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I find it quite interesting that pi+e and pi*e are not proven to be irrational (although it's proven that at least one of them is irrational [1]). It would be
by FartyMcFarter 2y ago
I find it quite interesting that pi+e and pi*e are not proven to be irrational (although it's proven that at least one of them is irrational [1]).
It would be mind-blowing if either of them were rational numbers, yet it's very hard to prove either way.
[1] https://math.stackexchange.com/a/159353 https://math.stackexchange.com/a/159353
- gosub100 2y agoSame for pi^pi^pi^pi https://youtu.be/BdHFLfv-ThQ?si=HhkJnLU3EVGAbwvz https://youtu.be/BdHFLfv-ThQ?si=HhkJnLU3EVGAbwvz
- tshaddox 2y agoOut of curiosity, why would it be mind-blowing if either of them were a rational number?
- charlieyu1 2y agoIf pi+e=a/b then you can write one as a/b minus the other Which is pretty insane because these two numbers are not supposed to be related
- hn_throwaway_99 2y ago> Which is pretty insane because these two numbers are not supposed to be related Not really, there is Euler's identity: https://en.m.wikipedia.org/wiki/Euler%27s_identity https://en.m.wikipedia.org/wiki/Euler%27s_identity
- sunshowers 2y agoThat one's "just" a special case of how complex numbers happen to work. I think the really cool relationship between e and pi is the fact that the Gaussian integral acts as a fixpoint/attractor when sampling and summing data from any distribution (this is the central limit theorem): ∫(−∞ to ∞) e^(-x²) dx = √π I think the attractor property makes it a little more fundamental in some sense, whereas Euler's identity is "just" one special case of e^ix. The Gaussian is kind of the "lowest energy" or "highest entropy" state of randomness, which I think is really cool.
- octachron 2y agoThis is only true for distributions with finite variance (and the edge case of distribution with slowly growing infinite variance). And for a given variance, gaussian distributions are exactly the maximal entropy distribution.
- pishpash 2y agoe and pi are highly related, both pop out of periodic phenomenon.
- Someone 2y ago> because these two numbers are not supposed to be related Says who? They’re not known to be related in that way, but it’s not like nature set out to prevent such a thing, or that large parts of mathematics would break down if it happened to be the case.
- cbm-vic-20 2y agoIs the result of the addition or multiplication of an irrational number with any other real number not equal to it (and non-zero in the case of multiplication) always irrational? ex: pi + e, pi * e, but also sqrt(2) - 1 or sqrt(3) * 2.54 ?
- aidenn0 2y agoDefinitely not; consider the formula for calculating the log of any base given only the natural logarithm. That can result e.g. in two irrational numbers, the ratio of which are integers.
- AlotOfReading 2y agoNo, sqrt(5)*sqrt(16*5)=20. More trivially, there's always a number y such that z = x*y for a given irrational x. You can give similar examples for all the other basic operations.
- LPisGood 2y agoTake any irrational a where 1/a is also irrational. Then a * 1/a = 1. Even moving from addition and multiplication to exponentials won’t save you: there are irrational numbers to irrational powers that are raational.
- pfdietz 2y agoThe nonconstructive proof of that is simple and fun: either sqrt(2)^sqrt(2) or (sqrt(2)^sqrt(2))^sqrt(2) is just such an example.
- School-Cotton 2y ago> Take any irrational a where 1/a is also irrational. In other words: any irrational at all
- School-Cotton 2y agopi and -pi are both irrational and their sum is zero.
- 2y ago
- rokob 2y agoI could believe pi*e rational but pi+e being rational would blow my mind.
- sunshowers 2y agoI would be shocked if either of them were proven to be rational.
- chongli 2y agoI mean you just have to get to the point where all of the trailing decimal places (bits) form a repeating pattern with finite period. But since there are infinitely many such patterns it becomes extremely hard to rule out without some mechanism of proof.
- programjames 2y agoWell, e^pi - pi = 20, is rational.
- hollerith 2y agoIt is not exactly 20.
- toth 2y agoVery nice, didn't know about that one! In a similar vein, Ramanujan famously proved that e^(sqrt(67) pi) is an integer. And obviously exp(i pi) is an integer as well, but that's less fun. (Note: only one of the above claims is correct)
- mvdtnz 2y agoYou didn't know that one because it's a lie. He's telling lies.
- toth 2y agoCharitably it was a joke, as was my quip about `e^(sqrt(67) pi)`. It is a funnier joke without a disclaimer at the end, but unlike GP I couldn't bring myself to leave one out and potentially mislead some people... What I meant was that I didn't know that `e^pi - pi` is another transcendental expression that is very close an integer. You might think this is just an uninteresting coincidence but there's some interesting mathematics around such "almost integers". Wikipedia has a quick overview [1]. I didn't realize it before, but they have GP's example and also the awesome `e + pi + e pi + e^pi + pi^e ~= 60`. [1] https://en.wikipedia.org/wiki/Almost_integer https://en.wikipedia.org/wiki/Almost_integer
- paulddraper 2y ago> although it's proven that at least one of them is irrational And not particular to e and pi. More generally, at least one of a+b and a*b must be irrational, if an and b are transcendental.