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In secondary school the most interesting idea before meeting complex numbers were the square root $\sqrt{ \phantom x}$ . Did anyone here have any idea about its origin and why they had to invent it?

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    $\begingroup$ The origin of $\sqrt{\phantom{a}}$ is probably related to the Pythagorean theorem, for obvious reasons. $\endgroup$ – Jack D'Aurizio Nov 15 '17 at 14:16
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    $\begingroup$ Quite natural: given a number $a$, what is the number $b$ that multiplied by itself produces $a$ ? I.e. $b \times b=a$ $\endgroup$ – Mauro ALLEGRANZA Nov 15 '17 at 14:17
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    $\begingroup$ And it is very natural, once a set of numbers closed with respect to $+$ and $\cdot$ has been introduced, to wonder what the inverse map of $x\mapsto x^2$ really is (and where it can be well-defined). $\endgroup$ – Jack D'Aurizio Nov 15 '17 at 14:18
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    $\begingroup$ Be careful! Functions and binary operators are a much later formulation of those principles. Historical questions require some sort of reference or a well founded theory based historical accounts. $\endgroup$ – String Nov 15 '17 at 14:21
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    $\begingroup$ The symbol $\sqrt{\text{ }}$ comes from a modification of "r". In Algebra books of Italian mathematicians of 1550 they used to write r.q.2025 instead of $\sqrt{2025}$ $\endgroup$ – Raffaele Nov 15 '17 at 16:13
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Here is the Babylonian method for finding square roots by hand https://blogs.sas.com/content/iml/2016/05/16/babylonian-square-roots.html But not the idea only the calculation

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The Babylonians knew how to solve some quadratic equations.

The Greek geometers knew the Pythagorean theorem. They knew there is no rational number whose square is $2$.

So both of these mathematical cultures knew some useful (to them) things about squares, and about whether particular quantities were squares. Whether they had the concept of "extracting a square root" is a question for historians of mathematics. I'm pretty sure that even if they did they didn't have a symbol for it, or think of it in contemporary terms as a function.

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    $\begingroup$ They knew what amounts to $\sqrt 2$ not being a rational number. But they regarded it slightly differently, namely as the side and the diagonal of a square being geometrical entities that were incommensurable, or in other words, a proportion of geometrical entities not expressible in terms of proportions of numbers (which to them would have meant the natural numbers only). $\endgroup$ – String Nov 15 '17 at 14:26
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    $\begingroup$ @String True. But they did find square roots geometrically. Euclid proves (somewhere) that (in modern terms) the altitude on the hypotenuse of a right triangle is the geometric mean of the segments into which it divides the hypotenuse. $\endgroup$ – Ethan Bolker Nov 15 '17 at 14:32
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    $\begingroup$ Cool! Yes, I am confident they knew techniques that are comparable to extracting square roots in modern terms. Also I am not a historian although I have had some training in the history of mathematics, so I am quite interested in reading what people that are more knowledgable than me in this subject have to say. I find it fascinating that Euclid, working in geometry, developed such important parts of number theory. $\endgroup$ – String Nov 15 '17 at 14:37
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    $\begingroup$ In this famous Babylonian tablet 1800 BC they computed the square root of $2$ up to six decimal places en.wikipedia.org/wiki/Babylonian_mathematics#/media/… $\endgroup$ – Raffaele Nov 15 '17 at 16:20
  • $\begingroup$ @String I had the impression that Euclid's Elements was about mathematics, not just geometry. It just happens that geometry dominated it, and also geometry makes prettier pictures. The proof that there are infinitely many primes is in there too. $\endgroup$ – David K Dec 1 '17 at 16:14

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