Complex Numbers

Why

We want to find the roots of negative numbers.¹

Definition

A complex number is an ordered pair of real numbers. The real part of a complex number is its first coordinate. The imaginary part of a complex number is its second coordinate.

We can identify the imaginary numbers with no complex part (i.e., the set $\Set*{(a, b) \in \R ^2}{b = 0}$) with $\R $ in the obvious way. For this reason, such a complex number is sometimes referred to as a purely real number. On the other hand, a complex number with zero imaginary part (i.e., an element of the set $\Set*{(a, b) \in \R ^2}{a = 0}$) is said to be a purely imaginary number.

Notation

When treating $\R ^2$ as the set of complex numbers, we denote it by $\C $. Let $z \in \C $ with $z = (a, b)$. The real part of $z$ is $a$ and its imaginary part is $b$. It is universal to denote $z$ by $a + ib$, and to call $i$ an (or the) imaginary number. Some authors use $j$, it is a matter of notation.

We denote the real part of $z$ by $\re(z)$, read “real of z,” and the imaginary part by $\im(z)$, read “imaginary of z.” So, in particular, $\re(z) = a$ and $\im(z) = b$.

1. Future editions will modify this, and will discuss the existence of solutions of algebraic equations. ↩︎