# -1 is not 1, so where is the mistake?

I know there must be something unmathematical in the following but I don't know where it is:

\begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} &= \frac1i \\ \\ \sqrt{\frac1{-1}} &= \frac1i \\ \\ \sqrt{\frac{-1}1} &= \frac1i \\ \\ \sqrt{-1} &= \frac1i \\ \\ i &= \frac1i \\ \\ i^2 &= 1 \\ \\ -1 &= 1 \quad !!! \end{align}

-
When dealing with square roots, it is important to remember that each number apart from 0 naturally has 2 different roots. This means that you have to be very careful when dealing with non-positive numbers – Casebash Jul 22 '10 at 4:22
@Casebash: I believe it's fairly standard to take sqrt(x) to mean the principal square root function of x--that is, if x is a nonnegative real number, the nonnegative square root. Defining which root is the principal root of nonreal complex numbers can be a little trickier (some texts use the one with argument in [0,π) while other texts and many calculators use the one with argument in (-π/2,π/2] ). – Isaac Jul 22 '10 at 4:48

Between your third and fourth lines, you use $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$. This is only (guaranteed to be) true when $a\ge 0$ and $b>0$.

edit: As pointed out in the comments, what I meant was that the identity $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ has domain $a\ge 0$ and $b>0$. Outside that domain, applying the identity is inappropriate, whether or not it "works."

In general (and this is the crux of most "fake" proofs involving square roots of negative numbers), $\sqrt{x}$ where $x$ is a negative real number ($x<0$) must first be rewritten as $i\sqrt{|x|}$ before any other algebraic manipulations can be applied (because the identities relating to manipulation of square roots [perhaps exponentiation with non-integer exponents in general] require nonnegative numbers).

This similar question, focused on $-1=i^2=(\sqrt{-1})^2=\sqrt{-1}\sqrt{-1}\overset{!}{=}\sqrt{-1\cdot-1}=\sqrt{1}=1$, is using the similar identity $\sqrt{a}\sqrt{b}=\sqrt{ab}$, which has domain $a\ge 0$ and $b\ge 0$, so applying it when $a=b=-1$ is invalid.

-
Well sqrt(-25) / sqrt(-1) = 5, so the explanation doesn't seem quite complete. – Wilhelm Jul 22 '10 at 4:21
Just because something is true for some cases, doesn't mean that it is true in other cases. The square root division law described by Isaac is the only defined way. sqrt(-25)/sqrt(-1) might = 5, but that's just a coincidence. There is only one rigorously defined way that keeps everything in order. – Justin L. Jul 22 '10 at 4:38
In your sqrt(-25)/sqrt(-1) example, this is only true because it is i/i * sqrt(25)/sqrt(1) = 1 * sqrt(25) = 5. If we allowed people to use Isaac's square root division rule for any values a and b, we'd get some pretty horrible messes...like what is stated in the question. – Justin L. Jul 22 '10 at 4:45
@Wilhelm, here @Isaac means that This is only true when a >= 0 and b>0 else it may be true may not be true. – Himadri Jul 22 '10 at 4:54
Thanks Justin and Himadri for clarifying my answer. Hopefully my edit will help as well. – Isaac Jul 22 '10 at 5:05
\begin{align*} \sqrt{-1} &= \hat\imath & \mathrm{LHS}&=i, \mathrm{RHS}=i \\ 1/\sqrt{-1} &= 1/\hat\imath & \mathrm{LHS}&=1/i=-i, \mathrm{RHS}=-i \\ \sqrt{1}/\sqrt{-1} &= 1/\hat\imath & \mathrm{LHS}&=1/i=-i, \mathrm{RHS}=-i \\ \textstyle\sqrt{1/-1} &= 1/\hat\imath & \mathrm{LHS}&=\sqrt{-1}=i, \mathrm{RHS}=-i \end{align*}
We can then see that the error must be assuming $\textstyle\sqrt{1}/\sqrt{-1}=\sqrt{1/-1}$.