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We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$

EDIT: I see this has been dealt with before but at least with this answer I'm not making the fundamental mistake of assuming an incorrect definition of $i^2$.

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    $\begingroup$ Why is $\sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1}$? We know that $\sqrt{ab} = \sqrt{a} \sqrt{b}$ but this is only true for $a,b \geq 0$... $\endgroup$ Jul 3, 2011 at 9:17
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    $\begingroup$ Duplicate of math.stackexchange.com/q/438/7850 $\endgroup$ Jul 3, 2011 at 11:49
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    $\begingroup$ @TheChaz: while I agree that deep down the two questions are duplicates, I also think that a person who understands that the two questions are the same would not be having this confusion. So closing this as a duplicate may not actually help directly. (This is one of those issues having to do with abstract duplicates, ideally we should have a master question dealing precisely with problems about incorrect algebraic manipulations of $\sqrt{-1}$.) $\endgroup$ Jul 3, 2011 at 12:29
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    $\begingroup$ In every math textbook! If there is a math textbook that says $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ without providing restrictions on $a,b$, then you should write to the author immediately! $\endgroup$
    – GEdgar
    Jul 3, 2011 at 12:43
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    $\begingroup$ $i^2=-1$ is the definition of $i$. $\endgroup$
    – Tpofofn
    Feb 8, 2013 at 18:00

9 Answers 9

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From $i^2=-1$ you cannot conclude that $i=\sqrt{-1}$, just like from $(-2)^2 = 4$ you cannot conclude that $-2=\sqrt 4$. The symbol $\sqrt a$ is by definition the positive square root of $a$ and is only defined for $a\ge0$.

It is said that even Euler got confused with $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$. Or did he? See Euler's "mistake''? The radical product rule in historical perspective (Amer. Math. Monthly 114 (2007), no. 4, 273–285).

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  • $\begingroup$ If I cannot conclude that $i=\sqrt{-1}$ then what good is it? Wouldn't that mean I couldn't use $i$ for what it's intended? Namely $\sqrt{-5} = \sqrt{-1\times 5} = i\sqrt{5}$ ? $\endgroup$
    – Greg
    Jul 4, 2011 at 13:04
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    $\begingroup$ @Greg, then only property of $i$ that you need is that $i^2=-1$. The symbol $\sqrt{-1}$ is abuse of notation. As for $\sqrt{-5}$, you get $(i\sqrt{5})^2= (-1)\cdot 5 = -5 =$, as you expect. $\endgroup$
    – lhf
    Jul 4, 2011 at 15:46
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    $\begingroup$ There's nothing wrong with defining(!) $\sqrt{-1}=i$, just as there's nothing wrong with defining $\sqrt{4}=2$ despite the fact that from $x^2=4$ you cannot conclude $x=2$. The difference to real numbers is that you cannot define the square root function so that it is continuous on all of $\mathbb C$. And the calculation in the question is just incorrectly applying power laws; you don't even need complex numbers to do something similar: $-1 = (-1)^1 = (-1)^{2\cdot1/2} = ((-1)^2)^{1/2} = 1^{1/2} = 1$ $\endgroup$
    – celtschk
    Jul 10, 2013 at 12:40
  • $\begingroup$ So the power laws like $a^n*b^n=(a*b)^n$ are only applicable, when $a,b > 0$? $\endgroup$
    – Mike
    Dec 11, 2019 at 23:17
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Any non zero number has two distinct square roots. There's an algebraic statement which is always true : "a square root of $a$ times a square root of of $b$ equals a square root of $ab$", but this does not tell you which square root of $ab$ you get.

Now if $a$ and $b$ are positive, then the positive square root of $a$ (denoted $\sqrt{a}$) times the positive square root of $b$ (denoted $\sqrt{b}$) is a positive number. Thus, it's the positive square root of $ab$ (denoted $\sqrt{ab}$). Which yields

$$\forall a,b \ge 0, \ \sqrt{a} \sqrt{b} = \sqrt{ab}$$

In your calculation, because $i$ is a square root of $-1$, then $i^2$ is indeed a square root of $1$, but not the positive one.

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The problem is this: the (polar) representation of a complex number depends on your choice of branch. Once you choose a branch of the square root, you cannot simultaneously represent $1$, and $-1$, because they are $\pi$ apart, so that $-1$ and $1$ will necessarily be in different branches of $z^{1/2}$. The choice of square root you make (if you want it to be well-defined) depends on the choice of branch you are working with. You are trying to combine numbers that live in different branches; it is as if you add $1+1$ and get $1-i\pi$, since 1 can also be represented as $i\pi$ ($~1$ can actually be represented as $ik\pi$, but when you operate, you are supposed to stay within a branch, and, in your case, you are not$~$).

In complex numbers, from the perspective of polar representations, when you multiply $z_1 \dot z_2$, you multiply the respective lengths, and add the respective angles (but you have to make up for the fact that the sum of the angles may be larger than $2\pi$ (or whatever argument-system you are working with. So in this sense, when you multiply $i$ by itself, you multiply the length of $i$ by itself, and add the argument to itself; i.e., you double the argument. In complex variables, you have many possible polar representations for a given number; specifically, given $z=re^{it}$, then $z=re^{i(t+2\pi)}$ is also a valid representation.

