In the spirit of the song "Echad - mi yodeya", please post interesting and significant Jewish facts about irrational numbers. I tried dealing with them, but it's just impossible ... They're completely irrational. Who knows? Maybe YOU do!
Please, no lazy gematria about these numbers. I'm looking for something clever.
The Mishnah in Eiruvin 1:5 says that the ratio of the circumference to the diameter is 3:1. Peirush HaMishnayos to that Mishnah notes that three is imprecise; the number (which the Tiferes Yisrael denotes as פ״י) actually goes on forever. Since no matter what value you use will be imprecise, the Tanna cut it off after the first digit.
The Tosfos HaRosh to Eiruvin 14a explains the Gemara's question of "how do we know this ratio of 3:1" as "how do we know that we can rely on this imprecise approximation?" We answer it from the passuk in Melachim 1:7:23 that claims the circumference of the ten-amah-diameter kiyor was thirty amos.
Tosfos to Sukkah 8a prove πr^2.
This topic of irrational mi Yodeya has been breached before: take a look here and at the links quoted in that discussion.
Check out my discussion here regarding relying on approximations to irrational numbers in Halacha, and my discussion here involving logarithms regarding taking away 1/60 of a sick person's disease.
Irrational is the value of לג (the Biblical unit). Allow me to explain:
We find five verses with logs in the Torah (Lev 14:10, 12, 15, 21, and 24), and they are all logs of oil. So if we let x be the value of לג, then x = log (oil).
To figure out what x is, we have to determine the value of oil. Oil, when not otherwise specified, must mean anointing oil (שמן המשחה), which was comprised of a total of 1500 shekels and 1 hin (Ex 30:23-24). Now, 1 hin is equal to 12 logs, which means that oil has the value of 1500 + 12x (by an abuse of units).
One final item before we solve for x: with what base do we use the logarithm? Well, in Lev 14:10, we find "ולג אחד שמן". Since לג is already in singular, do we not already know that one is meant? Rather, the אחד must come to tell us about the base (note also its placement, right after לג). Since log base 1 is meaningless, the natural logarithm must be intended, ie. base e.
Taking the exponential of both side of x = log (1500 + 12x) leads us to the equation e^x = 1500 + 12x. Whenever x is rational, the right-hand side is rational, whereas the left-hand side is irrational. Thus, we conclude that x is irrational.
Irrational is the standard deviation of the number of mishnayot in a perek.
The average is 4192/525, the variance is 1207203900/144703125, and the standard deviation is 2/525*√574859. Since the number inside the square root is prime, its square root is irrational.
Data from this answer.
One irrational thing is b'rachos. Being as they are essentially a statement of faith, and faith is an irrational thing, they epitomize the concept. This is why Chaza"l called them a "coin", or "matbe'a shetav'u chachamim". They gave this appellation carefully, knowing that most things in halacha involving money are indifferent to the form it comes in, but the b'rachos specifically must not be bills but change, which is the root of two.
𝜌 is the quality of generation at זמן המשיח.
How so? Well, we can make a recurrence relation describing the quality of one generation 𝜌(n) compared to the previous generation 𝜌(n-1). Indeed, to get from 𝜌(n-1) to 𝜌(n), we have to add one (it's the next generation of course), but then take the cubed root, due to ירידת הדורות (see, for example, here). Thus, 𝜌(n)^3 = 𝜌(n-1) + 1. As we let time pass (ie. n gets large), we arrive at 𝜌.
Why the cubed root, you might ask, and not the square root or some other root? Well, we assume that function describing the decrease in generational quality is independent of what generation we consider, so we might as well consider the first generation. We start with Adam and Chava, and consider their children. They have three named children (Gen 4:1, 2, 25), clearly indicating that the root is the cubed root.
