Mostly we will need to cover the following four topics to start learning quantum mechanics.

1. Complex Numbers
2. Vector Spaces
3. Matrices and Determinants
4. Probability Theory 101
5. Problems and Worksheets

Numbers

We all have an intuitive understanding of counting. We count by making a one-to-one pair of the given set of objects with an abstract set which we call “whole numbers.” These are commonly represented as the numbers $0,1,2,3,\dots$ and collectively denoted by $\mathbb{N}$.

Given the set of whole numbers we can naturally define the operation of addition. This operation has the following properties. For $a, b, c \in \mathbb{N}$, we have the following

1. If $a+b = c$, then $c \in \mathbb{N}$. This is known as closure.
2. $a+b = b+a$. This is known as commutativity property.
3. $a + (b+c) = (a+b) + c$. This is known as associative property.
4. The number $0$ is special in this operation. Since, $a+0=a$ for any $a$. We say the number $0$ is the identity element with respect to addition.

Once we know how to add numbers, a natural question is that can we ‘undo’ the operation of addition? If we go ahead and define an operation which is opposite or inverse of addition, then we soon run into trouble. The trouble occurs simply because our list of numbers $\mathbb{N}$ has a starting point, viz. $0$. So we cannot undo the following addition $a+b=0$ unless $a=b=0$.