# Square root of a negative number

Correct me if I am wrong, $$\sqrt{-4}=2i$$.

But how do you explain it to a student?

We know $$\sqrt{-1}=i$$, but one cannot say $$\sqrt{-4}=\sqrt{-1}\sqrt{4}=2i$$ as the laws of indices can only be applied to real numbers.

• Write it as $-4=(2i)^2$. – Lord Shark the Unknown May 14 at 5:09
• How would you answer a student if he ask why choose the positive root and not the negative? – LanaDR May 14 at 5:11
• Neither $2i$ nor $-2i$ are positive. – Lord Shark the Unknown May 14 at 5:12
• that is what I thought so as positive or negative does not make sense for complex numbers – LanaDR May 14 at 5:13
• What is $\sqrt{-4}$? Whatever it is, it is not "positive". – Lord Shark the Unknown May 14 at 5:15

Edit:

$$i^2 = -1$$

Therefore,

$$4i^2 = -4$$

and by taking square root on both sides, we have $$2i = \sqrt{-4}$$

• sounds good! thanks! – LanaDR May 14 at 5:18
• How is this any different from $\sqrt{-4}=\sqrt{-1}\sqrt{4}$, which was rejected in the question? – amd May 14 at 5:24
• Even if it isn't, why is it wrong? – NoLand'sMan May 14 at 5:28
• One needs to be careful to know what is defined for the complex numbers. For example, $\sqrt{ab} = \sqrt a\sqrt b$ is not necessarily true for complex numbers. – user1952500 May 14 at 5:31
• Okay, I have edited my answer, is this any better? – NoLand'sMan May 14 at 5:37

The square root of a negative number is usually given by \begin{align} (-4)^{1/2} &=e^{\ln{(-4)}/2}\\ &=e^{\ln{(4e^{i\pi})}/2}\\ &=e^{\ln{(e^{i\pi+2\ln{(2)}})}/2}\\ &=e^{\ln{(2)}+i\pi/2}\\ &=2e^{i\pi/2}\\ &=2\left(\cos{\left(\frac\pi2\right)}+i\sin{\left(\frac\pi2\right)}\right)\\ &=2i\\ \end{align} where the principal branch of the natural logarithm is taken instead of the 'positive' square root.

I would approach the topic dealing first with the $$2 \pi$$ periodicity of the angle.

Then I would pass to the convention used to split the $$2 \pi$$ angle: the "standard" is $$-\pi < \alpha \le \pi$$ thus the "principal" "standard" solution is $$2i$$.
But any other split can be adopted.