Is $i$ equal to $-i$?

Question

When I was in high school, I learned about $i$ in math class and I remember asking my teacher back then if $i$ was equal to $-i$ according to the simple following development :

\begin{equation} i=\sqrt{-1}=\sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}= \frac{1}{\sqrt{-1}}=\frac{1}{i}=-i \end{equation}

The teacher turned out to be unable to answer my question.

Even though I've learned since then that this equality is wrong somewhere, I have never understood where was the flaw in this simple thought exercise.

Answer 1

The problem is that the rule $$ \sqrt{\frac ab} = \frac{\sqrt a}{\sqrt b} $$ doesn't hold in general unless $a$ and $b$ are positive.

Answer 2

If $i=-i$, we would have $2i=0$ , so $4i^2=0$, which contradicts $4i^2=-4$.

Answer 3

If $i = -i$, then $2i = 0$, which is absurd.

Answer 4

As a couple of people have pointed out, the immediate problem in your computation is that $\sqrt{-1/1} \ne \sqrt{1}/\sqrt{-1}$. But I'd like to expand on that a bit. The deeper problem is that you have to use great care when applying the square root symbol to anything except nonnegative real numbers. If we apply it to a nonnegative real number, there is no ambiguity about what is meant: If $x\ge 0$, then $\sqrt{x}$ denotes "the unique nonnegative number $y$ such that $y^2=x$." However, if $x$ is anything but a nonnegative real number, there is not such a simple prescription, and it's safest to avoid using the square root symbol for such things.

There is one more or less standard convention that one could use: the principal branch of the square root function is generally defined for all complex numbers except negative reals as follows: For $z\in \mathbb C \smallsetminus \{y\in \mathbb R: y<0\}$, $\sqrt{z}$ is interpreted to mean the unique complex number $y$ such that $y^2=z$ and $\operatorname{Re} y > 0$. But this still doesn't apply to negative reals. Although it's less universally accepted, one could choose to extend it to negative real numbers by stipulating that $\sqrt{y} = i\sqrt{|y|}$ for $y<0$. But you should always explain exactly what you mean by the square root symbol if you're applying it to anything other than a nonnegative real number.

That said, the notation $\sqrt{-1}$ is actually quite common in the literature as a way of representing $i$. It is consistent with the "extended principal branch" that I explained above, but most people who use it are just using it because it's somewhat less likely to be confusing than the letter $i$, which is commonly also used as a subscript for sequences or as an index for vectors or tensors, among other things. Personally, I avoid the notation $\sqrt{-1}$ because of all the ambiguities I described above. But if you do encounter it in a book, it's a sure bet that the writer is simply using it as another notation for $i$.