March 14 is celebrated by nerds around the world as Pi Day. When written in digits, March 14 – 3/14 – represents the first three digits of the number traditionally represented by the Greek letter “π” (which of course is pronounced as “pie”).
Pi Day is celebrated by eating pie and discussing the significance of pi. Some people who have too much time on their hands compete at memorizing as many decimal digits of pi as possible. The record presently stands at 67,890 digits! While eating pie is optional, discussing the significance of π is truly mandatory.
So, what is π and why is it so important? π is arguably the most famous mathematical constant, expressing the ratio between the circumference of a circle and its diameter. If you take a circle with a diameter of 1, the circumference of this circle will measure approximately 3.14159265…. For a circle of radius (half of the diameter) “r,” the circumference of the circle measures 2πr, and the area of the circle is calculated as πr2. One can see why π is so important in measuring land, for instance.
However, π appears not only in geometry, but in number theory, algebra, analysis, statistics, and other mathematical disciplines. In physics, π is truly ubiquitous.
In antiquity, Babylonians approximated π as the closest whole number: 3. However, measuring land, architecture, and engineering require knowledge of π with reasonable precision. The Babylonians knew that the integer 3 was merely a rough approximation. A Babylonian tablet found near Susa (c. 19th-17th centuries BCE) gives an approximation of π as 25⁄8 (3.125), getting the first decimal right.
The Egyptian Rhind Papyrus (c. 1600 BCE) approximates π as 256⁄81 (≈ 3.16). Some Egyptologists claim that the builders of the pyramids used the ratio 22⁄7 (3.142…) for π. In the Shatapatha Brahmana (c. 6th century BCE), Indian astronomers estimated π to be 339/108 (which equals 3.139…). In the 3rd century BCE, Archimedes proved that π is greater than 223⁄71 (≈ 3.1408…) but less than 22⁄7 (≈ 3.1428…).
In the 2nd century BCE, Greek astronomer Ptolemy used an approximation of 377⁄120 (≈ 3.1416…). In the 3rd century CE, the Chinese mathematician Liu Hui very accurately approximated π to four decimals as 3927/1250 (= 3.1416). In the 6th century CE, Indian mathematicians obtained the same value of π as 62832/20000 (= 3.1416). By the 5th century CE, Chinese mathematicians improved the value of π till the seventh digit. Today, the value of π is known to trillions of digits.
The exact value of π was first discovered by German mathematician Gottfried Leibnitz. Leibnitz’s formula for π is an infinite series:
The fact that the exact value of π can only be expressed as an infinite series is highly significant, as we will see later.
The Jews who left Egypt must have been aware of the approximate value of π then used by the Egyptians. After all, Jewish slaves built the pyramids. Thus, some wonder why Tanach gives the crudest approximation of π. In I Kings (7:23) we read: “Then he made the molten sea of 10 cubits from brim to brim, round in compass, and the height thereof was 5 cubits; and a line of 30 cubits did compass it round about.”
In II Chronicles (4:2) we find an almost identical passage: “Also, he made the molten sea of 10 cubits from brim to brim, round in compass, and the height thereof was 5 cubits; and a line of 30 cubits did compass it round about.”
The round basin made out of cast metal for the Beit HaMikdash in Jerusalem by King Solomon had a diameter of 10 cubits and a circumference of 30 cubits. Accordingly, Tanach seems to be giving the value 3 for π. But even if we were to assume, arguendo, that the ancient Hebrews did not know the value of π beyond its integer value (3), they certainly knew how to measure. If they made the basin with a diameter of 10 cubits, they would have found the circumference to be approximately 31.41 cubits when they measured it. Rounding to the nearest integer would result in 31 cubits, not 30.
The Lubavitcher Rebbe, Rabbi Menachem Mendel Schneerson, suggested that the circumference of the basin was actually 30 cubits. Accordingly, the diameter was approximately 9.55 cubits, and the passuk simply gives us the rounded up integer, 10.
I don’t really see a problem to begin with, though. Tanach is not a mathematics textbook. Nowhere in it do the prophets set out to reveal the value of π. The verses describe the dimensions of the basin used in the Temple. As the Rebbe explained, given the circumference of the basin at 30 cubits, the diameter came to be a shade under 10 cubit and was simply rounded off to 10. The basin was already built, so there was no need for precise engineering drawings or exact dimensions. The Scripture simply informs us of approximate dimensions of the basin. There is nothing strange or unusual about it.
However, some of our sages thought that the verses cited above require an explanation. Thus, Mishnat HaMiddot seeks to solve this problem by positing a brim approximately 0.225 cubits thick, which was included in the measurement of the diameter, but not in the circumference. Thus, the verses approximate π to be 22⁄7, or 3.14135…. The dating of this source, however, is unclear. Some attribute it to the tanna Rabbi Nechmiah, in which case this work can be dated to c. 150 CE, while other authorities date this book to the late Gaonic period, c. 9th century CE.
Fast forward to the twentieth century. Rabbi Max Munk offered the following interesting insight. (This insight is often misattributed to the Vilna Gaon. See “Do Scripture and Mathematics Disagree on the Number π?” by Prof. Elishakoff and Dr. Pines in the article in B’Or HaTorah, 17, pp. 141-42.)
In I Kings, the word used for “diameter” is “kaveh,” which makes no sense in context. By contrast, in Chronicles II, the word for diameter is spelled as “kav,” meaning “line.” However, according to the mesorah, the word in I Kings is read (kri) differently than it is written (ksiv). It is read as kav, just as it is written in Chronicles II. Rabbi Munk points out that the gematria of kaveh (100+6+5) is 111 and the gematria of kav (100+6) is 106. He interpreted the ratio of these two values – 111/106 – as a correction factor: If you multiply the implied value for π (3) by this factor, you get 333/106 = 3.14150… – an approximation of π accurate to the fourth decimal point. This would be a far more accurate approximation of π than was known for many centuries thereafter.
Ultimately, no matter what ratio you find, it will always be just an approximation of the true value of π and therefore will always be open to criticism.
The real problem is that π can never be truly expressed as a ratio of two integers – it is an irrational number (rational numbers are those that can be expressed as a ratio of two integers). Interestingly, it was Maimonides who was the first to assert that π is irrational. In explaining a mishnah that posits 3 as the halachic (legal) value of π, Maimonides explains that 3 is as good an approximation as any other because we cannot compute π precisely anyway. Besides being an irrational number, π is also a transcendental number, meaning that it is not a solution or a root of an algebraic equation.
I once asked Rabbi Adin Steinzaltz (may he have a refuah shaleimah) where can we find irrational and transcendental numbers in the Torah if they are real. In response, Rabbi Steinzaltz quoted a famous statement by the 19th century Jewish-German mathematician Leopold Kronecker, who said: “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk” (G-d made the integers, all else is the work of man).
This is no trivial question. In fact, it is the crux of a debate in the philosophy of mathematics: Do mathematical objects “exist” in some ontological sense and are being “discovered” by mathematicians, or they are “invented” by mathematicians and exist only in our minds?
Plato founded the school of thought that mathematical objects exist as ideal forms and are discovered by us. Kronecker, on the other hand, espoused the opposite view. It seems to me that, by choosing an integer number (3) that is a poor approximation of π, the Prophets were not demonstrating ignorance of a better approximation well-known in antiquity; rather, they were revealing to us something else – a deep truth about the nature of mathematics. As Kronecker expressed it, “G-d made the integers, all else is the work of man.”
