## Two rabbis meet at the blackboard – basic research

By Edgard Pimentel

A stroll through mathematics in the Jewish tradition

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Thinking about math (or about it) is natural. As a result, several societies have dedicated themselves to the subject over time. For example, the contributions and advances made by the Greeks in Arabic mathematics are diverse. There are also the achievements of oriental mathematics and sub-Saharan mathematics that influence Egyptian mathematics or pre-Columbian mathematical knowledge. And there is the Jewish mathematical tradition.

The Talmud, a word that means study in Hebrew, is a series of books that cover various aspects of Jewish life. It was written in the 1st and 6th centuries and first printed in 16th century Venice. It consists of six large parts, which unfold into more than sixty treatises. In a very simplified way, the work records the oral Torah and its interpretation.

One of the discussions in the Talmud concerns the number pi, which is defined as the ratio of the length of a perimeter to its diameter. This number appears in a variety of situations, from civil engineering to aerospace navigation to telecommunications. In the Talmud it can be read that a circle whose circumference corresponds to three spans has a span in diameter. This means that the number Pi is equal to 3. The surprising thing is that the text seems to anticipate the inaccuracy of such a claim and cites the biblical book of Kings I as a justification. An important puzzle is installed here.

On the one hand, the biblical text says that pi is worth 3; On the other hand, sages or rabbis of the Talmud know approaches to pi that contradict this fact. In addition, they suspect that it is impossible to fully determine this number. How can the supposedly armored biblical statement be reconciled with the available knowledge on the subject? Countless attempts are presented, but the most curious claim that choosing the value 3 would simplify the calculations: it would be easier to understand a “round” value. A value of 3 would also be sufficient for ritual purposes.

It was not until 1168 that one of the leading intellectuals of the medieval Jewish tradition, Maimonides, offered an answer to the dilemma of his predecessors. In one of his comments on the Talmud, Maimonides states that the number pi is irrational, that 3.14 is a well-known approach “accepted by educated people”, and finally decides that once one has chosen the use of the number is impossible Whole number knows The entire part – that is, the number 3 – was justified.

Maimonides’ relevance to Jewish mathematics goes beyond this particular case. In his guide for the confused, he mentions – without demonstrating – geometric properties that he had learned from Apolônio’s Arabic version of the Conics. Since this work had not been translated into Hebrew, mathematicians from the Jewish tradition devoted themselves to establishing such properties on their own. Then came various demonstrations of the facts mentioned by Maimonides.

Jewish intellectuals’ interest in mathematics may have started with questions of religious observance – for example, knowing how to build structures according to the principles of tradition. But it quickly became independent. In 1321 Levi ben Gershon published the work Maaseh Hoshev [A arte de calcular]. In many ways this text looks like it does today: a theoretical part, followed by applications and a list of problems.

Even earlier, Ibn Ezra in his Book of Numbers, 1146, gives several examples to discuss important facts about numerical series. Also in the 12th century, it is Abraham Bar Hiya who teaches us how to calculate the area of ​​the circle by treating it like a … triangle! Bar Hiya imagined the circle as a record: if we cut it lengthways and opened its grooves, we would have a triangle whose height is the radius of the circle and whose base is its diameter. His method was strictly formalized only in the 90s of the last century.

In the Jewish tradition, mathematics appears in several ways: in studying the pi number, in the interest of sums and approximations of important numbers, in search of rigorous thinking, and on many other levels. Incidentally, as in any society, at any time.

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Edgard Pimentel is a mathematician and professor at PUC-Rio.

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