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Chazal often talk about irrational numbers in their math discussions. For instance, Sukkah 7b-8a uses both √2=1.4 and π=3 in its discussion, and Eiruvin 76a-b uses the same numbers in an almost identical sugya. And the list goes on.

But can we actually rely on such sources? That is, halacha l'ma'aseh, can we assume that π=3 for the sake of halacha? For instance, in the aforementioned sugya in Sukkah, discussing a round sukkah. If I actually wanted to make a round sukkah, can I take a 7x7 tefach square, multiply the side by 1.4 to get the diagonal of length 9.8, then multiply by 3 to get the circumference length of 29.4, and that would be a kosher sukkah? Or do I have to get as close to 7π√2 as humanly possible?

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    I remember learning about the yam shel shlomo in which the diameter and the circumference are mentioned with Pi equal to 3. Rashi goes through the calculation and shows that the shape is actually a cylinder resting on a square base so that the values do not round pi to 3. – sabbahillel Sep 9 '16 at 20:47
  • It is Gemara in eruvin – kouty Sep 10 '16 at 19:14
  • @sabbahillel π(r^2)h + lwh = 3(5^2)(2) + 10(10)(3) = 150 + 300 = 450 = (3*1*1)(150). The numbers only work if you round π to 3. – DonielF Sep 11 '16 at 1:05
  • If I recall correctly, the numbers are because the diameter is not actually the full side of the square. This would mean that the edge of the cylinder made is inset a little either because of construction or the thickness of the metal of which it is constructed. I think that Rashi calculates it using geometrical calculations and fractions. Using the exact value of pi, the actual diameter would be 9.54929 amos to get a 30 amah circumference. Also, it is possible that the calculation was accurate and the measurement of the actual side was rounded up to 10 or circumference rounded down from 31 – sabbahillel Sep 11 '16 at 1:44
  • Another point is based on the description in Melachim Alef 7:24 and 7:26, the rim was flared. There are those who say that the cylindrical portion was flared and the rope around the circumference had to be under the flared portion of the brim. – sabbahillel Sep 11 '16 at 2:06
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There seems to be three mainstream opinions in the Rishonim and Acharonim.

Only if you're not planning on cheating someone

The Sefer HaChinuch (Mitzvah 258) says explicitly that one may not rely on such estimations by Choshen Mishpat. In his words:

ותזכר...שלא אמרו זכרונם לברכה על הכיוון כי אם בקירוב, ולכן אל תסמוך בזה בחלוקת הדברים בין בני אדם. ואל תתמה איך יכתבו דבר בלתי מכוון...ובזה מה שלא כיוונו בו מביא אותו לידי חומרא ואינו מזיק לכל אדם בממנו.

And remember...that [the Chachamim], may their memory be for blessing, did not say [these previously stated mathematical principles] precisely, but were only estimating, and therefore, do not rely on this for separating things between people. And do not wonder, "How could they write something imprecise?" ... And on this that which they were imprecise would lead someone to act stringently and would not harm a person with his money.

The Shulchan Aruch (CM 231:16) and Rambam (Geneivah 8:1) say the same thing, that people are very careful with their money, and they don't want to part with it that quickly. They are quoting a Gemara in BM 91a; if anyone wants to argue with this opinion, they have to be more stringent. And indeed, they are.

Only if you're dealing with π

The Maggid Mishnah in Hilchos Shabbos (17:26) says that one may rely on π=3, but he says in Hilchos Eiruvin (3:2) that one may not rely on √2=1.4. What's the difference? Perhaps it's because the former has a passuk (Melachim Aleph 7:23; see Eiruvin 14a) while the latter doesn't.

The Sha'alos U'Teshuvos Tashbeitz (1:165) gives two explanations of these estimations: either they're Halachos L'Moshe MiSinai, or they're there to help explain the sugya better and one may not rely on them. The first side clearly holds like the Sefer HaChinuch, Shulchan Aruch, and Rambam, but perhaps the second side agrees with the Maggid Mishnah that one may not rely on √2=1.4 but can rely on π=3. Or maybe that side agrees with the even more stringent opinion below.

