$-1$ 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}

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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
@ALGEAN: In my defense, I searched for that question before I posted this. It doesn't seem to appear in the results for "1 = -1" –  Nick Jan 21 at 7:43
$i^2=(-i)^2$ doesn't imply $i=-i$. Simple but complex :-). –  copper.hat Jan 21 at 7:49

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.

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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

Isaac's answer is correct, but it can be hard to see if you don't have a strong knowledge of your laws. These problems are generally easy to solve if you examine it line by line and simplify both sides.

\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 $\textstyle\sqrt{1}/\sqrt{-1}=\sqrt{1/-1}$.

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Only problem I see with these, is that when one is shaky on the laws simplifying might be a problem. –  Wilhelm Jul 23 '10 at 0:28
@Wilhelm: It works pretty well if you use the most certain laws first –  Casebash Jul 23 '10 at 0:31

The rule $\sqrt{ab}=\sqrt a\sqrt b$ holds only for $a,b\ge 0$.

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and what happens to it when a or b is negative? Is there another formula or is this a dead end? –  Nick Jan 21 at 7:36

There is always a danger with dealing with roots of any kind, that one might not be dealing with the same number all the time. This comes in part from both $1^2=1$ and $(-1)^2=1$.

Writing these into an equation as $\sqrt{1}=1$ and $\sqrt{1}=-1$, gives a result that $a=-a$. Such might be true in some mantissa-space (mantissa here is a multiplication form of modulus: ie just as $a + bn = a \pmod{n}$, so would $a * n^b \operatorname{man} n$).

By taking square roots, one is effectively dealing in a potential mantissa-space where $+x=-x$, and some subtly is needed to distingiush the two. This is one of the reasons that $\sqrt{x} \ge 0$ is taken as convention.

The actual mistake in the calculations, is that $\sqrt{x}$ is let to vary by sign.

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given $a,b \in \mathbb R$
Rule 1:
$\sqrt {a b} \iff \sqrt {a} \sqrt {b}$ is valid only if $a,b \geq0$
Similarly, $\sqrt { \frac{a}{b}} \iff \frac{\sqrt {a}}{ \sqrt {b}}$ is valid only if $a\geq 0,b>0$
Rule 2:
And,if $a>0$
Case 1: $\sqrt{a}$ will give only one positive value (more precise: you will get purely real complex number with positive real part) eg. $\sqrt{4}=2$
Case 2: $\sqrt{-a} = i \sqrt{a}$ will give only one complex number with positive imaginary part. (more precise: you will get purely imaginary complex number with positive imaginary part) eg. $\sqrt{-4}= 2i$

So, you have done wrong in fourth line
Lets take a example, shows if you do not follows above rules you can create blunder. Take $a>0$
$\sqrt{-a} = i \sqrt{a}$
But if do not follow above rule you can do blunder as
$\sqrt{-a} = \sqrt{\frac{a}{-1}}$
$=>\sqrt{-a} = \frac{\sqrt{a}}{\sqrt{-1}}$
$=>\sqrt{-a} = -i \sqrt{a}$

All the above taking and discussion is done under the consideration that in general the notion of $\sqrt[n]{x}; x\in\mathbb C$ talking about only about principal root and principal root and it is unique.

But for all the roots of $\sqrt[n]{x}; x\in\mathbb C$ Some Mathematician generally uses $x^\frac{1}{n}; x\in\mathbb C$

For example:
$(4)^\frac{1}{2} = 2,-2$ { more precisely: $(4)^\frac{1}{2} \implies 2,-2$ but $2 \not\implies (4)^\frac{1}{2}$ or $(-2) \not\implies (4)^\frac{1}{2}$ }
Similarly, $(-1)^\frac{1}{2} = i,-i$ because $(-i)* (-i) = \sqrt{-1} = i *i$

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I think that there have been a number of approaches used to find 1/i. Here may be a few to consider:

The positive powers of i run in a cycle (i, -1, -i, 1). Projecting this back into zero and the negative exponents, i^0 = 1 and i^-1 should be -i.

If we "rationalize" 1/i by multiplying both sides by i, then 1/i * i/i = i/i^2 = i/-1 = -i.

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protected by MJDMay 2 at 16:34

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