1 = -1 Clearest way to explain why this proof is wrong.

Question

Say you are a high school student or a young undergrad. You are being taught about complex numbers and you are told that $i = \sqrt{-1}$.

You go home and you write this: \begin{equation} \begin{aligned} 1 & = 1 \\ 1 & = \sqrt{1} \\ 1 & = \sqrt{(-1)(-1)} \\ 1 & = \sqrt{-1}\sqrt{-1} \\ 1 & = i \times i \\ 1 & = i^2 \\ 1 & = -1 \end{aligned} \end{equation}

You are dismayed. The infamous imaginary numbers are inconsistent after all!

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The best answer would be the clearest explanation of why the "proof" is faulty, keeping in mind the level of the student. An explanation is extra clear if t presents insights to the student, as well as being correct and concise.

Answer 1

$$ \sqrt{(-1)(-1)} \neq \sqrt{-1}\sqrt{-1} $$

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One way to see this is to use the property $\sqrt{x^2}=|x|$ for all $x \in \mathbb{R}$, so that

$$ \sqrt{(-1)(-1)}=\sqrt{(-1)^2}=|-1|=1. $$

Only for non-negative $x$ and $y$ is it true that $\sqrt{xy}=\sqrt{x}\sqrt{y}$.

Answer 2

A priori, $\sqrt{-1}$ is just notation for some new object which, when squared, yields $-1$. It is not some operator $\sqrt{}$ applied to the number $-1$. There is no reason one should expect it to behave exactly like the traditional square root function, which isn't defined for negative numbers.