# Is +-5i the square root of -25 ?

May 19, 2017

Yes and no.

#### Explanation:

The notation:

$\pm 5 i$

is shorthand for the two values:

$5 i \text{ }$ and $\text{ } - 5 i$

So we can say that the polynomial:

${x}^{2} + 25 = 0$

has roots:

$x = \pm 5 i$

meaning that it has roots:

$x = 5 i \text{ }$ and $\text{ } x = - 5 i$

In other words, $- 25$ has two square roots, namely $5 i$ and $- 5 i$.

By common definition and convention, the symbols $\sqrt{- 25}$ denote $5 i$ (which is called the "principal square root"). So $- \sqrt{- 25} = - 5 i$. Both $\sqrt{- 25} = 5 i$ and $- \sqrt{- 25} = - 5 i$ are square roots of $- 25$.

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Remarks

The $\pm$ symbol is very useful for shortening expressions that describe multiple values, but can be unclear when overused.

For example, we might write:

${\left(a \pm b\right)}^{2} = {a}^{2} \pm 2 a b + {b}^{2}$

which is true, provided that the chosen signs match.

Then again, we might say that the roots of:

${x}^{4} - 10 {x}^{2} + 1 = 0$

are:

$x = \pm \sqrt{2} \pm \sqrt{3}$

but in that case we would intend all $4$ possible combinations.