Can you find the square root of -1/2 ?

2 Answers
Oct 10, 2017

You can't technically, but it's 1/2i

Explanation:

You can't square root a negative. Because we come up with a problem here it is. Remeber a when you square root a number you get a number when multiplied by itself gives us the number that we just square rooted

E. g. sqrt(16) = 4 * 4 or 4^2

Quick Example

When we try to square root a negative we realise this isn't possible. Because when you square root a number the root has to be the same number. So to complete the problem below we'll have to split the -16 into -1 and 16 (-1 * 16= -16).

E. g. sqrt(-16) = sqrt(-1*16)

Now we know that 16 is a square number, so when its square root is... 4 of course! However, we cannot square the negative one so we rewrite it as i which is called an imaginary number as there is no square root of any negative.

Now onto the question
sqrt(-1/2)

Let's split it into -1 and 1/2

sqrt(-1*1/2)

This gives us sqrt(1/2i)

Oct 13, 2017

sqrt(-1/2) = sqrt(2)/2i

Explanation:

Assuming you are asking about the square root of (-1/2) ...

In common with all non-zero numbers, -1/2 has two square roots, but since -1/2 is negative, those square roots are non-real Complex numbers.

The imaginary unit i satisfies i^2=-1

Hence we find:

(sqrt(2)/2i)^2 = (sqrt(2)/2)^2 * i^2 = 1/2 * -1 = -1/2

So one square root of -1/2 is sqrt(2)/2i

We also have:

(-sqrt(2)/2i)^2 = (-sqrt(2)/2)^2 * i^2 = 1/2 * -1 = -1/2

So the other square root is -sqrt(2)/2i

By convention, when we write sqrt(-1/2), we mean sqrt(2)/2i, which is known as the principal square root of -1/2.

Footnote

Why not simply write:

sqrt(-1/2) = sqrt(-1 * 1/2) = sqrt(-1) * sqrt(1/2) = isqrt(1/2)

I tend to avoid being too hasty with the "rule" sqrt(ab) = sqrt(a)sqrt(b), because it does not always work, once you are starting to deal with square roots of negative and/or complex numbers.

For example:

1 = sqrt(1) = sqrt(-1 * -1) != sqrt(-1) * sqrt(-1) = -1

On the other hand, the convention that if x < 0 then:

sqrt(x) = isqrt(-x)

is safe.