Sources related to Pi

March 14 is often colloquially referred to as Pi day (3/14). Pi is the ratio of the circumference of a circle to it's radius. What does Judaism have to say about Pi? Let's find out.

מלכים א ז׳:כ״ג

(כג) וַיַּ֥עַשׂ אֶת־הַיָּ֖ם מוּצָ֑ק עֶ֣שֶׂר בָּ֠אַמָּה מִשְּׂפָת֨וֹ עַד־שְׂפָת֜וֹ עָגֹ֣ל ׀ סָבִ֗יב וְחָמֵ֤שׁ בָּֽאַמָּה֙ קוֹמָת֔וֹ וקוה [וְקָו֙] שְׁלֹשִׁ֣ים בָּֽאַמָּ֔ה יָסֹ֥ב אֹת֖וֹ סָבִֽיב׃

I Kings 7:23

(23) And he made the molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line of thirty cubits did compass it round about.

The first reference to Pi is in the Tanach with the description of the Temple. Notice the dimensions given for the circumference and diameter of the pool.

משנה עירובין א׳:ה׳

(ה) ... כל שיש בהקפו שלשה טפחים, יש בו רחב טפח.

Mishnah Eruvin 1:5

(5) ... Anything [round] that has a circumference of three tefachim has a width of one tefach.

The Mishnah makes a statement about the ratio of the circumference of a circle to its diameter. Like the dimensions given in the Tanach, the ratio is given as 3:1.

עירובין י״ד א:נ״ב-נ״ד

א"ר יוחנן אמר קרא (מלכים א ז, כג) ויעש את הים מוצק עשר באמה משפתו עד שפתו עגול סביב וחמש באמה קומתו וקו שלשים באמה יסוב אותו סביב והא איכא שפתו

Eruvin 14a:19-20

Rabbi Yochanan said, "A verse - (I Kings 7:23) 'And he made the basin molten, ten ells from its edge to its edge and five ells of height and a line of thirty ells encircling it around.'" And behold, there is [the width of] its rim [taking away from the actual daimeter]?

When discussing this Mishnah, the rabbis in the Talmud seem to be aware of the fact that the ratio of the circumference to the diameter could not be 3:1. The explanation given is that the circumference was the inner circumference of the pool, while the diameter given was the outer diameter.

רמב"ם משנה עירובין א׳:ה׳:א׳

(ה) היתה של קש ושל קנים רואין אותה כו' - יש לך לדעת כי יחוס אלכסון העגולה אל המסבב אותה בלי ידוע ואי אפשר לדבר בו לעולם באמת וחסרון זו ההשגה אינה מאתנו כמחשבת הכת הנקראת גהלי"ה אבל הוא בטבעי זה הדבר בלי ידוע ואין במציאותו שיושג אבל ידוע זה בקרוב וכבר חברו חכמי התשבורת לזה חבורים לידע יחוס האלכסון אל המסבב בקרוב ודרך המופת בזה הקרוב אשר עליו סומכין חכמי החכמות הלמודיות הוא יחוס האחד לשלשה ושביעית וכל עגולה שיהיה באלכסון שלה אמה יהיה בהיקיפה שלש אמות ושביעי' בקרוב ולפי שזה לא יושג לעולם אלא בקרוב לקחו הם בחשבון הגדול ואמרו כל שיש בהיקיפו שלשה טפחים יש בו רחב טפח וסמכו על זה במה שהוצרכו אליו מן המדידה בתורה:

Rambam on Mishnah Eruvin 1:5:1

(5) If [the beam] is [made of] stubble or reeds, we view it, etc.: You should know that the ratio of the diameter of a circle to its circumference is not known and it is always impossible to speak about it in truth. And this lack of comprehension is not from us like the thought of the group called Gahlia, but (rather) it is in the nature of this thing that it is not known. And it is not [part of] its reality that it should be comprehended but it is known by approximation and the scholars of Mathematics have already written essays about this to know the approximate ratio of the diameter to the circumference. And the approximation that is the guideline that is relied upon by the scholars of applied wisdoms is that the ratio is one to three and one seventh (3.1428571429), such that any circle that has a diameter of one ell would have in its circumference approximately three and one seventh ells. And since this will never be comprehended except via approximation, [the Jewish sages] took a larger (coarser) calculation and said, "Anything that has three handbreadths in its circumference has one handbreadth in its width (diameter)." And they relied upon this [ratio] for that which they required [such a] measurement in the Torah.

Rambam, commenting on the same Mishnah, states that the true ratio of the circumference of a circle to its diameter is not only unknown, but unknowable. His idea that Pi is not discernible is possibly the first recorded hint that Pi might be irrational. Due to the impossibility of knowing the true ratio, he says that the sages simply used 3 as an approximation.