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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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    $\begingroup$ 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 $\endgroup$ – Casebash Jul 22 '10 at 4:22
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    $\begingroup$ @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] ). $\endgroup$ – Isaac Jul 22 '10 at 4:48
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    $\begingroup$ $i^2=(-i)^2$ doesn't imply $i=-i$. Simple but complex :-). $\endgroup$ – copper.hat Jan 21 '14 at 7:49
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    $\begingroup$ How $\sqrt{-1}=i$ if and only if $i^2=-1$? Or how can we conclude that $\sqrt{-9}=\pm 3i$? $\endgroup$ – daulomb Jan 21 '14 at 16:34
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    $\begingroup$ $\sqrt{-1}\neq \frac{1}{i} = -i$ $\endgroup$ – Alvin Lepik Mar 30 '16 at 6:54

10 Answers 10

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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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    $\begingroup$ Well sqrt(-25) / sqrt(-1) = 5, so the explanation doesn't seem quite complete. $\endgroup$ – Wilhelm Jul 22 '10 at 4:21
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    $\begingroup$ 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. $\endgroup$ – Justin L. Jul 22 '10 at 4:38
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    $\begingroup$ Thanks Justin and Himadri for clarifying my answer. Hopefully my edit will help as well. $\endgroup$ – Isaac Jul 22 '10 at 5:05
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    $\begingroup$ @workmad3: It's true that it works in both those cases, but it's misleading. It's comparable to saying that it's okay to cancel the 6s in 16/64 to get 1/4--it works, but for the wrong reasons. The identity sqrt(a) * sqrt(b) = sqrt(a * b) really does only work when a and b are nonnegative. $\endgroup$ – Isaac Jul 22 '10 at 13:33
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    $\begingroup$ @Isaac, I believe the key here originates from unit signs. Every number can be expressed in the form $N = e^{i \theta}*|N|$ where the $|N|$ gives the absolute value of the number and must strictly be greater than or equal to positive 0 while: $e^{i \theta}$ is the angle or orientation of the number in the complex number plane. The identity $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ holds true whenever a,b both share the same orientation ($e^{i\theta}$) (ie exist on the same line from the origin). $\endgroup$ – frogeyedpeas Jun 17 '14 at 23:14
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The simple reason this,

$\sqrt{\frac{-1}{1}} =\frac{\sqrt{1}}{\sqrt{-1}}$

Is not valid is because of a branch cut that must be taken as a result of the square root function.

Using exponentials:

$$ \tag{1} e^{i\pi} = -1 $$ and thus, $$ \tag{2} e^{-i\pi} = \frac{1}{-1} = -1 $$

However, in the complex plane, these represent a semicircle rotation from $+1 \to -1$ in both anticlockwise and clockwise directions. The fact that both end up at $-1$ on the Real Axis simply conceals the path taken from $+1 \to -1$ in the complex plane.

Now if we think of both (1) and (2) they are two opposite semi-circles that make a circle in the complex plane of radius 1 around the origin: enter image description here

However, from complex analysis, we know that there must be a branch cut from $0 \to \infty$ somewhere on this circle. Thus, you are crossing a branch cut on the Riemann Surface by doing,

$$ \sqrt{\frac{-1}{1}} \to \frac{\sqrt{1}}{\sqrt{-1}} $$

To correctly, dodge the Branch Cut issues, we can use the exponential form ensuring that we simplify the exponentials inside the square root before taking the square root, $$ \sqrt{\frac{e^{i\pi}}{1}\frac{e^{-i\pi}}{e^{-i\pi}}} = \sqrt{\frac{e^{0}}{e^{-i\pi}}} = \sqrt{\frac{e^{i\pi}}{1}} = i $$ or similarly, $$ \sqrt{\frac{e^{-i\pi}}{1}\frac{e^{+i\pi}}{e^{+i\pi}}} = \sqrt{\frac{e^{0}}{e^{+i\pi}}} = \sqrt{\frac{e^{-i\pi}}{1}} = -i $$ The fact we get two different answers from using (1) and then using (2) to invert the terms inside the square root demonstrate the chaos we give ourselves when crossing Branch Cuts. Morale of Story: Don't Cross Branch Cuts as "Here be multivalued-Dragons" see bottom for picture.

You can also be really sneaky and do two full rotations, which if you look at the picture below, will put you where you started on the Riemann sheet.

