# What is the square root of -2?

May 28, 2015

The answer your teacher will give depends on where you are in you mathematics education.

There is no positive or negative number that is the square root of $- 2$

If we square a positive number we get a positive answer.
If we square a negative number, we still get a positive number.

There is no positive or negative number (real number) whose square is negative.

But,

We know that, for positive numbers $a$ and $b$:
$\sqrt{a b} = \sqrt{a} \sqrt{b}$

Following the same reasoning we would expect to have:

$\sqrt{-} 2 = \sqrt{- 1} \sqrt{2}$

There is a problem with $\sqrt{- 1}$.

The solution is to invent a new number whose square is $- 1$.

Using tis new number, we can write $\sqrt{- 2} = \sqrt{2} \sqrt{- 1}$.

But, if we want to keep our usual arithmetic, then $\sqrt{- 1}$ needs an opposite, namely $- \sqrt{- 1}$ (These numbers add up to $0$.)

But we also have ${\left(- \sqrt{- 1}\right)}^{2} = - 1$. So, like every other number (except $0$), $- 1$ has two square roots.

Because it is a bother to write and say $\sqrt{- 1}$ over and over, we give this number a name. We call it $i$.
(In mathematics. we call it $i$. Electrical engineers call it $j$.)

$- 2$ has two square roots, $i \sqrt{2}$ and $- i \sqrt{2}$So we write

The square root symbol means the one without a minus sign in front, so $\sqrt{- 2} = \sqrt{2} i$ or $i \sqrt{2}$.