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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?

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How about ways one would use math practically for Halachic purposes? – Isaac Moses Dec 19 '12 at 3:52
I recommend you look through the archives of cheshbon.weeklyshtikle.com – Isaac Moses Dec 19 '12 at 3:54
If that won't impress him, you can tell him about this guy you know through an internet forum who frequently has to divide fractions at work for reasons that are difficult to explain in a comment box. :) – Seth J Dec 19 '12 at 4:08
Really, if a person doesn't find math intrinsically beautiful, it's probably hard to make a specific case for learning past arithmetic, given that most math is abstracted away from us in 21st c. Western society. Might want to look at the remarkable manifestations of math constructs in nature. Basic algebra is certainly related to logic though, so a good foundation in that is a good thing WRT Talmud. He should be able to cross over from e.g. kal va-chomer and represent that quasi-algebraically (if a > b etc.) – yitznewton Dec 19 '12 at 5:50
Related: matheducators.stackexchange.com/q/1113 – msh210 Apr 2 '14 at 1:42

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).

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It gets pretty complicated, so be warned. – Double AA Dec 19 '12 at 3:24
Good stuff. I find the hardest part about it to be the terminology, while in terms of math concepts it is right up there at high school level. – WAF Dec 19 '12 at 4:45
@WAF Makes you glad we have a decimal system! – Double AA Dec 19 '12 at 5:00
Yeah it's actually not that bad to take a calculator and verify the Gemara's result of "how many cubic fingers make a quarter-log of volume?" The problem is tracking the way various medieval commentaries do the math -- they use some very convoluted methods to avoid fractions until the bitter end! (And then Tosfos says the Yerushalmi has a different number, it meant a round cup instead of a cubed one ...) – Shalom Dec 19 '12 at 8:46

In Hakirah vol. 14, they published an article called "'Learning' Mathematics" which includes examples of different mathematical applications in classical Jewish literature.

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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, but I don't know where you could find it now. Perhaps interlibrary loan or such?

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Approaching Infinity seems to be on re-order from this site. I guess you could get it there later (or on Barnes & Noble now, but I doubt you want to pay $73) – b a Dec 19 '12 at 4:37
Another good book on this topic is Di Shemaya, by Alex Schutz. I bought a copy of it retail a few years ago in a bookstore in Baltimore. – Isaac Moses Dec 19 '12 at 4:45
@IsaacMoses Indeed it contains some interesting ideas. – Double AA Dec 19 '12 at 4:50
@DoubleAA Indeed! :) – Isaac Moses Dec 19 '12 at 5:03
@IsaacMoses If anyone is looking for a copy of Di Shemaya, I still have a few boxes stored in my garage. (Alex asked me to be his Lakewood "distribution" point back when he published.) – LazerA Dec 19 '12 at 6:48

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.

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Here's a fantastic article about this: aishdas.org/asp/2010/08/r-dr-eliezer-ehrenpreis-zl.shtml – Dave Jan 2 '13 at 4:51

A couple examples in Eruvin come to mind.

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.

Also 23b discusses the difference between 7.666 and sqrt(5000), with Rashi giving an interesting way of calculating the square root of 5000.

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+1 not so simple that equals 3 see shach 30.5 beta.hebrewbooks.org/tursa.aspx?a=yd_x3545 – hazoriz Apr 15 at 0:57

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.

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The links on that page are broken - the ".org" suffix needs to be changed to ".net". – Dave Apr 15 at 16:08

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