Complex Number

Math is complicated. Some can say it’s complex.

Numbers can be complex.

Let’s say we have two funny looking 2x2 Matrix. Just bear with me.

\begin{gather*} M_R=\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}\quad M_I=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix} \end{gather*}

Now we can put two scalars $a,b$ infront of these numbers and add it up like we did before:

\begin{gather*} z=a\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}+b\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}\\ \Rightarrow z=\begin{pmatrix}a & -b \\ b & a \end{pmatrix} \end{gather*}

We can write this weird looking matrix differently:

\begin{gather*} z=a+bi \end{gather*}

What is $i$ ? Well, we don’t know. It’s not a variable! But we know what $i^2$ is:

\begin{gather*} i^2=-1 \end{gather*}

Notice however much you can times $a$ , it won’t affect $b$ , and vice versa.