More Math
=========

There are lots of ways we can combine numbers or modify them.

.. doctest::

    >>> count = 5.0
    >>> count * 3
    15.0
    >>> count / 3
    1.6666666666666667
    >>> count % 3 # remainder
    2.0
    >>> count // 3 # quotient
    1.0
    >>> count * 3 + 2
    17.0
    >>> count * (3 + 2) # parenthesis group
    25.0
    >>> count / count
    1.0

Exponents
---------

Exponents are a way of saying "what would happen if I multipled a number X by itself Y times?"
So, for instance, what would happen if I multiplied 5 by itself 2 times?

.. doctest::

    >>> count = 5
    >>> count * count
    25
    >>> count ** 2
    25
    >>> count * count * count
    125
    >>> count ** 3
    125

this shows up in lots of computation, particularly when we're trying to 
figure out how fast a particular operation will be.

Exponents also allow us to answer the question "what number multiplied by itself Y times would give X?"
We do that by using a fractional exponent instead of a whole number exponent.

... doctest::

    >>> count = 25
    >>> count ** (1/2)
    5.0
    >>> root = count ** (1/2)
    >>> root ** 2
    25.0

.. note::

    You'll see count ** (1/2) a lot in your high school math classes, it's
    also something you'll see in geometric programming such as graphics.
    For instance it lets us figure out how big a square we need to show a given number 
    of pixels. It is normally called a "square root" and is written √25. 
    Cube root `**(1/3)` is less commonly used.

Exercise
--------

* Play with numbers, see if you can find an expression that crashes the interpreter

.. note::
    
    Hint: is there a number that you can't divide by?
    
Extra Exercise
------------------

.. note::

    While this is a fun brain teaser, you won't use this in this tutorial,
    bit-wise operators are less commonly used in high-level languages such 
    as Python, but are very common in lower level languages such as C, Go, or Rust.

* There are more operators we haven't seen here.
* See if you can figure out what these operators do.

.. doctest::

    >>> bin(1)
    0b1
    >>> bin(2)
    0b10
    >>> 2 >> 1 # called right-shift
    1
    >>> bin(4)
    0b100
    >>> 2 << 1 # called left-shift
    4
    >>> bin(8)
    0b1000
    >>> 2 << 2 
    8
    >>> 1 & 2 # called bitwise-and
    0
    >>> 1 & 3
    1
    >>> 1 ^ 3 # called bitwise-xor
    2
    >>> ~1 # called bitwise-not
    -2
    >>> 1|2 # bitwise or
    3