# Where does the mathematical use of the word 'root' come from?

Is there any connection between the mathematical and biological meanings of the word root? Where does the mathematical use come from?

• It's a translation of the latin word radix, which gave us the words radical (a synonym for root in the mathematical sense) . and radish (also a root). – Peter Shor Dec 12 '19 at 11:34
• You can get as far as '[root]: ... Figurative use is from c. 1200. Of teeth, hair, etc., from early 13c. Mathematical sense is from 1550s.' from the Online Etymological Dictionary (and questions may be put on hold if they lack any reasonable research). The transition from 'root of an equation' (ie 'find x') to 'root of the equation "x squared = 64" ' is transparent. But perhaps someone can discover more about the 1200 - 1550 era. – Edwin Ashworth Dec 12 '19 at 11:36
• @Peter Shor OP does not ask about the English origins of the word 'root'. – Edwin Ashworth Dec 12 '19 at 11:39
• @Edwin: The word radix in Latin had both the mathematical and biological meanings of the word root. The reason we use root for the mathematical sense is that we simply translated the Latin word radix to the English word root. So if you want to find the connection between the mathematical and biological meanings, you have to go back to Latin (and maybe Greek before that). – Peter Shor Dec 12 '19 at 11:40
• It's what they were smoking when they came up with the idea. – Hot Licks Dec 15 '19 at 2:58

In short, root is a translation of the Latin word radix, which is itself a mistranslation of the Arabic word jadhr. That word has multiple meanings in Arabic, one of which is indeed root. But the Arabic-speaking mathematicians1 who introduced it used it in its other meaning of 'basis', 'foundation', 'lowest part'.

1I say 'Arabic-speaking' because some of them were non-Arabs writing in Arabic, the lingua franca of their time and place. For example, Muḥammad ibn Mūsā al-Khwārizmī, who is traditionally regarded as the founder of algebra, was Persian. In particular, the word algebra comes from the Arabic word al-jabr, 'balancing', which appears in the title of his best-known work.

Discussion

According to Encyclopaedia Britannica,

In the 9th century, Arab writers usually called one of the equal factors of a number jadhr (“root”), and their medieval European translators used the Latin word radix (from which derives the adjective radical).

But why did the writers of mathematical treatises in Arabic use the word jadhr?

The reason has to do with the distinction between 'concrete' and 'abstract' numbers (see e.g. here), which was apparently very important to the mathematicians of the time:

concrete number: a number referring to a particular object or objects, as in three dogs, ten men
abstract number: a number that does not designate the quantity of any particular kind of thing

Collins Dictionary, here and here

(Nowadays these issues are usually relegated to the sciences and engineering, under the name of dimensional analysis.)

It seems that there was a tradition of thinking, going all the way back to ancient Egypt, that, in order to compute e.g. the area of a rectangle, one cannot simply multiply two abstract numbers denoting the lengths of the sides of the rectangle. Rather, one of these numbers has to first be multiplied by the 'basis' of area (I am not sure if there is any conceptual distinction between this sort of 'basis' and the modern concept of a unit of measurement). It is this 'basis' that Arab-speaking mathematicians referred to as jadhr, the Arabic word which means 'basis', 'foundation', 'lowest part'. However, jadhr also means 'root', and many translators of the Arabic words into other languages (including Latin and Chinese) mistakenly thought that jadhr was being used with that meaning in the texts. Here is the relevant excerpt from a source I will quote more fully below:

Thus the Egyptian computed the square by ﬁrst multiplying the one side (9 Kket) in the square unit. This was the square—basis (jadhr) to be multiplied by the other side, representing the pure number. The same process is also described by al-Khowarizmi who says: “And in every square ﬁgure one of its sides multiplied into the square unit is its jadhr … and we make the other side, namely, hj, three, and this is the number of its jadhr’s.”

S. Gandz, On the Origin of the Term "Root." Second Article,
the American Mathematical Monthly, vol. 35, p. 74, 1928.

Extended discussion

The following is taken from the article On the origin of the term "root" by S. Gandz (The American Mathematical Monthly vol. 33, 261–265, 1926):

The term “root” has its origin in the Arabic. “Latin works translated from the Arabic have radix for a common term, while those inherited from the Roman civilization have latus.”2 Radix (“root”) is'the Arabic jadhr, while latus (Greek, πλευρά, pleura, meaning “rib” or “side”) is the side of a geometric square.

