Bloch sphere
This is the big payoff of the chapter, understanding the Bloch sphere! The Bloch sphere, named after Felix Bloch, is a way to visualize a single qubit. From Chapter 1, Superposition with Euclid, we know that a qubit can be represented in the following way:
We did not say this before, but now that we have introduced complex numbers, we can say that α and β are actually complex numbers.
Now we know that complex numbers take two real numbers to represent, it looks as though we will need four real numbers to characterize a qubit state. This is very hard to graph as we cannot visualize 4D space. Let's see whether we can decrease the number of real numbers required to represent a qubit state.
First, let's replace α and β with their exponential form to get:
Now, let's rearrange the right side of the equation by taking out 
and distributing it. Notice that I need to subtract the phase of 
from the second term:
