A calculus of the absurd

Chapter 14 Complex numbers

This is a further maths topic.

14.1 Introduction

What happens when you square root a negative number? Well, according to a bunch of people in history, nothing. They claimed it was like dividing by zero, non possibilis102102 Not possible!!

Modern mathematicians, however, think that things do happen when you square root negative numbers (and this can easily be seen after looking at real-world applications).

To denote the square root of a negative number, we declare a new symbol, \(i\) 103103 Note that some people also write this using the letter \(j\)., which is defined as

\begin{equation} i^2 = -1 \end{equation}

Using this definition, we can (for example) write

\begin{align*} & \sqrt {-1} = i\\ & \sqrt {-15} = \sqrt {15}\sqrt {-1} = \sqrt {15} i\\ & \sqrt {-49} = \sqrt {49}\sqrt {-1} = 7i\\ & 15 + \sqrt {-49} = 15 + \sqrt {49}\sqrt {-1} = 7i\\ & x + y\sqrt {-1} = x + yi \end{align*}

In general, we can write a complex number as \(x + yi\). Here \(x\) denotes the "real" part of the complex number (as there is no \(i\) anywhere to be seen) and the \(y\) denotes the imaginary part (as there is an \(i\)).

For a complex number, \(z\), we can write the real part of \(z\) as \(\Re (z)\) and the imaginary part of \(z\) as \(\Im (z)\). Note that many (most?) people don’t use the weird symbols that this document uses, but instead \(\text {Re}(z)\) and \(\text {Im}(z)\) to denote the real and imaginary components of a complex number respectively.

Using this definition, we can work out what happens when we do some basic operations.

We can add complex numbers:

\begin{equation*} (a + bi) + (c + di) = (a + c) + (b + d)i \end{equation*}

We can also multiply them

\begin{align*} \frac {a + bi}{c + di} &= \frac {a+bi}{c+di} \cdot \frac {c-di}{c-di} \\ &= \frac {ac - adi + bci - bd i^2}{c^2 - d^2i^2} \\ &= \frac {ac + bd + (bc-ad)i}{c^2 + d^2} \end{align*}

14.1.1 The complex conjugate

The complex conjugate of \(x + yi\) is just \(x - yi\) - that is, when we take the complex conjugate of a number, the real part remains unchanged, but the sign of the complex part is flipped (i.e. positive numbers become negative and vice versa).

The complex conjugate can be denoted in a number of ways. For a complex number \(z\), we can write the conjugate as \(\overline {z}\) (it is also sometimes written as \(z^{*}\)).

  • Theorem 14.1.1 Let \(z\) be a complex number, then

    \begin{equation} z + \overline {z} = 2\RE (z) \end{equation}

This is a useful theorem (well it shows up sometimes), and the proof uses the standard method for proving identities (Section 6.2).

\begin{align} z + \overline {z} &= x + iy + x - iy \\ &= 2x + iy - iy \\ &= 2x \\ &= 2 \RE (x) \end{align}