#!/usr/bin/env python
# coding: utf-8

# # Studying Mathematics with Python
# 
# ## Preface
# 
# This handbook is an introduction to doing Mathematics with Open Source Software. 
# 
# 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:
# 
# 1. [Sympy](01 - Symbolic mathematics with Sympy.ipynb) for carrying out symbolic mathematical calculations;
# 2. [Numpy](02 - Linear algebra with Numpy.ipynb) for dealing with large numeric arrays: perfect for linear algebra;
# 3. [Pandas](03 - Data analysis with Pandas.ipynb) a brilliant tool for data analysis.
# 
# **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.
# 
# **Note** that there is a choice between python 3 and python 2. **Python 3 is recommended and assumed in this handbook.**
# 
# ## Interacting with Python
# 
# Once you have installed Anaconda, you will now have Python on your machine. You can interact with Python in multiple ways:
# 
# ### The Python shell
# 
# - On Windows open a "Command Prompt":
# 
#   ![Image of a command prompt Window](./img/01-00_command_prompt_on_windows.png)
#   
# - On Mac OS or Linux open a "Terminal":
# 
#   ![Image of a terminal Window](./img/01-01_terminal_on_linux.png)
#   
# This is a simple utility that allows you to give commands to your computer. In there type:
# 
# ```bash
# python
# ```
# 
# This should then look something like:
# 
# ```python
# Python 3.5.2 |Anaconda 4.2.0 (64-bit)| (default, Jul  2 2016, 17:53:06) 
# [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)] on linux
# Type "help", "copyright", "credits" or "license" for more information.
# >>> 
# ```
# 
# The `>>>` is a prompt for you to type a command. Let us carry out a straightforward addition:
# 
# 
# ```python
# >>> 2 + 2
# ```
# 
# and press `ENTER`:
# 
# 
# ```python
# >>> 2 + 2
# 4
# ```
# 
# **This is not a very efficient and practical way of using Python.** We will instead learn to use a [Jupyter notebook](http://jupyter.org/).
# 
# ### Jupyter notebooks
# 
# If you are still in the Python shell (with a `>>>` waiting for you to give a command) then type:
# 
# ```python
# >>> exit()
# ```
# 
# This will leave the prompt and you will now be back at the Command Prompt or the Terminal window. Here simply type:
# 
# ```bash
# jupyter notebook
# ```
# 
# 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.
# 
# In your browser window you should have a tab that looks like the following:
# 
# ![The Jupyter browser window](img/01-02_jupyter_home.png)
# 
# 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 ...`.
# 
# 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:
# 
# - Click on "play" symbol;
# - Press `SHIFT + ENTER`.
# 
# #### Writing code in a Jupyter cell
# 
# Here is an example:
# 
# ![A simple Python command in a Jupyter cell](img/01-03_command_in_jupyter_cell.png)
# 
# ---
# 
# **EXERCISE**: Create a cell and run the following:
# 
# ```python
# 2 + 2
# ```
# ---
# 
# #### Writing text/mathematics in a Jupyter notebook
# 
# 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:
# 
# 1. Markdown
# 2. Code
# 3. Raw NBConvert (not covered in this handbook)
# 4. Heading (not covering in this handbook)
# 
# ---
# 
# **EXERCISE**: Create a **markdown** cell and type the following:
# 
# ```markdown
# Here is a simple list of topics we will cover in this handbook:
# 
# - Symbolic mathematics
# - Linear algebra
# - Basic data analysis
# 
# Note that I can also use LaTeX:
# 
# $$a x ^ 2 + b x + c$$
# ```
# 
# ---

# We have gone through an overview of using Jupyter, we will now move on to basic Python syntax.
# 
# ## Basic Python
# 
# Python is a fully fledged professional programming language. Used across a wide number of sectors:
# 
# - Commercial products like Dropbox, Instagram and others;
# - Research areas like machine learning, astrophysics and others.
# 
# 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/
# 
# 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.
# 
# ### Variables
# 
# We can assign values to variables for future use:

# In[1]:


# Anything after a `#` will be ignored. This is what we call a 'comment'
a = 20  # Assigning a value to a
b = 21  # Assigning a value to b
a * b  # Calculating the product of a and b


# In[2]:


a / (a + b)  # Calculating a / (a + b)


# In[3]:


a ** b  # Calculating a raised to the power of b


# ---
# **EXERCISE** For $a=15$ and $b=16$, use Python to find the values of the following:
# 
# - $a\times b$
# - $a + b$
# - $a - b$
# - $\frac{a}{a + b}$
# - $a ^ b$
# 
# ---
# 
# ### Functions
# 
# Similar to variables we can create functions that can be easily reused.

# In[4]:


def proportion(a, b):
    """
    We can use triple " to describe what our function does.
    
    Here for example: we're creating a function to calculate 
    the proportion of a of a + b
    """
    return a / (a + b)


# We can read our description by typing:

# In[5]:


get_ipython().run_line_magic('pinfo', 'proportion')


# We can **use** our function by passing it variables:

# In[6]:


proportion(20, 21)


# In[7]:


proportion(3, 1)


# ---
# 
# **EXERCISE** Create a function exponent that raises $a$ to the power of $a + b$ and use it calculate:
# 
# - $3 ^ {(3 + 2)}$
# - $2 ^ 6$
# 
# ---

# ### Repeating things with for loops
# 
# We can use a `for` loop to repeat bits of code with Python. For example the following will calculate:
# 
# $$\sum_{i=0}^9i$$

# In[8]:


total = 0
for i in range(10):
    total = total + i
total


# There are a couple of things happening there:
# 
# 1. We're creating a variable `total` and assining it to have value `0`.
# 2. We are using `range(10)` which is a Python function to get a generator of numbers from `0` (inclusive) to `10` (exclusive).
# 3. The `for loop` simply repeats all the block of code that is indented.
# 4. The code that is repeated is `total = total + i` so we are incrementally adding `i` to `total`.
# 
# We know that:
# 
# $$\sum_{i=0}^Ni=\frac{N (N + 1)}{2}$$
# 
# Let us verify this in our given case:

# In[9]:


N = 9
expected_sum = N * (N + 1) / 2
total == expected_sum


# The double `==` is used to check if two values are equal and creates what is called a `Boolean` variable.
# 
# ---
# 
# **EXERCISE** For $N\in\{10, 20, 352\}$ verify the following identity:
# 
# $$
# \sum_{i=0}^Ni^2=\frac{N(N + 1)(2N + 1)}{6}
# $$
# 
# 
# Bonus points if you use a function (or two!) :)
# 
# ---

# ## Summary
# 
# This section has discussed:
# 
# - Using Python via the shell and Jupyter;
# - Writing code and markdown in Jupyter;
# - Some basic features of writing Python code:
#     - Variables;
#     - Functions;
#     - For loops.
#     
# There are a number of other things that can be done in Python but this will do for now.
# 
# Let us move on to using [Sympy](01 - Symbolic mathematics with Sympy.ipynb) for carrying out symbolic mathematical calculations.