# Imaginary unit

The **imaginary unit**, , is the fundamental component of all complex numbers. In fact, it is a complex number itself. It has a magnitude of 1, and can be written as . Any complex number can be expressed as for some real numbers and .

## Contents

## Trigonometric function cis

*Main article: cis*

The trigonometric function is also defined as or .

## Series

When is used in an exponential series, it repeats at every four terms:

This has many useful properties.

## Use in factorization

is often very helpful in factorization. For example, consider the difference of squares: . With , it is possible to factor the otherwise-unfactorisable into .

## Problems

### Introductory

### Intermediate

- The equation has complex roots with argument between and in the complex plane. Determine the degree measure of . (Source)

### Olympiad

- Let and with no real roots. If , show that . <url>viewtopic.php?t=78260 (Source)</url>