Can there be a real solution to the square root of -1?

Question

Basically what the title says:

Can there be a real solution to the square root of $-1$ (or any negative number in fact) or is it defined to be unreal?

Because of this: $$ \begin{align} \sqrt{-1} & = (-1)^\frac 12 \\[6pt] & = (-1)^\frac 24 \\[6pt] & = \sqrt[4]{(-1)^2} \\[6pt] &= \sqrt[4]{1} \\[6pt] &= 1 \end{align}$$

I truly wish that we called it an unreal number. – RougeSegwayUser Oct 15 '13 at 18:55 — RougeSegwayUser, Oct 15 '13 at 18:55
$\sqrt[4]{1} = i$, also. – Spine Feast Oct 15 '13 at 19:02 — Spine Feast, Oct 15 '13 at 19:02

score 2 · Accepted Answer · answered Oct 15 '13 at 19:00

Nope, unfortunately. Your logic seems to imply that $1^2 = -1$ which isn't the case.

It also implies that root 4 of 1 is only 1.

This is kinda why complex numbers were invented.

Now, you could invent your own number system and call them "real numbers" or re-define the square root to mean something else.

Can there be a real solution to the square root of -1?

1 Answers1