# Can -5 be the square root of 25?

Sep 15, 2015

Yes

#### Explanation:

$5 \times 5 = 25$
Hence 5 is the square root of 25.

Sep 15, 2015

$- 5$ is a square root of $25$ in that ${\left(- 5\right)}^{2} = 25$

When we say the square root of a positive number, we usually mean the positive square root, also known as the principal square root.

#### Explanation:

Any non-zero number has exactly $2$ square roots.

Actually in a technical sense $0$ has $2$ square roots, but they are both $0$.

If $x > 0$ then $\sqrt{x}$ denotes the positive square root and $- \sqrt{x}$ denotes the negative one.

If $x < 0$ then $\sqrt{x} = i \sqrt{- x}$ denotes the square root with positive coefficient of $i$ (the imaginary unit). $- \sqrt{x} = - i \sqrt{- x}$ is the other square root.

If $z$ is Complex, then it still has exactly two square roots, but different people use different conventions as to which one is the principal square root (i.e. the one represented as $\sqrt{z}$).

In terms of polar representation, this boils down to choosing whether you prefer to use angles in the range $\left(- \pi , \pi\right]$ or $\left[0 , 2 \pi\right)$ for $z$, resulting in angles in the range $\left(- \frac{\pi}{2} , \frac{\pi}{2}\right]$ or $\left[0 , \pi\right)$ for $\sqrt{z}$.