# Rabbinic Mathematics

יַּ֥עַשׂ אֶת־הַיָּ֖ם מוּצָ֑ק עֶ֣שֶׂר בָּ֠אַמָּה מִשְּׂפָת֨וֹ עַד־שְׂפָת֜וֹ עָגֹ֣ל׀ סָבִ֗יב וְחָמֵ֤שׁ בָּֽאַמָּה֙ קוֹמָת֔וֹ ׳וּקְוֵה׳ ״וְקָו֙״ שְׁלֹשִׁ֣ים בָּֽאַמָּ֔ה יָסֹ֥ב אֹת֖וֹ סָבִֽיב׃
מלכים א 7:23

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
I Kings 7:23

This Hebrew Bible passage from I Kings—along with a similar one from II Chronicles—forms the biblical basis for Talmudic scholar Matityahu Hacohen Munk’s suggestion that “some of the geometrical rules did not hold in King Solomon’s temple,” a heavenly ‘‘world of truth’’ beyond our own, mathematical historians Tsaban and Garber write [1].

What’s so heavenly about the Molten Sea, a putative basin created by King Solomon in the ancient Temple of Jerusalem for ritual ablution? And why do the Rabbis Johanan and Papa discuss it extensively in the Babylonian Talmud, bickering in particular about its brim—“[as thin as] the flower of a lily… a handbreadth thick… wrought like the brim of a cup” [2, Eruvin 14a:29-31]?

The simple answer is that this particular snippet of the Word of God contains an oddity, asserting that this circular basin’s circumference is thrice its diameter—or that the geometrical constant π, rather than an irrational number, with an infinite and unpredictable decimal expansion, is in fact rational, and indeed an integer—the number 3, to be exact.

Rabbi Papa offers a solution. “When [the measurement of the circumference] was computed it was that of the inner circumference,” [2, Eruvin 14a:32-34] he argues, while the measured diameter included, in addition, the brim. Given the (actual) value of π, this forces a brim of ≈.225 cubits, or, assuming an 18-inch cubit, just over four inches. A “handbreadth”, anyone?

Undeterred, Rabbinic scholars have offered other solutions. Rabbi Haim David Z. Margaliot suggested in 1938 that “the circumference of the circle was measured from inside using a stick of length equal to the radius of the needed circle” [1]. The measured quantity, then, was not the perimeter of the circle, but rather that of an inscribed regular hexagon; this quantity is, indeed, thrice the circle’s diameter.

Rabbi Shimon Ben Tsemah (1361–1444) had a simpler, more pedagogical explanation: “One should always teach his student in the easiest way,” [1] he quipped.

In fact, it was apparently known already to Maimonides (1135-1204) that π is irrational. Citing what they claim is one of the earliest historical articulations of this fact, Tsaban and Garber quote from the Perush Ha-Mishnah:

You need to know that the ratio of the circle’s diameter to its circumference is not known and it is never possible to express it precisely. This is not due to a lack in our knowledge, as the sect called Gahaliya [the ignorants] thinks; but it is in its nature that it is unknown, and there is no way [to know it], but it is known approximately. The geometers have already written essays about this, that is, to know the ratio of the diameter to the circumference approximately, and the proofs for this. This approximation which is accepted by the educated people is the ratio of one to three and one seventh. Every circle whose diameter is one handbreadth, has in its circumference three and one seventh handbreadths approximately. As it will never be perceived but approximately, they [the Hebrew sages] took the nearest integer and said that every circle whose circumference is three fists is one fist wide, and they contented themselves with this for their needs in the religious law. [1]

The irrationality of π wasn’t mathematically proven, on the other hand, until 1761; Johann Heinrich Lambert’s ultimate proof hinged upon the continued fraction expression

$\text{tan}(x) = \frac{x}{1 - \frac{x^2}{3 - \frac{x^2}{5 - \frac{x^2}{7 - \ddots}}}}$

and its properties.

Interestingly, Maimonides’ suggested approximation, $\frac{22}{7}$ (accurate to 2 decimal places), is exactly the first convergent of the continued fraction expansion of π, succeeded by $\frac{333}{106}$ (accurate to 4 decimal places), $\frac{355}{113}$ (accurate to 6 decimal places), $\frac{103993}{33102}$ (accurate to 9 decimal places), and so on.