See this image for a pictorial representation of the manifold which you are attempting to abuse (This is 3D version of the above and the branch cut can be seen where the sheet intersects itself). This picture also shows why we can take the Branch Cut anywhere as long as it is within one rotation as we can spin the shape in the vertical axis and it will remain the same.

enter image description here

A demonstration of what happens to an unprepared Math Student, (Here called 'Bob') who unknowingly crossed a Branch Cut without being prepared. enter image description here (credit also to @marios-bounakis for some exciting discussions about Euler identities)

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    $\begingroup$ This answer is flawed. It states implicitly that $\sqrt{1/-1}\ne \sqrt{-1/1}$. But in fact, $\sqrt{1/-1}=\sqrt{-1/1}$. The error is the OP was in writing $\sqrt{1}/\sqrt{-1}=\sqrt{1/-1}$. $\endgroup$ – Mark Viola Feb 18 '17 at 16:55
  • $\begingroup$ @MarkViola - incorporated your comment to make it clear that it is due to the square root and not simply the movement of signs from numerator to denominator within the square root. $\endgroup$ – Alexander McFarlane Mar 12 '18 at 13:51
  • $\begingroup$ Great. And you took less than 13 months to augment. ;-)) $\endgroup$ – Mark Viola Mar 12 '18 at 15:30
  • $\begingroup$ I'm sorry but I spent the last year recovering in the serious burn unit $\endgroup$ – Alexander McFarlane Mar 12 '18 at 16:41
  • $\begingroup$ Oh my. I am sorry to hear that and hope that you are recovering nicely. I had 4 surgeries from April 2014 to June 2015 and have received medical treatments since April 2014 to the present. I can truly empathize. $\endgroup$ – Mark Viola Mar 12 '18 at 16:45
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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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  • $\begingroup$ Only problem I see with these, is that when one is shaky on the laws simplifying might be a problem. $\endgroup$ – Wilhelm Jul 23 '10 at 0:28
  • $\begingroup$ @Wilhelm: It works pretty well if you use the most certain laws first $\endgroup$ – Casebash Jul 23 '10 at 0:31
  • $\begingroup$ Why do you sometimes use $\hat{\imath}$ and sometimes use $i$? $\endgroup$ – L. F. Apr 12 at 10:05
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You use the rule $\sqrt{ab}=\sqrt a\sqrt b$, which indeed holds for $a,b\ge 0$: If $u,v$ are nonnegative real numbers with $u^2=a$ and $v^2=b$ then clearly $uv$ is a nonnegative real number with $(uv)^2=ab$. However, without the assumption $a,b\ge0$, this rule no longer holds in general.

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    $\begingroup$ and what happens to it when a or b is negative? Is there another formula or is this a dead end? $\endgroup$ – Nick Jan 21 '14 at 7:36
  • $\begingroup$ @Nick Simply separate the roots in terms of positive ones and i. Also, it is not incorrect to say that $\sqrt{\frac{1,-1}}=1/i=-i$, as $(-i)^{2}=-1=\frac{1,-1}}=1$, it's just that square roots have two possible values always. $\endgroup$ – Anonymous Pi Apr 12 '15 at 19:05
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    $\begingroup$ @HagenvonEitzen, adding "only" to the condition makes it necessary however it is a sufficient condition. Do you see the Isaac's answer contains a statement "This is only(guarenteed to be true) when ..."? This sentence show that it is a sufficient condition and not necessary. In particular, addition of the term "only" made your answer incorrect. $\endgroup$ – Sufyan Naeem May 10 '15 at 11:35
  • $\begingroup$ @Hagen von Eitzen) I think your answer is wrong...Statement is TRUE when at least one of $a$ and $b$ is $\ge 0$. $\endgroup$ – Empty Oct 5 '15 at 19:00
  • $\begingroup$ As said in the comments, the phrasing makes this answer wrong. $\endgroup$ – YoTengoUnLCD Mar 30 '16 at 3:30
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As others have mentioned, one approach is to be consistent about using principal roots, and make sure that the identities you're using are actually applicable for manipulating those principal roots. And your mistake there lies in equating the 3rd and 4th lines. But to me, that type of approach feels about as intuitive as memorizing a phone book.

Personally I find it much easier to think of square roots (or roots in general) as set-valued operators. Every number (other than zero) has two square roots, three cube roots, four cube roots, etc (which in general can be complex).

Define the $n^{th}$ root to be the set-valued mapping: $\sqrt[n]x \rightarrow \{ y : y^n = x \}$

Examples: \begin{gather} \sqrt{4} \rightarrow \{ +2 , -2 \} \\\\ \sqrt{1} \rightarrow \{ +1 , -1 \} \\\\ \sqrt{-1} \rightarrow \{ -i, +i \} \\\\ \sqrt[3]{1} \rightarrow \left\{ 1 , \tfrac{-1 + i \sqrt{3}}{2} , \tfrac{-1 - i\sqrt{3}}{2} \right\} \end{gather}

