# Who invented the square root?

## Why we pull roots

Calling solutions to polynomial equations the "roots" of the polynomial has to do with one of the most bizarre mistranslations in mathematical history

In German we pull square roots. In English one speaks of "square root extraction". In Spanish, a literal translation is used: "extraer la raiz cuadrada". Where does this phrase come from and what do equations have to do with the plant world?

The expression "taking the square root" and referring to solutions of polynomial equations as the "roots" of the polynomial has to do with one of the most bizarre mistranslations in mathematical history. Some words and symbols (think of today's Arabic numerals) found their way into the mathematical language of Europe through borrowings from Arabic mathematical works. It was the hour of birth of algebra and also of certain misunderstandings.

Three different epochs are identified in the origin of algebra: With the Babylonians and Egyptians, mathematical problems were formulated simply in language and there was usually no excellent word for the solution sought (or unknown variables as we would say today). This "rhetorical algebra" was followed by an "annotated" system in which special marks or letters enriched the rhetorical formulation. Ultimately, the well-thought-out doctrine of symbols and abstract transformation rules that we know today emerged.

At the beginning of this development, mathematicians from the Persian, Indian and Arab cultures made important contributions. Even the term "algebra" is of Arabic origin. It was used by Muhammad ibn Mūsā al-Khwārizmī in a mathematical compendium from the year 820.

### The word algebra

Between the years 800 to 1300 the Arabs inherited the science and technology of the Egyptians, Babylonians, Greeks and Romans. All kinds of scientific writings were collected and translated in Baghdad, the center of the Arab empire from the ninth century onwards. The Bait al-Hikma (House of Wisdom) in Baghdad became the think tank of the caliphate. It was the meeting place for the most important scholars of that time.

Al-Khwārizmī also worked in the House of Wisdom. His fame was legendary even during his lifetime - he was considered the Euclid and Diophantine of the Arab world. Born in the region between Persia and Uzbekistan, Al-Khwārizmī was a universal scholar. Together with other mathematicians, he popularized the Indian numerals and wrote didactic explanations on algebraic procedures. Even his name, Al-Khwārizmī, turned into our "algorithm" over time.

The best-known book by Al-Khwārizmī was entitled "Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala", which could be translated as "Compendium of Compensations by Completing and Adjusting". The book was a kind of "user guide" for solving math problems. Many translate the word "Al-Jabr" in the title as "complement". The first Latin version of the work (1145) was called "Liber algebrae et almucabala", with which the word "algebra" entered the European vocabulary.

However, more than 80 years ago, Solomon Gandz and Otto Neugebauer showed that the above-mentioned interpretation of "algebra" was incorrect.1 Centuries earlier, Babylonians and Egyptians had developed the basic equalization techniques for solving equations with one variable.

Gandz and Neugebauer were able to trace Al-Khwārizmī's sources back to the Babylonians, Assyrians and Sumerians. "Gabru-maharu" means in Assyrian *Oppose* or *Be equal*. The Arabs took over the mathematical technique but also the phonetics of the word and this resulted in the borrowed word "al-jabr", which was identified with the Arabic word "al-muqabala". Gandz deduced from this that the title "Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala" could be translated as the "science of equations". *Two is Better* - and so in the title "al-jabr" and "al-muqabala" actually designate equations, one time in Assyrian, another time in Arabic.

### The mistranslation

However, before the Arabs took over the math relay, they had to deal with the witty Greek mathematicians. The Greeks had a preference for geometrical proofs, since these can be carried out quite clearly for "the eye".

The mathematical problems of that time were reduced to constructions with rulers and compasses. For example, if you want to find the square root of 2 geometrically, you draw a square with edge length one. The diagonal of the square then has the desired length. The corresponding geometric methods are available for addition, subtraction and even multiplication and division of segment lengths. The fact that many of these geometric quantities could no longer be written as simple fractions was a thing *computational* Industrial accident discovered by the Pythagoreans. Tracking down the irrational numbers was so important to them that they offered a hecatomb (a hundred slaughtered cattle) to the gods. "Since then, the oxen have been trembling whenever a new truth comes to light," said Ludwig Börne.

But now: The Arab mathematicians (including al-Khwārizmī in Baghdad) used two words for the unknown quantity in a problem: "mal" and "jidr". The first word was used for the square of the unknown, the second for the unknown quantity itself.

The Arabs wanted to solve the algebraic equations that the Greeks mastered in a geometric way. For the Greeks, the "side" or "edge" of a geometric construction was the size to be searched for, the "pleura" in their language. The Islamic mathematicians translated "pleura" as "jidr" because this word means "base" or "lowest part", but in plants it describes the "root". When the first European mathematicians translated the word "jidr", they changed it into the Latin "radix" (like radish), the word for root, a mistake with serious consequences.

Among other things, the translators Johannes Hispaniensis in Seville, Gerhard von Cremona in Toledo, and Leonardo di Pisa (better known as Fibonacci) popularized the term "root" in Italy and this became a popular phrase.2 Chapter 14 of Fibonacci's "Liber Abaci" ( von 1202) had this title, for example: "De reperiendis radicibus quadratis et cubitis ..." (On finding square and cube roots).

From then on, the word "root" was used both for solving quadratic equations and for solving any algebraic equations. In the first case the solution became a "square root", in the second simply the "root" of the equation.

Not all mathematicians made the same mistake. The great algebraic artist François Viète in France and others translated directly from the Greek originals and therefore used the word "latus" (page) for the unknown. The alternative terminology had long since established itself in Europe. Since then, the "root finding" has made the pupil sweat - an exciting story that would be worthy of being told by Scheherazade in the Thousand and Second Nights. (*Raúl Rojas*)

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