2See Smith, History of Mathematics, vol. II, p. 150.

It is certainly rather strange that such a term as “root” should be used in this connection. It suggests that if the basic number is a root, the square might be a bush, and so on up in a kind of a mathematical garden.

The Chinese, indeed, do use the word kun to mean root, grass, and shrub, and the Hindus also use the word mūla for the root of a plant, but this was very likely due to the Arabic influence, which is so often seen in China and which may have spread into India by way of China. It is, however, a fact that thus far we have no satisfactory explanation as to why this botanical term should have found place in the theory of numbers. …

Therefore the problem before us, as already set forth clearly by Professor Smith, is to to find out whether the medieval Latin authors were correct in their translation of the Arabic word jadhr by radix (“root”). …

The writer therefore started out to investigate the real meaning of the word word jadhr, not depending upon lexicons or the Latin versions of the thirteenth century, but seeking the meaning as it it appears in the manuscripts of the old Arabic mathematicians themselves.

Muḥammad ibn Mūsā al-Khwārizmī (c. 825), is the oldest Arabic mathematician of much prominence, and it was his ’Ilm al-jabr wa’l mnuqabalah that gave to Europe both the name and the foundation principles of algebra. His chapter Bab al Misacha ("on geometry”) begins as follows: “Know that the meaning of the expression ‘one in one’ is an ‘area’; and its meaning is one cubit (in length) in one cubit (in breadth). And every roof of equal sides and angles which has one cubit for each side is a (square) unit. But if it has two cubits for every side and has equal sides and angles, then the whole roof is four times the roof which has one cubit in one cubit. … And in every square roof of equal sides one of its sides (multiplied) in a square unit is its jadhr; or if the same be multiplied in two (square units) then it is like two of its jadhrs, whether this roof be small or great.” …

Jadhr not only means “root” but also “basis,” “foundation,” “lowest part.” Mohammed ibn Musa … begins his chapter on areas by introducing the new notion of a square unit. Then he says that in order to get the area of any ﬁgure we must ﬁrst multiply the one side by the square unit; this is then the basis to be multiplied by the other side. We do not multiply one side by one side, but one jadhr, representing the square basis, by one side representing the number. The same deﬁnition and idea is also to be found elsewhere in his algebra. …

The term [jadhr] does not mean “root,” but “square basis,” that by whose multiplication we get the square area. This was the reason why jadhr was used by later writers, such as Omar Khayyam, as the basic number of a square number. The latter did not know anything more about the original meaning of jadhr, and he used the word loosely for dil‘ (Greek, πλευρά), which means “rib” or “side.” But he, like al-Khowarizmi, still knew that it was a concrete number in opposition to an abstract number.

By the time of Beha Eddin (c. 1600) the original meaning was entirely forgotten. In his Kholdsat al-Hisdb (“Essence of Arithmetic”) he says: “What is multiplied into itself is called jadhr in arithmetic, dil‘ (“side”) in geometry, and shai’ (“thing,” “cause”) in algebra.” He certainly understood by jadhr the abstract “root.” In this misunderstood form the term jadhr found its way into medieval Latin as radix.

And in On the Origin of the Term "Root." Second Article by the same author (The American Mathematical Monthly, vol. 35, 67–75, 1928), we find a further explanation:

It is now interesting to ﬁnd that this rather strange process of squaring the area is preserved in the Rhind Mathematical Papyrus as the oldest method of computing the area used by the Egyptians as early as 1650 B.C. Thus it is quite possible that in formulating this geometric deﬁnition of the jadhr al—Khowarizmi was prompted, not only by mathematical reasons, but also by old historical reminiscences of the Egyptian geometry.

The Egyptian unit of length in the measurement of land was the Khet of 100 cubits. The commonest unit of area was the setat or square Khet which contained 10,000 square cubits. For practical purposes in land measuring they used a unit called the cubit—of-land or the cubit—strip. It was a narrow strip of land 100 cubits long and one cubit broad. Smaller portions of a setat were expressed in such cubit-strips. To get the area of a rectangle they sometimes multiplied its length in cubits by its width in Khet. This gave them the correct answer in cubit strips. Still, more clearly we see in Problem 48 that for the multiplication of 8 Khet into 8 Khet and of 9 Khet into 9 Khet “the working actually written looks like the multiplication of 9 ‘setat by the pure number 9.”