Then your question can be written as follows (treating multiplication and division as Kronecker products too see all the possible outcomes):

\begin{gather} \sqrt{-1} \rightarrow \{ i, -i \} \\\\ \frac{1}{\sqrt{-1}} \rightarrow \left\{ \frac{1}{i} , \frac{1}{-i} \right\} \\\\ \frac{1}{\sqrt{-1}} \rightarrow \left\{ \frac{1}{i} , \frac{1}{-i} \right\} \\\\ \frac{\sqrt{1}}{\sqrt{-1}} \rightarrow \frac{ \left\{ 1 , -1 \right\} }{ \left\{ i , -i \right\} } = \left\{ \frac{1}{i} , \frac{-1}{i} , \frac{1}{-i} , \frac{-1}{-i} \right\} = \left\{ \frac{1}{i} , -\frac{1}{i} \right\} = \left\{ -i , i \right\} \\\\ \sqrt{\frac{1}{-1}} = \sqrt{-1} \rightarrow \left\{ i , -i \right\} \\\\ \sqrt{-1} \rightarrow \left\{ i , -i \right\} \\\\ \left( \sqrt{-1} \right)^2 \rightarrow \left\{ i^2 , \left(-i\right)^2 \right\} = \left\{ -1 , -1 \right\} = \{ -1 \} \end{gather} so \begin{gather} \left( \sqrt{-1} \right)^2 = -1. \end{gather} Also note that \begin{gather} \left( \frac{\sqrt{1}}{\sqrt{-1}} \right)^2 \rightarrow \left\{ (-i)^2 , i^2 \right\} = \left\{ -1 , -1 \right\} = -1. \end{gather}

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    $\begingroup$ A very nice step by step illustration. $\endgroup$ – Ken Draco Feb 8 '18 at 4:09
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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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I think that there have been a number of approaches used to find $\frac{1}{i}$. 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" $\frac{1}{i}$ by multiplying both sides by $i$, then $\frac{1}{i} * \frac{i}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$.

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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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Some of the confusion here is because there are two types of roots: roots and principle roots. The principle roots always return only one value. However symbolically they are almost never different. You can very, very rarely encounter $_+\sqrt{\;}\;$for the principle root. So confusion is often created because people confuse these two types of roots as they are represented by exactly the same symbols. $$_+\sqrt1=1$$ $$\sqrt1=-1,+1$$ $$\sqrt{-1}=-i,+j$$ If we are doing something like this $$\sqrt{-1}=\frac{1}{i}$$ it means we have to take the correct roots; otherwise, we can have even this: $$\sqrt{1}=-1\;\ and \;\sqrt{1}=1$$ Hence, $1=-1\,.\;$ That's not the way to go. We have multiple roots when dealing with complex numbers, even square roots from real numbers have two answers. To avoid this, the principle root is used but it doesn't differ symbolically from just square root. People extremely rarely write $_+\sqrt{\;}\;$. Therefore there's a lot of confusion. The rules of extracting roots from complex numbers don't strictly follow the rules used with principle roots, so you may easily arrive at a wrong answer. So when we see, for example, the formula: $$\sqrt[n]{z}=\sqrt[n]{p}\, \bigg(\cos \frac{φ+2πk}{n}+i\sin\frac{φ+2πk}{n}\bigg)$$ We have to understand that what is meant is this $$\sqrt[n]{z}=\,_+\sqrt[n]{p}\, \bigg(\cos \frac{φ+2πk}{n}+i\sin{\frac{φ+2πk}{n}}\bigg)$$ Another example. We can't reduce a multiple root $\;\sqrt[nk]{z^k}\;$ to $\;\sqrt[n]{z}\;$ because the first one has $nk$ different root values and the second--only $n\,$.

Formulas such as $\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$ will work but not always. Generally, we can't use rules for principal roots for real numbers when we are dealing with complex numbers.

To recap:

First, $\sqrt{-1}=-i,+i\;\;$ ( NOT JUST $\;i\,$)

Second, when dealing with complex numbers, $\sqrt1=-1,+1\;\;$ ( NOT JUST $\;1\;$)

And third, $\sqrt{-1}\cdot\sqrt{-1}=1\;$ if we take $\;i,-i\;$ or $\;-i,i\;$ as roots, and $\sqrt{-1}\cdot\sqrt{-1}=-1\;$ if we take either $\;i,i$ or $\;-i,-i\;$ as roots. And even if we deal with real numbers and extract roots (not principal roots) we can still arrive at different values.

In very old books $\sqrt{-1}$ is sometimes used instead of $i$. It can only add to confusion.

Most crucial here: Roots must be distinguished from principle roots. When dealing with complex numbers we don't extract some principal root of them, but we have a set of different root values. So, almost every single line is wrong in the OP's 'proof'. Kevin Holt provided a very nice step by step illustration.

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what you are saying is similar to $1^2=(-1)^2$ so $1=-1$, math does not work that way

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protected by MJD May 2 '14 at 16:34

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