Absolutely never

In the aforementioned sugya in Eiruvin 14a, the Gemara opens up with the cryptic line: "How do we know [that the circumference of a 1-tefach circle is 3 tefachim]?"

Um, how do we know? Take a tape measure! (Or at least take a string, wrap it around a circle, and compare it to a ruler, or three fists lined up next to each other.)

The Tosfos HaRosh explains: Since π≠3, how do we know we can rely on this halacha l'ma'aseh? The Aruch HaShulchan (OC 363:22, YD 30:13) explains: Since π=3, the diameter is less than 1. How do we know we can make the mavoi permissible to carry in anyway?

These seemingly similar explanations could be arguing about this very point. The Aruch HaShulchan, who emphasizes the permissibility of the mavoi, would seem to hold that one may not rely on these estimations whatsoever. The Tosfos HaRosh, who asks how we know we can rely on π=3 but is silent about √2=1.4, could hold like either of the above opinions.

R' Moshe Feinstein (Igros Moshe, YD 3:120:5) asks whether one is obligated to use a microscope and other scientific tools to make his tefillin as square as possible. R' Moshe paskened that as long as one's tefillin match the 1:1.4 ratio mentioned by the Gemara, it's okay. He quotes a Brisker who implies that ideally one should not rely on such estimations, but technically it's okay. R' Moshe concludes in astonishment of how such a gadol like the Brisker Rav could possibly have said such a thing.

As they are discussing √2=1.4, it seems clear that R' Moshe holds like the above Shulchan Aruch, and while me'ikar hadin the Brisker agrees, ideally he would like you to hold like the Aruch HaShulchan.

TL; DR

CYLOR.

The longer TL; DR

  • The Sefer HaChinuch, Shulchan Aruch, Rambam, one tzad in the Tashbeitz, R' Moshe Feinstein, technically the Brisker Rav, and maybe the Tosfos HaRosh all allow you to rely on π=3 and √2=1.4 as long as you're not dealing with Choshen Mishpat.
  • The other tzad in the Tashbeitz and maybe the Tosfos HaRosh allow you to rely on π=3 by non-Choshen Mishpat but not √2=1.4 even by the other three areas of halacha.
  • The Aruch HaShulchan and ideally the Brisker Rav prohibit relying on these estimations.
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    He quotes a Brisker Was it the Brisker Rav? Someone else quoting him? Not him at all? Consider clarifying. – mevaqesh Sep 9 '16 at 18:32
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    +1 i usually don't like these types of question+ answer combos, even though it's completely acceptable, but this is very nice. I was keeping an eye out to see if you mentioned that Reb Moshe. Btw mevaqesh's issue with your phraseology is spot on. Struck me as weird. – user6591 Sep 9 '16 at 19:02
  • @mevaqesh I don't have his exact lashon in front of me, but he says he heard in the name of the Brisker Rav and doesn't believe he actually said it. – DonielF Sep 11 '16 at 1:06
  • @user6591 here is the original: ולכן כתבתי שהוא פלא מה שאמר אחד שהגרי"ז הלוי זצ"ל היה סבור שיש מעלה כשימדדו ריבוע דתפילין במיקראסקאפ, אך אם הוא אמת יש אולי לתרץ דחשש שמא עושי הבתים לא טרחו כפי יכלתם למדוד היטב שא"כ ליכא אף מדה דהאינשי, שהם אף שא"א לצמצם והתורה הכשירה כל היכא דמצית למעבד הרי צריך למדוד כל היכי דמצי בכל הטירחא וראיית העין שאפשר, ולא יוכשר כשיתעצלו לטרוח ולעיין היטב שזה הא אינו אמת אף למדת האינשי, שלכן החמיר למדוד במיקראסקאפ שזה שייך להחמיר לחוש לזה אף שלדינא אין מחוייבין לחשוש דסומכין על האומדנא דמומחין. ולשון שעה"צ סימן שע"ב אות י"ח הוא תמוה וברור שסומכין אף במדות [cont.] – mevaqesh Sep 11 '16 at 3:34
  • דאורייתא כדסובר בעצמו. – mevaqesh Sep 11 '16 at 3:34

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