Thus the Egyptian computed the square by ﬁrst multiplying the one side (9 Kket) in the square unit. This was the square—basis (jadhr) to be multiplied by the other side, representing the pure number. The same process is also described by al-Khowarizmi who says: “And in every square ﬁgure one of its sides multiplied into the square unit is its jadhr … and we make the other side, namely, hj, three, and this is the number of its jadhr’s.”

This archaistic way of computing the square ﬁnds its justiﬁcation not only in the “peculiar Egyptian system of multiplication” but in the very nature of primitive computation. The ancient Egyptians did not compute areas and measure their ﬁelds according to abstract rules. They originally found the area in an experimental practical way by taking a small square measure and trying out how many times it was contained in the ﬁeld to be measured, “as we today measure cloth by the yard.” For this purpose the mere length-measure, the Khet or cord, could not be of any use. They had, therefore, to create the square-unit, the cubit-strip, being a Khet long and one cubit broad. This was the ﬁrst jadhr, square—basis, used in practical life.

This cubit-strip might also, in all probability, be the underlying idea for the Egyptian conception of the square root. The idea of the square root existed in Egypt and the technical term for it was Knbt, literally “corner” or “angle.” Peet has the explanation for that, that the length of each of the two sides of a square containing any corner of it was its square root. Yet it is not the pure, one-dimensional side which contains a “corner” but only the jadhr, the side multiplied with the square unit, contains a “corner.” “Corner” or “angle” is the primitive word for a small square unit. Apollonius (c. 225 B.C.) deﬁnes the angle as the contracting of a surface at one point under a broken line. Since the word “corner,” “cornerstone” usually implies also the meaning “lowest part,” “basis, foundation” we might see in the Egyptian Knbt the source and origin of the Greek pythmenes (bases), the Arabic ass and jadhr, the Hebrew iqqar and ash and the Hindu mūla.

Thus weare enabled to pursue the origin of our term “root” to an algebraical term denoting the concrete number with a basis, the ﬁrst power (x) and to a geometric term denoting the basic square unit (x·12). All the three conceptions are contained in the Arabic jadhr and are preserved in the deﬁnitions of al-Khowarizmi.

• Very relevant. +1 in spite of the awful pun. – Edwin Ashworth Dec 15 '19 at 14:48
• "It suggests that if the basic number is a root, the square might be a bush, and so on up in a kind of a mathematical garden". Or maybe the Arabic already had the metaphorical meaning: root of a problem. – Lambie Dec 15 '19 at 16:58

We use the word root for both the mathematical and biological meanings because the word radix was used in Latin for both these meanings. When people started translated mathematics from Latin into English, they used the word root (which has had the biological meaning since Old English) for the translation of radix.

Why did Latin use the same word for the biological and mathematical meanings of root? I don't know. It doesn't look like they got it from Greek. Anyway, the question of why radix had both these meanings in Latin doesn't seem like it's on-topic for this stack exchange ... this possibly should be asked on the Latin stackexchange, where people would be more knowledgeable about the other meanings radix had in Latin.

• Your claim that 'Latin use the same word for the biological and mathematical meanings of root' is lacking support and directly contradicts that given in the OEtD. – Edwin Ashworth Dec 12 '19 at 13:35
• @Edwin: It is supported by wiktionary. It is supported by the Online Latin Dictionary. And as I read the OEtD, I don't see where it says it's not a calque from Latin. – Peter Shor Dec 12 '19 at 15:15
• I repeat, we're talking about the broadening into the maths usage. When this particular usage first occurred, way after the appearance of the literal sense; which group of mathematicians. Not the entry of the word into the English lexicon (which is LMGTFY). – Edwin Ashworth Dec 12 '19 at 15:30
• OED confirms that it is from the corresponding post-classical senses ‘starting point’ and ‘square root’ of classical Latin rādīx. Additionally, it compares to Old French, Middle French, French racine (13th cent. in this sense, originally in racine cube cube root). In the etymology of radix, it says: "in post-classical Latin also square root (frequently from c1120 in British sources)". – ermanen Dec 12 '19 at 19:18
• @EdwinAshworth The two papers I dug up and presented in my answer might be of interest. – linguisticturn Dec 15 '19 at 9:37