Question

What is the square root of -2?

Answer 1

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(ab) = sqrta sqrtb`

Following the same reasoning we would expect to have:

`sqrt -2 = sqrt (-1) sqrt2`

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) = sqrt2 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 `(-sqrt(-1))^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 sqrt2` and `-isqrt2`So we write

The square root symbol means the one without a minus sign in front, so `sqrt(-2) = sqrt2 i` or `i sqrt2`.