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I'm currently tutoring someone in math that has an atitude of "what's the point of all this... why would I ever need this..." (which is typical of many high-schoolers). To try to pique his interest, I figured it would be great if I could provide him with examples from the Gemara that actually make use of some of the regent-level math he's learning.
So my question is what are some examples from the Talmud (Rashi and Tosfos included) that make use of (or refer to) high school level math?
Try Pesachim 109a-b where the Gemara (and more elaborately in Rashi and Tosfot) tries to work out the volume of a Reviit in Etzba^3 based on its knowledge of the volume of a Mikva in Amah^3 (ie lots of basic algebra and unit conversion).
I remember doing the gemara on Sukkah 8a in high school while I was also in a geometry class in the afternoons. It's pretty basic high-school geometry stuff. Squares and circles. It's the Tosfos there, though, that go all out.
It's particularly ingenious how Tosfos (bottom of the page) demonstrates that the ratio of the diagonal of a square to its side (which we know is sqrt(2)) is not exactly 7/5 as the gemara asserts.
You might want to show him Rambam Hil. Kiddush HaChodesh and the diagrams in the back, as well as Chazon Ish on Kiddush HaChodesh and the attendant illustrations. At the very end the Chazon Ish even includes a handy sine table! In R' Chaim Kanievsky's Shekel HaKodesh there is an appendix that explains the trigonometric underpinnings of the numbers given by the Rambam.
The Mirkeves HaMishneh wrote a kuntres called Breichos B'Cheshbon (included in some editions of Mirkeves HaMishneh) which provides advanced mathematical explanations for various sugyos. About 30 years ago it was translated into English and explained, in a book called Approaching Infinity. It was a fascinating read and is linked as a PDF.
In Hakirah vol. 14, they published an article called "'Learning' Mathematics" which includes examples of different mathematical applications in classical Jewish literature.
The gemara (Y'vamos 82b) discusses iteratively replacing mikva water with fruit juice.1 The case involves adding one se'ah of fruit juice to a 40 se'ah mikva, and then removing one se'ah of the solution. R' Yochanan rules that this iterative process may be repeated so long as more than 50 percent (or at least 50 percent) of the solution remains water.
Rashi (s.v. mai lav) seems to comment that no more than 19 iterations may be performed.2 Tosafos Y'shanim (ad loc.) points out that 20 iterations does not mathematically get you below 50 percent (assuming some mixing of the solution occurs during this process), but concludes that 20 iterations would still render the mikva rabbinically invalid because it "appears like a majority" of the solution is fruit juice.
Rashi's wording ("d'lo nishkol ruba, aval ad palga shapir dami") does not strike me as amenable to this approach. My guess is that Rashi limited the iterations to 19 to account for a worst case scenario of virtually no mixing,3 in which case more than 19 iterations would still involve at least a safeik that the mikva is biblically valid.
However, if we could assume that the solution becomes perfectly mixed after each addition of fruit juice, we could use high school math to determine the maximum number of iterations after which the mikva would still remain biblically valid:
Suppose we want strictly more than 20 se'ah of water in the mikva solution. Denote the maximum number of allowable iterations n (∈ R). We can set up the inequality 40*(40/41)^n > 20 ⇒ (40/41)^n > .5 ⇒ log(base 40/41) of .5 < n ⇒ 1/(log(base 2) of 41 - log(base 2) of 40) < n. In this case n is slightly more than 28, so we can denote the integer-valued number of iterations as n' = sup{q ∈ Z | q ≤ n}, i.e. the largest integer less than or equal to n, namely 28.4
(Although I don't think logarithms were in use during Rashi's time, a close enough approximation for n could have been computed numerically in at most a matter of hours).
The halachic ramifications of this perfect mixing scenario are questionable, but I think this case could be used as a construct for a math problem.
1 Or temed, as the case may be. There are different interpretations as to what temed is, one being that it is a form of dilute wine made by soaking the sediment particulates from old wine in water. See Tosafos (s.v. nasan se'ah) for how this could affect the halacha. For the purposes of this answer, fruit juice is assumed to be the substance in question.
2 If Rashi meant that the solution must contain less that 20 se'ah of juice, he presumably would not have used the phrasing "he may do this until 19 se'ah," which implies an integer valued restriction.
3 Such as where 19 se'ah of fruit juice are added to one side of the mikva and 19 se'ah of solution are removed from the other side of the mikva in almost instantaneous succession. Although this seems to be an unfeasible feat, the limit would presumably be set at the edge of the possible rather than the feasible.
4 Please let me know if I made a math error. Incidentally, I wish Mi Yodeya had LaTeX support at times like this.
You may be interested in my Sefer which includes the Sugyos Eruvim Daf 14,57,76, Pesachim 109, Succos 7,8. It explains Gemara, Rashi, Tosfos, Marsha, Maharom, Gra, others, and presents every step in equation form and diagrams; it also makes corrections and clarifications on diagrams found in the Shas. It includes an Appendix on Basic Algrebra, 21 definitions of Symbols, 32 Laws. Eruvim has 25 equations, Pesachim 106 equations, and Succos 184 equations. This sefer is a small part of what I am preparing to publish Bezras Hashem. In that larger sefer I have treated many other math discussions in Shas and Mishnyos. I am also preparing to present an online course on this topic.
A couple examples from Eruvin come to mind:
Eruvin 14a brings proof that for halachic matters, pi = 3, and then continues to explain the dimensions of Solomon's pool. This contains some prealgebra and simple geometry.
Eruvin 23b discusses the difference between 7.666 and sqrt(5000), with Rashi (s.v. "אלא אמרה תורה טול חמשים") giving an interesting way of calculating the square root of 5000.
I just discovered this Youtube channel The Math Rebbe. He takes sugyas from different gemaras and commentaries and explains how they use modern high-school and beyond math. I've only just started watching, but it seems interesting. There's currently one (10 episode) season.
The first episode is actually about what i posted here previously.
But beware, he makes some pretty bad puns.
No idea if a high school student ought to be taught this, but I think some of them might be interested, and they could understand the question. It is actually a math problem that I don't know how to solve (roughly posted on a math forum and they didn't either though they had suggestions), and I doubt that the kadmonim used my approach, but still:
https://drive.google.com/file/d/1I0o-bJ1v2LnKhAYLo58HLZ5RTwmlBuV9/view?usp=share_link
Shorter version: If a Jew's and a non-Jew's flour are mixed together and you made dough, and the Jew's flour had enough for a shiur challah, the Yerushalmi (Challah 3(5)) says that the Jew's dough would need to be connected, but it is: "connected by strings (גידין)", whatever that means, so it's chayav in challah (brought in Beis Yosef Yoreh Deah 330).
I wondered: What should actually happen if there's a bunch of flour grains, half Jewish and half non-Jewish, and you add water so they swell up and bump into each other and make a mass? Do the Jewish grains all end up connected, or are there isolated patches (which would mean the whole lot would not be chayav in challah)?
To try and approach it, I imagined a lot of closely packed spheres, half red and half blue randomly assigned. Each sphere is touching many neighbors, twelve actually. Are all the red spheres connected via red neighbors?
Not a great model, but it was a start. You could vary it in different ways.
I ran a computer program to try it out.
Anyhow, not your usual math-in-the-gemara, but still! Comments welcome.
For an easy cool one, see Tosefos Menachos 106a d"h שהן They explain how to add up an arithmetic sequence of numbers, like פרי החג, by working from the ends to the middle. Skill for life!
