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When researching square roots I found that $\sqrt{x}$ is the principal square root and $\pm\sqrt{x}$ is the square roots, with the reason for why being given through an example equation by user9464 of:

What is the positive real number $x$ such that $x^2=π$?

The reason given was that without $\sqrt{}$ meaning the principal square root, " $x=\sqrt{\pi}$ would be the WRONG answer."

But this does not seem conclusive as a reason why it is as such when you can simply answer with $x=\left\vert\sqrt{\pi}\right\vert$

Just by adding an absolute value grouping the equation is fixed, so I was wonderring if there was any more conclusive reason other than it messing up the mordern convention of functions?

Arturo Magidin
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BlueT
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    For over 100 years and in well over 100 "regional math locations" in the world, $;\sqrt{;;};$ has represented the nonnegative square root of a nonnegative real number. Before then, the symbol usually denoted both square roots. I think the transition to the single-valued notion was due to a more standardized meaning of "function" around the late 1880s to early 1900s to mean what used to be "single-valued function", as well as an increase in rigor in calculus and analysis textbooks (during the same period of time) that was easier to provide using a single-valued meaning of $\sqrt{;;}.$ – Dave L. Renfro Oct 03 '24 at 20:10
  • "without √ meaning the principle square root, " x=π−−√ would be the WRONG answer."" I don't think its a matter that it'd be a wrong answer so much as it wouldn't be an answer at all. $\sqrt \pi$ is supposed to refer to a single number. If we don't have it refer to the principal (positive) square root it refers to two different numbers at the same time. That isn't what we want a symbol representing a single number to be. – fleablood Oct 03 '24 at 20:14
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    Nobody has mentionned the fact that for a complex $z$, $\sqrt{z}$ is ... a "complex" being... – Jean Marie Oct 03 '24 at 20:17

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You could say it's just a question of convention: $\sqrt x$ is defined to be the non-negative square root of $x$.

But then what would it mean to say that $\sqrt x$ is either the positive square root or the negative square root? Or alternatively to say that $\sqrt x$ is both the positive square root and the negative square root? Either of these alternatives would lead to chaos!

TonyK
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$\sqrt \pi$ if used as a specific real number has to be some real number. There are exactly two real numbers $x,-x$ that when squared have $x^2=(-x)^2=\pi$. One of them, $x$ is postive and the other $-x$ is negative and $|x|=|-x|$.

So which one do we declare is THE number refered to as $\sqrt \pi$? By convention it is the positive $x$.

The problem with declaring $\sqrt \pi = \pm x$ and $x=|\sqrt \pi|$ is that $\pm x$ is NOT a number. It is an indication that there are two numbers that could work but it, itself is not a number. If we want $\sqrt \pi$ to be a meaningful reference to an actually number we must declare that $\sqrt \pi = x$ or $\sqrt \pi = -x$. One or the other. Not both. Not neither. So which one do we choose.

Well, for convenience we hold a meeting and discuss it and we universally all agree that although it is totally arbitrary and we could go either way, we declare that for obvious (it is obvious, isn't it?) convenience we will all agree that $\sqrt \pi = x$. The vote is unanimous and universally accepted.

fleablood
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I do not follow the statement that the non-principal value for $\sqrt{\pi}$ would be the "wrong answer." After all, the only real requirement on $\sqrt{}$ is that if $x = \sqrt{y}$, then $x^{2} = y$. Now, given a particular context, perhaps $x = -\sqrt{\pi}$ could seem absurd, say if taking the square root $\sqrt{\pi}$ should give you the length of the side of a square with area $\pi$.

As to your more general question, it is indeed simply a matter of convention. It very well could've been that mathematicians hundreds/thousands of years ago define the principal square root to give the negative value. However, again, remember one reason it is called the "square" root is that it connects the area of the square to its side length, and the length should definitely not be negative.

DiracComb16796
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