# What is the square root of negative one?

Sep 20, 2015

The principal square root of minus one is $i$.

It has another square root $- i$.

#### Explanation:

I really dislike the expression "the square root of minus one".

Like all non-zero numbers, $- 1$ has two square roots, which we call $i$ and $- i$.

If $x$ is a Real number then ${x}^{2} \ge 0$, so we need to look beyond the Real numbers to find a square root of $- 1$.

Complex numbers can be thought of as an extension of Real numbers from a line to a plane. The unit in the $x$ direction is the number $1$. The unit in the $y$ (imaginary) direction is the number $i$. So $i$ is called the imaginary unit.

$i$ has the property that ${i}^{2} = - 1$.

If $a \ge 0$ then $\sqrt{a}$ means the non-negative square root of $a$, which lies on the part of the Real line at and to the right of the origin $0$.

If $a < 0$ then we define $\sqrt{a}$ to be the principal square root of $a$, lying on the positive part of the imaginary ($y$) axis.

So $\sqrt{- 1} = i$ and $- \sqrt{- 1} = - i$.

This all looks like it is working well, but some things break down for square roots of negative numbers. For example, the identity $\sqrt{a b} = \sqrt{a} \sqrt{b}$ does not hold in general:

$1 = \sqrt{1} = \sqrt{- 1 \cdot - 1} \ne \sqrt{- 1} \cdot \sqrt{- 1} = - 1$