Complex Numbers

Complex numbers are elements of a number system that contains the real numbers, $\R$, and a “special” element, $i$ (or $j$), which is commonly called the imaginary unit. $i$ satisfies the equation $i^2 = -1$. Complex numbers make up the so-called set $\mathbb{C}$ of Complex Numbers.

Complex numbers are particularly useful in signal processing, Physics, etc., because they allow us to make computations that are impossible using real numbers only. Of course, when dealing with real-life systems (as we do in Physics), there are some constraints on complex numbers — namely, they always “come in pairs” — such that we always end up with real results.

Rectangular form

A complex number is a number of the form:

This is the so-called rectangular form. Complex number are usually represented on a plane: as well as a real number fits on a line, complex numbers (which we express using two real numbers, one of which combined with the imaginary unit) fit on a plane:

• the real part, $\Re$, is on the horizontal axis;
• the imaginary part, $\Im$, is on the vertical axis.

Example

Let’s take the number $c = 1 + 2j$. If we represented it on the complex plane, it would be a point located at $(1, 2)$: one unit to the right of the origin (due to the real part $\Re$) and two units on the top (due to the imaginary part $\Im$).

Polar form

Another way to represent complex numbers is by using the polar form — that is, the distance from origin and the angle formed with the horizontal axis. A complex number in polar form can be expressed as:

The distance from the origin, $\lvert c \rvert$, can be computed from the rectangular form as $\sqrt{\Re^2 + \Im^2}$.

We call $\theta$ the phase.

Euler’s Formula and Euler’s Identity

Using Euler’s Formula, the same number can be expressed in yet another way:

Let’s ignore the “length”, $\lvert c \rvert$, for a moment. If we substitute $\pi$ for $\theta$, we get $(\cos\pi + j\sin\pi) = e^{j\pi}$.

We know that $\cos\pi = -1$ and that $\sin\pi = 0$. We can then rewrite the previous as $(-1 + j 0) = e^{j\pi} \iff (-1) = e^{j\pi}$.

If we rearrange the terms, we get:

This is considered as one of the most beatiful equations in Maths, since it combines many of the fundamental constants: $e$ (base of the natural logarithm), $\pi$, $j$ (or $i$); numbers $1$ (neutral element in multiplications) and $0$ (neutral element in additions).