# Imaginary unit

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In mathematics, the imaginary unit allows the real number system $\mathbb{R}$ to be extended to the complex number system $\mathbb{C}$ , which in turn provides at least one root for every polynomial P(x) (see algebraic closure and fundamental theorem of algebra). The imaginary unit is denoted by i, j, or the Greek ι (see alternative notations). Although its precise definition varies, the imaginary unit's core property is that i 2 = −1.

There are in fact two square roots of −1, namely i and −i, just as there are two square roots of every non-zero real number.

For a history of the imaginary unit, see Complex number: History.

## Contents

### Definition

The imaginary number i is defined solely by the property that its square is −1:

With i defined this way, it follows directly from algebra that i and −i are both square roots of −1.

Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is perfectly valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers by treating i as an unknown quantity while manipulating an expression, and then using the definition to replace any occurrence of i 2 with −1. Higher integral powers of i can also be replaced with −i, 1, i, or −1: