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In the Mathematics Educators Stack Exchange, a poster asked whether $\sqrt 4$ could be considered as both $+2$ and $-2$. However, in modern mathematics, $\sqrt 4$ is conventionally defined as $+2$. I thought that exploring the history of mathematics could help answer their question: who decided to choose the positive value as the standard definition of the square root, when did this convention become established, and why was this choice made?

This might seem like a naive question, but it's actually quite common among students and those learning mathematics.

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    $\begingroup$ I believe square roots already appeared in classical Greek mathematics, which did not have negative values at all. $\endgroup$ Commented Mar 29 at 20:49
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    $\begingroup$ And why is there a vote to close? I find it a fair question for the site $\endgroup$ Commented Mar 31 at 5:15
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    $\begingroup$ Is it a "convention" that only the positive case is considered the answer? Even in elementary algebra, I was always taught that the square root of √4 is ±2. I don't recall anywhere in my education where just +2 was considered to be the correct (or conventional) answer. $\endgroup$ Commented Mar 31 at 20:47
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    $\begingroup$ FWIW, "Square root defined as a positive value" exception: IEEE754 defines the sqrt(negative zero) as a negative: negative zero. $\endgroup$ Commented Apr 1 at 9:04
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    $\begingroup$ Would it be more accurate to say "standard definition of the square root symbol" (radical symbol) or "standard definition of the term 'square root'"? Those might have different answers. $\endgroup$ Commented Apr 2 at 0:43

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This answer only provides a rough outline, as I do not possess comprehensive knowledge of the history of European mathematics, let alone the global history of mathematics. The gist of the development process seems to have been be that square roots were exclusively positive for three millenia, then the square root began to be considered as a multi-valued operation for several centuries, until we mostly focused back on the principal branch at some time during the 20th century.

The Babylonian clay tablet YBC 7289 dated to about 1700 B.C. provides the first known evidence of a square root computation, giving $\sqrt{2}$ to three sexagesimal places: $1;24,51,10$. The exact process by which this result was computed is not known. The Greeks likely learned of square roots from the Babylonians. Hippasus of Metapontum (5th century B.C.) is credited with proving that $\sqrt{2}$ is irrational. Book X of Euclid's Elements (3rd century B.C.) demonstrates how to manipulate expressions involving square roots, although on a purely geometric basis. In the 1st century A.D. Heron of Alexandria gave a numerical method for computing square roots, likely of much older origin. Up to this point there was no notion of negative numbers in mathematics, thus all square roots were positive by default.

The Chinese book The Nine Chapters on the Mathematical Art from the 1st century A.D. contains the first known appearance of negative numbers, a concept further developed by the Chinese mathematician Liu Hui in the 3rd century. But it would be a long time before the concept was introduced into European mathematics. The 15th century French mathematician Nicholas Chuquet is credited with the first use of negative numbers, limited to exponents only [4].

From what I could find, Simon Stevin's L'Arithmetique (1585), contains the first examples of negative roots in the history of European mathematics, in the context of quadratic equations. One example is found in article II on p. 333, where he shows that $x^{2}=4x+21$ has a solution $-3$. The first direct presentation of the square root as a multi-valued operation that I could locate is in Colin MacLaurin's A Treatise of Algebra, published posthumously in 1748. On p. 43:

It appears from what was said of Involution, that "any Power that has a positive Sign may have either a positive or negative Root, if the Root is denominated by any even Number." Thus the square Root of $+a^{2}$ may be $+a$ or $−a$, because $+a \times +a$ or $−a \times −a$ gives $+a^{2}$ for the Product.

It should be noted that Indian mathematics had reached the equivalent notion by the 12th century. Henry Thomas Colebrooke (tr.), Algebra, with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bháscara. London: John Murray 1817, p. 135 quotes from the Bījagaṇita (Algebra) of Bhāskara II (c. 1114-1185) as follows:

The square of an affirmative or of a negative quantity is affirmative; and the root of an affirmative quantity is two-fold, positive and negative. There is no square-root of a negative quantity: for it is not a square.

I was able to trace the presentation of the square root as a multi-valued operation through German textbooks [1,2,3] from 1768 right to the end of the 19th century, which suggests that this was a commonly and consistently held notion. J. Brunotte, Lehrbuch der Arithmetik und Algebra für Gymnasien, Real- und Handelsschulen. Nürnberg 1894, p. 78:

Da sowohl $(+a)^{2}=a^{2}$ als auch $(-a)^{2}=a^{2}$, so folgt $$\sqrt{a^2}=\begin{cases} +a \\ -a \end{cases} \ \ \text{ ebenso } \sqrt{4}=\begin{cases} +2 \\ -2 \end{cases} \ \ \ \ \sqrt{\frac{64a^{4}}{49x^{2}}} = \pm \frac{8a^{2}}{7x} $$

d.h. jede Quadratwurzel gibt zwei, nur dem Vorzeichen nach entgegengesetzte Werte, ist also im allgemeinen zweideutig. In den folgenden Untersuchungen und Rechnungen werden jedoch nur die absoluten Werte in Betracht gezogen werden.

This notes that the square root operation is multi-valued, but advises the reader that the further text will consider the absolute values only.

Google Books does not constitute a good source for literature past the 1920s, presumably due to copyright issues, so I have had no luck in tracing the presentation of the square root in textbooks for middle schools or high schools any further. I do not recall what happened in my own math classes, but located among my books one widely used mathematical textbook from Germany (Lambacher-Schweizer) from 1981, where the "Quadratwurzelfunktion" (square root function) is defined as $\mathbb{R}_{0}^{+} \mapsto \mathbb{R}_{0}^{+}$, that is, covering the principal branch only.

Some commenters under this question have opined that the focus back on the principal branch was a consequence of the introduction of electronic calculators and programming languages for computers (in particular since about 1970) whose $\sqrt{}$ keys and sqrt() math library functions are limited to that. I consider that plausible, but not conclusive evidence.

[1] Heinrich Wilhelm Klemm, Mathematisches Lehrbuch. Stuttgart 1768, p. 127: "Eine jede quadratische Gleichung hat 2 Wurzeln, dann von $x^{2}$ kann $+x$ und $−x$ die Wurzel seyn."

[2] Joseph Melchior Danzer, Mathematisches Lehrbuch. Erster Teil. Munich 1781, p. 66: "Alle Wurzeln von geraden Exponenten können also zweyerley seyn, positiv und negativ. $\sqrt{9}=\pm 3$. Die Wurzeln von ungeraden Exponenten aber haben nur das Zeichen, welches ihre Würde hat. Z.B. $\sqrt[3]{−8}=−2$."

[3] Franz Minsinger, Lehrbuch der Arithmetic und Algebra. Augsburg 1832, p. 100: "Das Ausziehen der Quadratwurzel aus Zahlen (ganzen, gemischten und gebrochenen) ist bekanntlich das Zerfällen der gegebenen Zahlen in zwei gleiche Faktoren, daher die gegebene Zahl stets positiv sein muß, die Wurzel aber positiv und negativ sein kann. [...] z.B. $\sqrt{81}=\pm 9$

[4] Chuquet used $\bar{\mathrm{p}}.$ for $+$ and $\bar{\mathrm{m}}.$ for $-$. One of his examples is the multiplication of $8^{3}$ by $7^{1 \bar{\mathrm{m}}.}$ resulting in $56^{2}$, meaning (I think) $8x^{3} \cdot 7x^{-1} = 56x^{2}$.

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As this question has already been validated I will only append a bit to other responses by converting my comment into an answer:

Augustin-Louis Cauchy is known for having modernised the concept of functions and continuity. In the preliminary chapter of his Cours d'analyse from 1821, he introduces the double radix (surd) √√$x$ and the full radix $\sqrt{x}$ (p.7), such that $$\text{√}\!\!\!\text{√}x=\pm\sqrt{x}$$ then when discussing functions (pp.20-21) in his first chapter he implies that only $\sqrt{x}$ is a (explicit) function (while the double radix is what he calls an implicit function).

He says (p.8),

A l'aide de ces conventions, on évite la confusion que pourrait entraîner l'emploi de signes dont la valeur n'aurait pas été déterminée d'une manière assez précise.

(Google Translate) By using these conventions, we avoid the confusion that could result from the use of signs whose value has not been determined in a sufficiently precise manner.

(Gallica link here)

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  • $\begingroup$ With this Cauchy establishes (or at least popularizes) the modern convention of having $\sqrt{x}$ be singled valued (rendering the nonnegative root). The ambiguity of the historical √x is avoided, and confusion with that historical notation is avoided by using the full radix. If this is indeed the introduction of the full radix (with overbar, e.g. $\sqrt{x}$) notation, then he is defining it as the single nonnegative root, and thus requiring either √√ or ± to specify both roots, thus answering the question. $\endgroup$ Commented Apr 5 at 5:58
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Not an answer, just a small (but maybe influential) data point, which njuffa probably also looked up:

Euler in his "Elements of Algebra" ~1765 seems somewhat inconsistent:

Before introducing the notation $√$ he declares that square roots can be positive or negative:

  1. What we have said in the preceding chapter amounts to this: that the square root of a given number is that number whose square is equal to the given number; and that we may put before those roots either the positive or the negative sign.

He then introduces the $√$ operation without being explicit as to whether it is multivalued:

  1. […] a particular sign has been agreed on to express the square roots of all numbers that are not perfect squares ; which sign is written thus $√$ and is read square root. Thus, $√ 12$ represents the square root of $12$, or the number which, multiplied by itself, produces $12$ ; and $√ 2$ represents the square root of $2$ ; […] and, in general, $√ a$ represents the square root of the number $a$.

Somewhat later, it seems like he considers $√ a$ to be only positive:

  1. When the number before which we have placed the radical sign $√$, is itself a square, its root is expressed in the usual way; thus, $√ 4$ is the same as $2$ ; $√ 9$ is the same as $3$; $√ 36$ the same as $6$.

But a bit later he treats it as multivalued:

  1. We have before observed, that the square root of any number has always two values, one positive and the other negative; that $√4$, for example, is both $+2$ and $-2$, and that, in general, we may take $√ a$ as well as $-√ a$ for the square root of $a$.

According to Cajori, the radical sign $√$ was first introduced 1525 in Die Coss by Christoff Rudolff like this "In this algorithm the square root, for reasons of brevity, will be denoted with the character √ , so √4 is the square root of 4." but Rudolff's book does not seem to consider negative numbers, so the question of multivaluedness is obsolete.

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More modern commentary can be found in

  • Dieudonne, J. (1971) Infinitesimal Calulus. Hermann.

on p.223f:

This situation may seem shocking; however it is no remedy to say, as unfortunately many books do on this problem, that we can define a so-called “multivalued function” as the inverse of the function $z > \rightarrow z^2$ by admitting that this so-called “function” has two values at each point $\neq 0$. Such a “definition” is mere verbiage, for the authors take good care not to give the slightest rule for calculating with these new mathematical objects which they pretend to define. This of course makes the so-called definition useless; it is easy to see why they avoid this problem, since by admitting two values, between which no distinction is made, for the so-called function $\sqrt z$, one is compelled to admit four values for $\sqrt z + 2\sqrt z$, eight values for $\sqrt z + 2\sqrt z + 4\sqrt z$, and so on, which makes all calculations impossible.

There is in fact a solution to the paradox of “multivalued functions”, the profound and powerful theory of “Riemann Surfaces”, which is beyond the level of this book.

This is then brought into the context of computation and calculators by

Hubbard, J. H., & Hubbard, B. B. (2009). Vector calculus, linear algebra, and differential forms: a unified approach. Matrix Editions.

Refering to Dieudonne above they claim that computers have vindicated Dieudonne's perspective (p. 11):

Remark. In the past, some textbooks spoke of "multi-valued functions" that assign different values to the same argument; such a "definition" would allow arcsin to be a function. In his book Calcul Infinitesimal, published in 1980, the French mathematician Jean Dieudonne pointed out that such definitions are meaning less, "for the authors of such text s refrain from giving the least rule for how to perform calculations using these new mathematical objects that they claim to define, which makes the so-called 'definition ' unusable."

Computers have shown just how right he was. Computers do not tolerate ambiguity. If the "function" assigns more than one value to a single argument, the computer will choose one without telling you that it is making a choice. Computers are in effect redefining certain expressions to be functions When the authors were in school, $\sqrt 4$ was two numbers, $+2$ and $-2$. Increasingly, "square root" is taken to mean "positive square root" because a computer cannot compute if each time it lands on a square root it must consider both positive and negative square roots.

As an ahistorical comment, it seems to me that Dieudonne's example does not actually lead to a cascade of numbers (more "independent variables" could lead to some though). I would also contest the notion that a computer cannot compute pairs (or sets) of results...

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    $\begingroup$ I also don‘t think that Dieudonne or Hubbard have a real argument here. They could instead just admit that they don’t know how to formalize the idea in the current foundations of mathematics. But either way, this suggest that the shift to single valued roots was adopted after the focus of mathematicians on modern functions and set theoretic foundations after 1930. $\endgroup$ Commented Mar 31 at 18:31
  • $\begingroup$ @MichaelBächtold While it is clear enough from those quotes that neither Dieudonne nor Hubbard find the prospect of such formalization as having any mathematical interest, it is rather amusing to assert that don't know how to formalize such an idea. $\endgroup$ Commented Apr 16 at 19:01
  • $\begingroup$ @LeeMosher to me it’s clear that the way Euler et. al. thought of this multivaluedness does not correspond to the „set valued formalisation“ that Dieudonne seems to have in mind (nor to Riemann surfaces). In that sense I believe these modern commentators are missing the point. And dismissing it as „not mathematically interesting“ is maybe the easiest way to get rid of the problem. $\endgroup$ Commented Apr 16 at 21:41
  • $\begingroup$ @LeeMosher As to why I think this is an interesting problem: have a look at discussions on mse, matheducators etc., trying to make sense of indefinite integrals (see, for instance, matheducators.stackexchange.com/q/2338/590). The lack of a formalization of "multivaluedness" in its original sense appears to be the cause of these difficulties. $\endgroup$ Commented Apr 17 at 7:56

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