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I remember (rightly or wrongly) learning at school in the 80s that the square root of a number had two roots, positive and negative. For example, √25 = ±5. That is, the square root as I learnt it was quite literally the reverse operation of the square: both 5² and (−5)² are equal to 25, so it followed the roots of 25 were 5 and −5. And that seemed entirely logical.

Some years later, however, I encountered the concept of the Principal Square Root, and was told that by definition the square root function, √, referred only to the positive root. That posed no problem mechanically; I just remembered the definition as a rule, and it was enough to get me through engineering and computing degrees. However, the justification always pestered me: if we know logically that two roots exist, why was this function definition of a square root introduced?

I have since heard from others of my era and before, including those with mathematics majors, that they too learnt that the square root function had two roots, positive and negative. I wonder, therefore, whether this is a relatively new (albeit decades-old) convention that has been introduced for some reason.

Any way I look at it, the definition seems to be introducing arbitrary limitations on our calculations: (−5)² = 25 but √25 is only +5. Is there a logical, mathematical justification for this that I am not seeing?

Will Jagy
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POD
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    If you want $\sqrt{\phantom{x}}$ to be a function, it better have only one output for each input. – Randall Nov 25 '24 at 17:51
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    The answer as I see it is that both definitions are used (by different people), but the convention that $\sqrt{25}=5$ and $\sqrt{25}\neq-5$ is the less confusing and more useful definition. In the cases where $\pm5$ is the correct answer you can just write $\pm\sqrt{25}$. – David K Nov 25 '24 at 18:00
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    In terms of usefulness, the "principal root" definition lets us specifically name the two square roots of any number: the square roots of $x$ are $\sqrt x$ and $-\sqrt x$. Note that the $\pm$ notation also is problematic: sometimes it means "both values occur" and sometimes it means "either the positive or negative value occurs and not the other, but I don't know which one occurs (or I choose not to say)." – David K Nov 25 '24 at 18:04
  • @Randall : That, I suspect, is actually the answer (and one which I am surprised I did not see immediately). It begs the question, though, must √ be a function, or what purpose does it serve for it to be one? – POD Nov 25 '24 at 18:16
  • @DavidK : My question is very different, as I believe I have made clear if you read my question carefully. I am not asking what the definition is; I am asking whether the justification for that definition is arbitrary or mathematical, and if it is the latter, what that justification is. – POD Nov 25 '24 at 18:20
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    Nope, $\sqrt{}$ need not be a function. But functions are useful in general. And there is a useful function $y=f(x) \iff y^2=x$ and $y \ge 0$. The mathematical notation for this function that the mathematical community uses is $y=\sqrt{x}$. This can be too much for young folks in school first learning about square roots, so usually mathematics teachers do not go into all those formalities at that time. But, by the time you get to a precalulus or calculus course, it's time to grow up and smell the flowers... er... functions. – Lee Mosher Nov 25 '24 at 18:21
  • @DavidK : In response to your second comment, you make a good point, and perhaps that extends on the comment by Randall above, in relation to its reasoning. – POD Nov 25 '24 at 18:24
  • @LeeMosher : That is what I was beginning to suspect, that it is largely a dedactic decision rather than one founded in mathematical logic. I suspect that given the confusion that it often creates amongst students of algebra, it may do more damage than good presented as a "rule". However, that is a philosophical discussion for another time. Thank you for your comment. – POD Nov 25 '24 at 18:30
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    It gets worse. You should look at all the posts on this site regarding principle $n^{\text{th}}$ roots of the complex numbers. I don't think there's any way around this other than to eventually learn what it all means. – Lee Mosher Nov 25 '24 at 18:32
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    @LeeMosher : As is so often the case, it seems to boil down to conventions and definitions, and as a teacher I find it most problematic that the educational establishment settles on conventions that are not necessarily universal, then spends undue effort testing those conventions instead of focusing on the underlying mathematical logic and communication. – POD Nov 25 '24 at 18:37
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    There's a convention some people use in complex-number arithmetic where $\sqrt z$ means the principal square root (a function of $z$) and $z^{1/2}$ is a multifunction of $z$ giving both square roots. Then we have to understand what a multifunction is and how to interpret it notationally. – David K Nov 25 '24 at 18:37
  • Thank you, @DavidK. That is a distinction that I had not encountered. – POD Nov 25 '24 at 18:39

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I'll expand my comments into an answer.

First, the literal answer to your final question is Nope! The symbol $\sqrt{}$ need not represent a function. But functions are useful in general. And there is a useful function $y=f(x) \iff y^2=x$ and $y \ge 0$. The mathematical notation for this function that the mathematical community uses is $y = \sqrt{x}$. But these subtleties regarding functions are overkill for young folks in school first learning about square roots, so usually mathematics teachers do not go into all those formalities at that time.

By the time a student gets to a precalulus or calculus course, they encounter concepts of functions, and one-to-one functions, and inverse functions of one-to-one functions. The function $y=x^2$ with domain $\mathbb R$ and range $\mathbb [0,\infty)$ defined by $y=x^2$ is an excellent example to start with: it is onto, but it is not one-to-one, and so it does not have an inverse. However, an amazing thing happens: if you restrict its domain to $[0,\infty)$ then one gets a function that is onto and one-to-one. So its inverse exists. The notation that the mathematical community uses for that inverse function is $y = \sqrt{x}$ (all notation is established by community agreement). You can also restrict its domain to $(-\infty,0]$, and if you do that then you also get a function; and with the meaning of $\sqrt{x}$ already established in the community, you can then go on to derive derive a formula for that new inverse function, namely $y=-\sqrt{x}$.

There is a very concrete logical structure to the community definition of the square root function, and learning that structure is very useful for discovering and applying many, many other useful inverse functions. It is not so hard to teach; I have taught it in precalculus and calculus many times. And it is useful to learn, because there are many, many situations in mathematics where one encounters a function that is not one-to-one on its originally given domain, but which becomes one-to-one on a suitably restricted subset of the domain. This happens, for example, when one defines inverse trig functions $\cos^{-1}(x)$, $\sin^{-1}(x)$, $\tan^{-1}(x)$.

Lee Mosher
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