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I was asked to find $\sqrt{-121}$.

$11i$ is the right answer, of course, but I'm wondering why $-11i$ is not an answer.

If I square it,

$(-11i)^2 = (-11)^2 \times i^2 = 121 \times (-1) = -121$

So, $-11i$ also seems to be an answer.

Where am I mistaken here?

A.G.
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user67275
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  • Does your \sqrt mean principal square root or multivalued? – peterwhy May 05 '24 at 00:10
  • @H.sapiensrex This is misleading without specifying the context. In the context of functions, both $\sqrt{x}$ and $x^{1/2}$ require a choice of branch. It's true that $\sqrt{x}$ may imply a canonical choice of branch for some people while $x^{1/2}$ may not, but unless we enter the realm of multivalued functions (uncommon, as most people don't know how to deal with them without paradoxes), that just means that $x^{1/2}$ might be ambiguous, not that it's both solutions. – Brian Moehring May 05 '24 at 00:20
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    @H.sapiensrex Your convention on $\sqrt x$ is only for positive real numbers. It does not apply to complex numbers, so your comment is mis-leading. – Kavi Rama Murthy May 05 '24 at 00:22
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    $11i$ is a square root and $-11i$ is another square root. All non-zero complex numbers have two square roots. If $w$ is one of the square then $-w$ is always the other. The question is which one is the "principal" square root is another question altogether. And that is completely arbitrary. We had a meeting and we all voted and we chose that $11i$ is the principal and $-11i$ is the secondary. – fleablood May 05 '24 at 00:27
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    You know. It's not really fair when a novice has an earnest question about a confusing and ambiguous issue and we then complain that the question doesn't take into account all the distinctions that an advanced student would know to explain the ambiguity but which the novice would never have heard of and that's exactly why they are asking the question in the first place.... "Does your \sqrt mean principal square root or multivalued" I don't think it's reasonable for us to assume the OP is experienced enough to even know what this is talking about. (But the reference to the link IS good.) – fleablood May 05 '24 at 00:32
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    The thing is for a positive real number $w$ and natural (positive, integer) $k$ there is a unique positive real number $c$ so that $c^k = w$. This is so useful to us to refer the the unique and positive number where $c^k =w$ that we gave it the name THE $k$-th root of $w$. But this assumes two things. 1) $w$ is a positive real and 2) we are only interested in the positive answer that works. If $k$ is even there is also a negative answer and there are also complex answers as well. When we talk of THE square root of positive $w$ we are not interested in any of the others. TBC... – fleablood May 05 '24 at 00:44
  • ... cont... it gets less clear cut when $w$ isn't positive (or isn't real). For $x^k=w$ there are $k$ different answers but there isn't such a very strong incentive to pick one of them and say "There, that one-- that one is THE $k$th root". We still do it but it feels more arbitrary and theres some variation. It's easier to say $i$ is more "basic" a square root than $-i$ (although both are square roots) but what about the $5$th root of $-1$. Should it be $-1$ or should it $0.309 + 0.951i$? (Both to the 5th power is $-1$) – fleablood May 05 '24 at 00:53
  • @peterwhy - I guess it comes down to whether or not you thought the OP would be able to answer that question? Or if you hadn't considered that issue before, what would think now, in light of your last comment, which points to the idea that the OP came here from a confusing social network post? – JonathanZ May 05 '24 at 02:36
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    @JonathanZ I deleted my reply comment, as that seemed to give an impression that I think "the OP came here from a confusing social network post". I asked about such possible confusion on the symbol and "principal" long ago, without targeting this OP or the latest social networks. – peterwhy May 05 '24 at 02:51
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    The latest revision changed the title from "Square-root of -121" to "the square root of -121". Did that change the intention of the OP? I don't know. Wikipedia describes "the principal square root" also as simply the square root (with a definite article). – peterwhy May 05 '24 at 03:02
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    My own feeling is that whoever posed the question stated by the OP has not really taken into their heart that $-11i$ is equally correct. It drives me nuts (not really; it's an expression) when people say $i$ is "the" square root of $-1$. – user43208 May 05 '24 at 03:11
  • "My own feeling is that whoever posed the question stated by the OP has not really taken into their heart that −11i is equally correct. It drives me nuts (not really; it's an expression) when people say i is "the" square root of −1. " It always saddens me a little to see an earnest and sincere OP try to "play polite" believe they are trying to abide by the rules and say things like $11i$ is "the right answer" and "of course" and "where am I mistaken". $11i$ is not "THE" right answer, there nothing "of course" about it, and the OP isn't mistaken anywhere. – fleablood May 05 '24 at 13:54
  • These comments just show that there are different conventions. E.g. how Wikipedia would define the $\sqrt~$ symbol as the principal square root function that prioritises the square root with argument in $(-\pi/2, \pi/2]$, and still some can have another choice; or maybe the choice is too arbitrary and $\sqrt~$ should be left undefined for non-(non-negative reals); or maybe $\sqrt~$ should mean square roots (no definite article) and not just the principal square root (whichever it means). TBC – peterwhy May 05 '24 at 21:25
  • "we then complain that the question doesn't take into account all the distinctions [...]" I trust that the OP is earnest and sincere. My first comment question only requested clarification on the convention that the OP (or their instructor) is using, and I hope that was not perceived as complaining or unfair. The post even had \sqrt(-121), not $\sqrt{-121}$. But to @user67275, any clarification, even one that says you are not able to answer, would be helpful here. – peterwhy May 05 '24 at 21:25

1 Answers1

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$\sqrt{x}$ for $x>0$ implies the positive root.

For $x<0$ the convention is to choose the complex number with the positive imaginary part.

So if you were to write $\sqrt{-121}$ with no clarification, the convention would be to say this is $11i$.

But as you say, $(-11i)^2=-121$ as well, just as $(-11)^2=121$ but we say that $\sqrt{121}=11$.

Red Five
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  • Nice so far as it goes but maybe we should go further and explain about which is the principal square root of general complex number of $a+bi$ (If $c_1,d_1,c_2,d_2$ are such that $c_1=-c_2, d_2=-d_1, c_1^2-d_1^2=c_2^2-d_2^2=a$ and $2c_1d_1=2c_2d_2=b$ so that $(c_1+id_1)^2 =(c_2+id_2)^2=a+bi$ and $c_1+id_1=-(c_2+id_2)$ which one is the principal square root? $c_1+id_1$ or $c_2+id_2$ and why?) And maybe we should touch on determining for $k$th powers which of the $k$ roots should be considered the principal $k$th root of a value. – fleablood May 05 '24 at 13:47