*                *                *

Modernity has succeeded in approximating π more closely—that is, to more than 0 digits. Quoting variously from Wikipedia, we have infinite series representations of π that would have made Maimonides reaffirm his commitment to G-d:

• François Viète (1593): $\frac{2}{\pi} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} \cdot \frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2} \cdot \cdots$
• John Wallis (1655): $\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdot \cdots$
• Ramanujan (1910): $\frac{1}{\pi} = \frac{2 \sqrt{2}}{9801} \sum_{k = 0}^\infty \frac{(4k!)(1103 + 26390k)}{k!^4 (396^{4k})}$
• Chudnovsky brothers (1987): $\frac{1}{\pi} = \frac{12}{640320^{\frac{3}{2}}} \sum_{k = 0}^\infty \frac{(6k)!(13591409 + 545140134k)}{(3k)!(k!)^3(-640320)^{3k}}$

The latter of these formulas are based on Ramanujan’s constant, $e^{\pi \sqrt{163}}$, about which I’ll say a bit more.

Ramanujan’s constant yields the approximation

$\frac{\text{ln}(640320^3 + 744)}{\sqrt{163}} \\ = 3.14159\ 26535\ 89793\ 23846\ 26433\ 83279^+,$

which approximates π to an astounding 30 decimal places. This approximation follows immediately from the extreme nearness of Ramanujan’s constant (which is transcendental) to the integer $640320^3 + 744$, a much subtler phenomenon that finds its roots deep in algebraic number theory and the theory of modular forms.

Indeed, the first key here is that 163 is a Heegner number, a positive integer $d$ for which the imaginary quadratic field $\mathbb{Q}[\sqrt{-d}]$ has class number 1; in fact, it is the largest such number (the fact that there are only finitely many of these was proven by Heegner). Things only get crazier from here. Identifying elliptic curves over $\mathbb{C}$ with lattices $\Lambda = \mathbb{Z}[1, \omega] \subset \mathbb{C}, \text{Im}(\omega) > 0$ via Weierstrass’ elliptic functions, it turns out that the j-invariant $j(\omega)$ of $\omega$ a quadratic integer is algebraic over $\mathbb{Q}$, and in fact is exactly the primitive element of the Hilbert class field $E$ of $K = \text{Frac}(\Lambda) / \mathbb{Q}$; moreover its degree over $\mathbb{Q}$ is also the class number of $K$ (see [5, Thm. 4.3, (b)]). In our case $\Lambda = \mathcal{O}_{\mathbb{Q}[\sqrt{-163}]} = \mathbb{Z}[1, \frac{1 + \sqrt{-163}}{2}]$, this class number is 1 (by definition of a Heegner number), and we see that $j \left( \frac{1 + \sqrt{-163}}{2} \right)$ is an integer, $640320^3$ in this case.

It turns out, now, that the j-invariant, viewed (via the above identification) as a function on the open upper half-plane, is modular, and thus has a Fourier expansion; this expansion is moreover well-approximated by its truncation to its first two terms, which takes the form

$j(\omega) \approx \frac{1}{\text{exp}(2 \pi i \omega)} + 744.$

Plugging in $\omega = \frac{1 + \sqrt{-163}}{2}$ into the above and solving for $\omega$ yields the approximation $e^{\pi \sqrt{163}} \approx 640320^3 + 744$, which is accurate to 12 digits.

Modern supercomputers equipped with the Chudnovsky algorithm have now computed π to around 22 trillion digits [6].

*                *                *

The juxtaposition of divinely ordained authority with a patent mathematical falsehood pits various worlds against each other in an amusing way. Of course, π does not equal 3, whatever the bible claims. And though the sophistication with which we can now approximate π perhaps doesn’t exceed that of other areas of modern mathematics, it does become startling when compared to the (modest) biblical state of the art.

Yet despite the ridiculousness of the claim, and of the attending Rabbinic debate, our redoubtable modern understanding—at the end of the day—fails to penetrate the humanistic realm, where the Hebrew tradition succeeds at every page. The juxtaposition—of Maimonides and Ramanujan—while hilarious, is also sad. Even more than revealing the limits of tradition, it also reveals the limits of mathematics.

References:

1. B. Tsaban, D. Garber, On the rabbinical approximation of π, Historia Mathematica 25 (1) (1998) 75–84.
2. The Soncino Babylonian Talmud, Eruvin.
3. Rabbi Yosef David Kappah, Mishna with Maimonides’ Commentary, Mo’ed section, Jerusalem: Mosad Harav Kook 1963, 63–64. [In Hebrew]
4. Belding, Bröker, Enge, Lauter. Computing Hilbert Class Polynomials.
5. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves.
6. Trueb, Peter (30 November 2016). “Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi”. arXiv:1612.00489