You must then choose a specific representation for your $z$, specifically; you will need to specify the range of the argument you will be working with. So, say you work with the "standard" range of $[0,2\pi)$. Then the expression for $i$ (me) is as $i=1e^{i\pi/2}$, so that $i^2= (1)(1).e^{i(\pi/2+\pi/2)}=e^{i\pi}=-1$. But the way backwards from multiplying to taking roots is more complicated if your base is non-negative, and/or your exponent has non-zero imaginary part. When this last is the case, you define:

$$z^{{1}/{2}} \:=e^{{\Large{\log(z)}}},$$ where we define:

$$\log(z):=\ln |z|+i \arg(z)$$

(This choice of definition has to see in part with wanting to have the complex log agree with the standard real $\log$--though this agreement is possible only for one choice of "branch", as we will see.) But because of the infinitely-many possible choicesfor the argument of a number, the $\log(z)$ itself--defined locally as the inverse of $e^z$ is somewhat-ambiguously-defined, since $e^z$ does not have a global inverse (since it is not $1-1$, for one thing, but $e^z$ does have local inverses, e.g., by using the inverse function theorem). So when we mention $\log$, we are referring just to one of (infinitely-) many possible local inverses of $e^z$ .Each possible local inverse to $e^z$ is called a "branch" of the $\log$. So once we choose a branch for the $\log$, which is a choice of an open set (technically, it is half-open) of width $2\pi$ from which we will choose the argument we will use. So, say we choose the standard branch $(0,2\pi)$, which we call Log(z). We then define :

$$z^{1/2}:=e^{\large{\operatorname{Log}(z)/2}}$$

But, in this branch , $(-1)^{1/2}$ is not even defined, because the argument for $(-1)$ is $0$ , which is outside of the allowable values $(0,2\pi)$. So, in this sense, the expression $(-1)^{1/2}$ is not well-defined, i.e, does not really make sense. ($~$Note that this particular branch of $\log$ reduces to the standard one when you select an argument of $t=0~$).

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  • $\begingroup$ Maybe I overdid it, trying to understand the log better. Could someone help with the spacing? I am saturated on formatting, sorry. $\endgroup$
    – gary
    Jul 3, 2011 at 10:58
  • $\begingroup$ Help exactly with what spacing in your answer? $\endgroup$
    – night owl
    Jul 4, 2011 at 2:34
  • $\begingroup$ Well, I try to space out the paragraphs, but they do not come out with the spacing of the original; specifically, in spreading the text out some more, and in separating paragraphs. $\endgroup$
    – gary
    Jul 5, 2011 at 17:26
  • $\begingroup$ @ gary: Alright got it. $\endgroup$
    – night owl
    Jul 7, 2011 at 13:48
  • $\begingroup$ Another downvote without a constructive comment. Is it personal? If there is something specific you do not like, it would be nice to hear; I may be able to learn from it. Still, it is up to you, whomever downvoted me without an explanation. $\endgroup$
    – gary
    Jul 12, 2011 at 1:12
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Following Willie Wong's suggestion, I will put the comments I made before into an answer. The problem with this argument is that the formula $\sqrt{ab} = \sqrt{a} \sqrt{b}$ is valid for $a,b \geq 0$. In particular, it is not valid in the case when $a=b=-1$ so that the step $\sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)}$ is incorrect.

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Don't take me the wrong way, all of you are right from a certain point of view.

Let me show you the following. Let $a$ and $b$ be complex numbers. If $\sqrt a$ is a square root of $a$ and $\sqrt b$ is a square root of $b$, then $(\sqrt a\sqrt b)^2 = (\sqrt a)^2 (\sqrt b)^2 = ab$ by assumption, so indeed $\sqrt a\sqrt b$ is a square root of $ab$. Notice that it is some square root; there is no canonical choice over $\mathbb{C}$!

From this point of view, your arguments work until $i^2 = \sqrt1$. All you've shown is that $-1 = i^2$ is a square root of 1 and no one will disagree with that fact.

So what I'm trying to say is, if you want to write something like $\sqrt a\sqrt b = \sqrt{ab}$, which isn't the worst idea, you can't write something like $\sqrt 1 = 1$.

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@Greg: $i^2=-1$ is a matter of definition. It was introduced to handle a negative sign under the radical when solving polynomial equations. The best example is $x^2 +1 =0$, which yields $x = \pm\sqrt{-1}$. Using your manipulation above would yield $x^2 = 1$ which clearly does not solve the equation.

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You may only apply $\sqrt{a} \sqrt{b} = \sqrt {ab}$ when $a,b \ge0$.

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There are many ways to show that your second equality is incorrect. Just for the sake of argument and a refreshing change of pace, suppose your point is true. That is, suppose $i^{2} = 1$. Then \begin{align} (x + i)(x-i) = x^2 - i^{2} = x^{2} - 1. \end{align} Use the Descartes Rule of signs to derive a contradiction. Hint: $i$ is not real. Can you finish the line of reasoning and derive an absurdity?

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    $\begingroup$ Isn't a quicker absurdity $-1 = i^2 = 1$? $\endgroup$
    – user92843
    Jul 3, 2011 at 9:33
  • $\begingroup$ Hmm. if $1+i=0$ and $1-i=0$ then you can set them equal to each other. So $1+i=1-i$ which means $i=-i$. But I'm not sure why you conclude $(1+i)(1-i)=0$. $\endgroup$
    – Greg
    Jul 3, 2011 at 9:33
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    $\begingroup$ @user02138: The OP is assuming $i$ is the square root of $-1$ in the derivation. If you're assuming nothing about $i$ then there's no contradiction, since for all you know $i$ could equal 1. $\endgroup$
    – user92843
    Jul 3, 2011 at 9:46
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In general,

$\forall a,b \in \mathbb{R} , ab\geq 0, (\sqrt{ab} =\sqrt{ |{a}| }\sqrt{|b|}) $

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