### Complex Numbers

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 $$\CC = \RR \oplus \RR \, i$$ That is, a complex number $z$ is a pair of real numbers ($a$,$b$), which is usually written as $$z = a + bi$$ 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 $$i^2 = -1$$ Third, we have used associativity, that is $$(xy)z = x(yz)$$ for any complex numbers $x$, $y$, $z$. Finally, we have used commutativity, i.e. $$xy = yx$$ to replace $bic$ with $bci$. 1)

We define the complex conjugate $\bar{z}$ of a complex number $z=a+bi$ by $$\bar{z} = a - bi$$ 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 $$|z|^2 = z\bar{z} = a^2 + b^2$$ The only complex number with norm $0$ is $0$. Furthermore, any nonzero complex number has a unique inverse, namely $$z^{-1} = {\bar{z} \over |z|^2}$$ Since complex numbers are invertible, linear equations such as $$y = x z \label{linear}$$ can always be solved for $z$ in terms of $y$, so long as $x\ne0$.

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

Thanks to the identity 2) $$e^{i\theta} = \cos\theta + i\sin\theta \label{cis}$$ 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!} $$z = r e^{i\theta}$$ where $$r = |z|$$ 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 $\pi\over2$.

1) We are really assuming that multiplication is linear over the reals. This not only implies distributivity, but also commutativity between real numbers and the complex unit, $i$, which in this case is enough to ensure full commutativity.
2) This identity is usually proved by comparing the power series expansions of each side. An alternative proof is obtained by noticing that both sides of this equation satisfy the differential equation \begin{align} {d^2\!f\over  dx^2} = - f \nonumber \end{align} with the same initial conditions.