n",
    "$$\\sum_{i=0}^9i$$"
   ]
  },
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    "total = 0\n",
    "for i in range(10):\n",
    "    total = total + i\n",
    "total"
   ]
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    "There are a couple of things happening there:\n",
    "\n",
    "1. We're creating a variable `total` and assining it to have value `0`.\n",
    "2. We are using `range(10)` which is a Python function to get a generator of numbers from `0` (inclusive) to `10` (exclusive).\n",
    "3. The `for loop` simply repeats all the block of code that is indented.\n",
    "4. The code that is repeated is `total = total + i` so we are incrementally adding `i` to `total`.\n",
    "\n",
    "We know that:\n",
    "\n",
    "$$\\sum_{i=0}^Ni=\\frac{N (N + 1)}{2}$$\n",
    "\n",
    "Let us verify this in our given case:"
   ]
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       "True"
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   "source": [
    "N = 9\n",
    "expected_sum = N * (N + 1) / 2\n",
    "total == expected_sum"
   ]
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   "source": [
    "The double `==` is used to check if two values are equal and creates what is called a `Boolean` variable.\n",
    "\n",
    "---\n",
    "\n",
    "**EXERCISE** For $N\\in\\{10, 20, 352\\}$ verify the following identity:\n",
    "\n",
    "$$\n",
    "\\sum_{i=0}^Ni^2=\\frac{N(N + 1)(2N + 1)}{6}\n",
    "$$\n",
    "\n",
    "\n",
    "Bonus points if you use a function (or two!) :)\n",
    "\n",
    "---"
   ]
  },
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   "source": [
    "## Summary\n",
    "\n",
    "This section has discussed:\n",
    "\n",
    "- Using Python via the shell and Jupyter;\n",
    "- Writing code and markdown in Jupyter;\n",
    "- Some basic features of writing Python code:\n",
    "    - Variables;\n",
    "    - Functions;\n",
    "    - For loops.\n",
    "    \n",
    "There are a number of other things that can be done in Python but this will do for now.\n",
    "\n",
    "Let us move on to using [Sympy](01 - Symbolic mathematics with Sympy.ipynb) for carrying out symbolic mathematical calculations."
   ]
  }
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