Begin with the real numbers, $\RR$. Add “the” square root of $-1$; call it $i$. You have just constructed the complex numbers, $\CC$, in the form \begin{equation} \CC = \RR \oplus \RR \, i \end{equation} That is, a complex number $z$ is a pair of real numbers ($a$,$b$), which is usually written as \begin{equation} z = a + bi \end{equation} which can be thought of as either a point or a vector in the (complex) plane. How do you multiply complex numbers? Simply multiply it out, that is \begin{align} (a+bi) (c+di) &= (a+bi) c + (a+bi) di \nonumber\\ &= (ac-bd) + (bc+ad)i \end{align}

What properties of the complex numbers have we used? First of all, we have distributed multiplication over addition. Second, $i$ is “the” square root of $-1$, that is \begin{equation} i^2 = -1 \end{equation} Third, we have used associativity, that is \begin{equation} (xy)z = x(yz) \end{equation} for any complex numbers $x$, $y$, $z$. Finally, we have used commutativity, i.e. \begin{equation} xy = yx \end{equation} to replace $bic$ with $bci$. 1)

We define the complex conjugate $\bar{z}$ of a complex number $z=a+bi$ by \begin{equation} \bar{z} = a - bi \end{equation} thus changing the sign of the imaginary part of $z$. Equivalently, complex conjugation is the (real) linear map which takes $1$ to $1$ and $i$ to $-i$. The norm $|z|$ of a complex number $z$ is defined by \begin{equation} |z|^2 = z\bar{z} = a^2 + b^2 \end{equation} The only complex number with norm $0$ is $0$. Furthermore, any nonzero complex number has a unique inverse, namely \begin{equation} z^{-1} = {\bar{z} \over  |z|^2} \end{equation} Since complex numbers are invertible, linear equations such as \begin{equation} y = x z \label{linear} \end{equation} can always be solved for $z$ in terms of $y$, so long as $x\ne0$.

The norms of complex numbers satisfy the following identity: \begin{equation} |yz| = |y| |z| \end{equation} Equivalently, there is an identity involving the real “components”, namely \begin{equation} (ac-bd)^2 + (bc+ad)^2 = (a^2+b^2) (c^2+d^2) \end{equation} (where, say, $z=a+bi$ and $y=c+di$), which is called the 2-squares rule.

Thanks to the identity 2) \begin{equation} e^{i\theta} = \cos\theta + i\sin\theta \label{cis} \end{equation} polar coordinates can be used to write complex numbers in terms of their norm and a phase angle $\theta$. (A factor of the form $e^{i\theta}$ is called a phase.) That is, any complex number can be written in the form \footnote{Note in particular the famous equation \begin{equation*} e^{i\pi}+1=0 \end{equation*} which relates 5 of the most basic symbols in mathematics!} \begin{equation} z = r e^{i\theta} \end{equation} where \begin{equation} r = |z| \end{equation} since $|e^{i\theta}|=1$. Each complex number thus has a direction associated with it in the complex plane, determined by the angle $\theta$. Note finally that multipliation by $i$ rotates a complex number counterclockwise by $\pi\over2$.