I was wondering when in history did people start use the $\sqrt{}$ sign for square root, what did they use before, and why this curious nomenclature is adopted.
[Math] How did the square root get its shape
math-historynotation
Related Solutions
The "root" of "square root" is from latin radix.
From Florian Cajori, A history of mathematical notations (1928), page 361 of I vol of Dover reprint :
The principal symbolisms for the designation of roots, which have been developed since the influx of Arabic learning into Europe in the twelfth century, fall under four groups having for their basic symbols, respectively, $R$ (radix), $l$ (latus), the sign $\surd$, and the fractional exponent.
The sign $R$; first appearance.-In a translationa from the Arabic into Latin of a commentary of the tenth book of the Elements of Euclid, the word radix is used for "square root." The sign $R$ came to be used very extensively for "root," but occasionally it stood also for the first power of the unknown quantity, $x$. The word radix was used for $x$ in translations from Arabic into Latin by John of Seville and Gerard of Cremona. [...]
With the close of the seventeenth century [the symbol $R$] practically passed away as a radical sign; the symbol $\surd$ gained general ascendancy.
See page 366 :
Origin of $\surd$.-This symbol originated in Germany. L.Euler guessed that it was a deformed letter $r$, the first letter in radix [see page 213 of II vol : L.Euler says in his Institutiones calculi differentialis (Petrograd, 1755), p.100 : "in place of the letter $r$ which first stood for $radix$, there has now passed into common usage this distorted form of it $\surd$."]
This opinion was held generally until recently. The more careful study of German manuscript algebras and the first printed algebras has convinced Germans that the old explanation is hardly tenable. [...] The oldest of these is in the Dresden Library, in a volume of manuscripts which contains different algebraic treatises in Latin and one in German. [...] They [the main facts found in the four manuscripts] show conclusively that the dot was associated as a symbol with root extraction.
Christoff Rudolff was familiar with the Vienna manuscript which uses the dot with a tail. In his Coss of 1525 he speaks of the Punkt in connection with root symbolism, but uses a mark with a very short heavy downward stroke (almost a point), followed by a straight line or stroke, slanting upward. As late as 1551, Scheubel, in his printed Algebra, speaks of points.
See page 144 :
In 1553 Stifel brought out a revised edition of Rudolff's Coss. Interesting is Stifel's comparison of Rudolff's notation of radicals with his own, and his declaration of superiority of his own symbols. We read: "How much more convenient my own signs are than those of Rudolff, no doubt everyone who deals with these algorithms will notice for himself. But I too shall often use the sign $\surd$ [...]."
Added
From John Fauvel & Jeremy Gray (editors), The History of Mathematics: A Reader (1987).
Page 229 :
Al-Khwarizmi (ca.780 – ca.850) : "A square is the whole amount of the root multiplied by itself."
Page 250 :
Luca Pacioli (1445–1517) regarding quadratic equations : "Now we must see in how many ways they can be made equal, one to the other, and the other to the one, and two of them to one of them, and one to two of them. On this I say that they can be made equal to each other in six ways. First, the square to the things. Second, the square to the numbers. Third, thing or things to numbers. [...]"
Page 260 :
Gerolamo Cardano (1501 – 1576), Ars Magna, on the cubic equations : "Cube the third part of the number of 'things', to which you add the square of half the number of the equation, and take the root of the whole, that is, the square root, which you will use, in the one case adding the half of the number which you just multiplied by itself, in the other case [...]."
Finally :
René Descartes (1596 – 1650), Geometry, page 299 of 1637 edition (see Dover reprint) : "if I wish to extract the square root [racine quarrée] of $a^2+b^2$, I write $\sqrt{a^2+b^2}$; if I wish to extract the cube root [racine cubique] of $a^3 - b^3 +abb$, I write $\sqrt{C.a^3 - b^3 +abb}$ [...]"
Here is a short answer to the question in the title of OP:
Well, if we don't do so, what could a better alternative be?
What is the notation $\sqrt{}$?
The confusion seems to be from understanding of the notation $\sqrt{}$. When writing, for instance $\sqrt{16}$, one pronounces it as "square root of $16$". However, what one really means is "the principal square root of $16$".
Let's go back to the definitions. A square root of a real number $a$ is a number $y$ such that $y^2 = a$; in other words, a number $y$ whose square is $a$. For example, $4$ and $−4$ are square roots of $16$ because $4^2=(-4)^2=16$. Note carefully that the notation $\sqrt{}$ is not involved in this definition at all.
Now, for every given positive real number, say $16$ again, there are two "square roots" (note carefully again that we don't write $\sqrt{x}$ for "square roots of $x$" yet) of it. What if one wants specifically to refer to the positive one? Instead of explicitly saying "I'm refering to the positive square root of $16$", one uses the notation $\sqrt{}$ to define $\sqrt{16}$ as the positive square root of $16$. Here comes the notation $\sqrt{}$. Of course you are losing "information" when you write $\sqrt{16}$ to mean "the positive square root of $16$". Because it is by definition so. What does one do for the "lost information"? One naturally has $-\sqrt{16}$ as the negative square root of $16$.
One can put two definitions together to see what is really going on:
A "square root" of a real number $a$ is a number $y$ such that $y^2=a$;
Given a positive real number $x$, the notation $\sqrt{x}$ is defined as a positive real number $y$ such that $y^2=x$. And in this case, we write $y=\sqrt{x}$.
Why is $\sqrt{}$ defined in the way above?
If one does not define $\sqrt{a}$ as the positive square root of $a$ and instead as the "square roots of $a$", then one would have $\sqrt{16}=\pm 4$. Now how would you write the answer to the following question?
What is the positive real number $x$ such that $x^2=\pi$?
[Added: ]Compare the following two possible definitions for the notation $\sqrt{}$:
- I. For any positive real number $a$, define $\sqrt{a}$ as the square roots of $a$;
- II. For any positive real number $a$, define $\sqrt{a}$ as the positive square root of $a$;
Now, if one uses definition I, then $\sqrt{16}=\pm4$. With this definition, you have perfectly what you might want: $$ x^2=16\Rightarrow x=\pm 4;\quad\text{and }x=\sqrt{16}=\pm4. $$
If one uses definition II instead, on the other hand, one would have $\sqrt{16}=4$.
You might be happier with definition I and ask why on earth one prefers II. Here is "why". Suppose you are asked to solve the following problem.
Find the solution to the equation $x^2-\pi=0$ such that $x>0$.
If one uses definition II, then one immediately has $x=\sqrt{\pi}$.
Now if one uses definition I, $x=\sqrt{\pi}$ would be the WRONG answer.
One more lesson from Terry Tao:
It’s worth bearing in mind that notation is ultimately an artificial human invention, rather than an innate feature of the mathematics one is working on; sometimes, two writers happen to use the same symbol to denote two rather different concepts, but this does not necessarily mean that these concepts have any deeper connection to them.
Best Answer
You can find information about the history of the usage in Jeff Miller's page here: Earliest Uses of Symbols of Operation.
Quoted: