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This question arose in a conversation with a teacher who was introducing square roots to her students.

I know from the website Earliest Uses of Symbols of Operation that the symbol $\sqrt{}$ has its origin in medieval times. But I have not found why the word "radical" is used (in English) for this symbol. The English connotation of "far-reaching" or "extreme" strikes students as odd (and might add to the mystery of roots). Perhaps in other languages there is not an analogous association?

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    $\begingroup$ i believe it comes from "radix" meaning "root" in Latin but my memory is a bit fuzzy, if i can find a source or two to back me up i'll post an answer $\endgroup$ – celeriko May 2 '15 at 12:23
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    $\begingroup$ See the corresponding History section of its wikipage, and their corresponding sources: "A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436–1476). An R was also used for Radix to indicate square roots in Giralamo Cardano's Ars Magna... The symbol '√' for the square root was first used in print in 1525 in Christoph Rudolff's Coss..." $\endgroup$ – Benjamin Dickman May 2 '15 at 15:44
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    $\begingroup$ More generally, see Florian Cajori's text A History of Mathematical Notation (2013). The radical sign is discussed here. $\endgroup$ – Benjamin Dickman May 2 '15 at 15:49
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    $\begingroup$ "Radix" indeed means "root" in Latin (see Google translate). Then the next question is why a solution of an equation, especially $x^n=a$ is called a "root" :-) $\endgroup$ – Alexandre Eremenko May 2 '15 at 22:46
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    $\begingroup$ You could say it's because Galois worked extensively with this symbol, and he was a political radical.... $\endgroup$ – Trogdor May 4 '15 at 1:27
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"Radical" comes from Latin "radicalis", having roots, an equivalent "radix" was also commonly used earlier. While Rudolff did use the radix in 1525 his did not have the overbar on top, now called vinculum, Latin for bond. That innovation was added by Descartes in La Geometrie (1637). Before that all sorts of tricks had to be resorted to for grouping terms "under the root". Even before Rudolff Luca Pacioli in Summa Arithmeticae (1494) would write $\mathcal{RU}$ for "radix universalis", indicating that it applied to everything that follows. Today radix is still occasionally used to refer to the base ("root") of a positional system, the separation point is called the radix point, and fractions of the form $\frac1{b}+\frac3{b^2}-\frac5{b^3}$ are called radix fractions with radix $b$, see Eves's Introduction to the History of Mathematics and Math SE Name for “decimals” in other bases?.

This explains why $x$ in $x^n=a$ is called root or radical, it is the hidden "base" of $a$, and the name likely spread to general polynomial equations by association. To counter the negative connotation it might help to point out to students that "radical" means not generically "far reaching" or "extreme" , but foundational, reaching to the root causes. One may also point out that in medicine, for example, a radical treatment like surgery, aimed at the root causes of an ailment rather than its symptoms, is often the best course of action, if available. And of course in mathematics the radical symbol is a long happy marriage of radix and vinculum, root and bond.

Another Math SE thread How did the square root get its shape? is relevant. The commonly repeated idea that the shape of the radical comes from Latin "r" is Euler's speculation from Institutiones Calculi Differentialis (1775). Cajori, the author of the classic History of Mathematical Notations, is sceptical of it. The resemblance is only there when the vinculum is attached, without it Rudolff's radix resembles "v" more than "r", and the vinculum was attached only over 100 years later. One of the posters offered an alternative speculation, "Arabic letter jeem ج which is the first letter in the Arabic Jathr جذر meaning root".

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  • $\begingroup$ Are you sure about: "This explains why $x$ in $x^n=a$ is called root or radical, it is the hidden "base" of $𝑎$"? According to Jeff Miller, already al-Khwarizmi used the word "root", while the notation with exponents appeared for the first time with Vieta or Descarte, according to Cajori. $\endgroup$ – Michael Bächtold Apr 29 '19 at 4:52
  • $\begingroup$ @MichaelBächtold There is no explicit reference that I know of, the inference is from the timeline. Al-Khwarizmi, or rather his Latin translators, only apply "radix" to fractional powers, "res" or "cosa" are used for solutions of equations more generally. Only later "root" is extended to more general equations. But you are right, it is not tied to the notation. $\endgroup$ – Conifold Apr 29 '19 at 5:33

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