Complex numbers

This article is the first on a series about complex variables and functions on them. In this instalment we provide an introduction to complex numbers, assuming no prior knowledge on the subject. Familiarity with real numbers, plots of real functions on the $xy$-plane, vectors in $\R^2$ and polar coordinates will prove useful throughout the series.

What are complex numbers?

Without further ado, let us give a formal definition of what a complex number is.

A complex number is a number of the form $a + ib$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, for which we have that $i^2 = -1$. The set of all complex numbers is denoted by $\C$.

Perhaps you are wondering what the meaning of this new number $i$ is. It is clear that $i$ doesn't belong in $\R$, for there is no real number such that its square is $-1$. Instead of a number, you can think of $i$ as a mere symbol, a placeholder, for which we have the rule $i^2 = -1$. Using this new symbol, we can express the roots of other negative real numbers in a new way. For example, $\sqrt{-9} = 3i$, for we have that $(3i)^2 = 3^2 i^2 = 9 \cdot (-1) = -9$.

Note that every real number $r$ is also a complex number, since it can be written as $r + i0$, so we can think of $\C$ as an extension of $\R$. In fact, if we think of $\R$ as the real line, the addition of this imaginary unit creates a completely new dimension and so we can picture $\C$ as the complex plane. Analogous to the $xy$-plane, we represent the value of $a$ on the $x$-axis and the value of $b$ on the $y$-axis. For some complex number $z = a + ib$, these valued are called the real part of $z$ (written as $\re z$) and the imaginary part of $z$ (written as $\im z$), respectively.

Here is an interactive representation of the complex plane. You can left-click on it to plot a complex number. Use the mouse wheel to zoom and the middle button to pan (this can also be done using SHIFT + left button).

$z = 1+i$

As you can see, the similarities with the $xy$-plane are evident and it will often be useful to think of complex numbers as vectors in $\R^2$. For this, we simply associate the vector $(a,b)$ and the number $a+ib$.

Operations with complex numbers

As you might have expected, we can perform arithmetic with complex numbers. For addition and subtraction, this is done in a similar way to vectors in $\R^2$.

Let $z$ and $w$ be two complex numbers with $z = a + ib$ and $w = c + id$, where $a, b, c, d \in \R$. We can add and subtract them as follows:

$z + w = (a + ib) + (c + id) = (a+c) + i(b+d)$
$z - w = (a + ib) - (c + id) = (a-c) + i(b-d)$

Note how in our definition we had to specify that $a, b, c, d \in \R$. The reason for this is that the representation $z = a + ib$ is not unique if we let $a$ and $b$ be arbitrary complex numbers. For example, $1 + 0i$ could also be written as $0 + i(-i)$. For our purposes, however, we will always assume that, when we write $z = a + ib$, $a$ and $b$ are taken to be real.

The case of multiplication is not so simple. We can't borrow multiplication from of vectors, plainly because there is no multiplication between two-dimensional vectors that yields another two-dimensional vector. Instead, we use the rule $i^2 = -1$ to multiply two complex numbers algebraically.

Let $z = a + ib$ and $w = c + id$ be two complex numbers. We have: $$zw = (a + ib)(c + id) = ac + ibc + iad + i^2bd$$ $$ = (ac - bd) + i(bc + ad)$$

At first sight, this definition has no apparent geometric meaning, but we will soon uncover the secrets of complex multiplication. For now, have a go at multiplying two complex numbers on the complex plane.

$z=1$

$w=i$

$zw=i$

Do you notice anything interesting? Try setting $w$ to $i$ and move $z$ around. How about now? Perhaps visualising both $z$ and $zi$ in the same plot will help you see what's happening:

$z=1$ and $zi=i$

It is easy to see the pattern now: multiplying by $i$ rotates $z$ by a right angle counterclockwise. In fact, complex multiplication has a lot to do with rotation, but in order to explain this relation we will need to introduce the system of polar coordinates. Before we do this, however, we will introduce two more operations on complex numbers.

Let $z = a + ib$. We define the conjugate of $z$, writen $\bar{z}$ and read z bar, as $a - ib$.

Geometrically, the conjugate is the reflection of $z$ about the $x$-axis. One of the most important properties of $\bar{z}$ is that $z\bar{z} = (a + ib)(a - ib) = a^2 - b^2 + i(ba - ab) = a^2 + b^2$ is always a real number. We can now use this fact to introduce division of complex numbers.

Let $z$ and $w$ be as usual. We can divide them as follows: $$\frac{z}{w} = \frac{a + ib}{c + id} = \frac{a + ib}{c + id}\frac{c - id}{c - id} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}$$

Polar coordinates

So far we have determined a point $z$ on the complex plane by specifying its real and imaginary parts ($x$ and $y$ values if you wish), but there are other ways of uniquely defining a point on the plane. Seen as a vector, each complex number has a modulus (its distance to the origin) and it describes an angle with the $x$-axis, which we call its argument. These two properties are actually sufficient to determine a single point on the plane and thus we could use them as a coordinate system, which we call polar.

Let $z = a + ib$ be a nonzero complex number. Its modulus $|z|$ is given by $\sqrt{a^2 + b^2}$. Its argument, denoted $\arg z$, is any $\theta \in \R$ such that $z = |z| (\cos \theta + i \sin \theta)$.

Note that $\arg z$ is not unique; in fact, if we have such a $\theta$ then $\theta + 2\pi$ is also a valid value for $\arg z$. It will sometimes be useful to restrict $-\pi < \theta \leq \pi$, then this number is unique and we call it $\Arg z$. We can then write $z$ in polar form as $(r, \theta) = (|z|, \Arg z)$.

An Introduction to Complex Numbers

What are complex numbers?

Operations with complex numbers

Polar coordinates

Adrián Doña Mateo