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I was curious to know more on the history of the term "root", in the sense of "a value that results in a true statement, when substituted into an equation" (e.g., the roots of $2x^2 + 1= 9$ are $\pm2$).

I saw this question, which is also about the origin of "root", but in the sense of a zero (i.e., rearranging the previous example into the form $2x^2 - 8 = 0$). I'm guessing the answer is the same, although, based on dictionary & encyclopaedic entries, it seems the "valid value" meaning predates the "returns zero" meaning.

Interestingly, Wolframalpha does not even list the "returns zero" meaning in its 14 definitions for "root", despite the fact that Mathematica's "root" function is related to that sense of the word.

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I think there are two separate issues here: the origin of "root" and of the custom to arrange polynomial equations in descending powers equated to zero. The latter was initiated by Stifel and made standard by Descartes's La Geometrie (1637), which demonstrated its utility by bringing out relations between equations, roots, and factoring, see Manders, Algebra in Roth, Faulhaber, and Descartes. Prior to that the custom was to arrange equations so as to avoid negative coefficients, as even Stifel still called negatives numeri absurdi, see Scott, The Scientific Work of René Descartes:

"Stifel, in his Arithmetica Integra (Nurnberg, 1544) introduced the practice of arranging all the terms of an equation in descending powers of the unknown and then equating them to zero. In this he was followed by Descartes, and after him the practice became increasingly common... Where possible, Stifel avoided negative roots, which he called numeri absurdi."

The notion of "root" as solution to an equation was simply transferred to this form of writing them, so it has little to do with the "root" as such, as it was previously transferred to general equations from the special case $x^n=a$, which explains the name, see Why is the radical symbol √ called "radical"? The word, in its Arabic version jathr, goes back to al-Khwarizmi himself, his famous treatise on algebra (c. 820). Robert of Chester, in his Latin translation Liber algebrae et almucabola (1145), translated jathr as radix. There is even speculation that the first letter in Arabic jathr gave rise to the symbol for root introduced by Rudolff in Europe c. 1525. Recorde in The Whetstone of Witte (1557) translated radix as root, but only as in square root. Although al-Khwarizmi occasionally calls unknowns in equations "jathr" too, see MathForum, this extension did not take in Europe until 17th century; res or cosa were more commonly used.

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  • $\begingroup$ Thank you so much for the incredible answer: so many interesting references provided -- I look forward to reading them! Thank you for your time and for sharing your knowledge. $\endgroup$
    – Rax Adaam
    Nov 13, 2021 at 15:46
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    $\begingroup$ Just to complicate things a little, the English word for the root vegetable "radish" derives from the Latin word "radix", which means "root". $\endgroup$ Nov 14, 2021 at 1:28
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    $\begingroup$ And this reminds me of an old cartoon showing several square-root symbols with triangles on top of them. The caption was "radicals supporting change". $\endgroup$ Nov 14, 2021 at 1:30

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