It would be mind-blowing if either of them were rational numbers, yet it's very hard to prove either way.
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)
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`.
In a 1975 April Fool article in Scientific American magazine,[8] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it – hence its name.
It is not an integer of course.
FYI, the reason you get these almost integers is related to the `n`s being Heegner numbers, see https://en.wikipedia.org/wiki/Heegner_number.
Of course? I’m not aware that we have some theorem other than “we computed it to lots of decimals, and it isn’t an integer” from which that follows.
Basically e^(sqrt(163)*pi) is the leading term in a Laurent series for an integer, and the other (non-integer) terms are really small but not zero.
Edit: looks like I swallowed the bait, hook like and sinker
e^(i theta) = cos theta + i sin theta
That formula gives infinitely many trivial relationships like this due to the symmetry of the unit circle
e^(i 2 pi) = 1
e^(3i/2pi)/i=1
e^(5i/2pi)/i=-1
e^(i 2n pi) = 1 for all n in Z ...
etc
Even moving from addition and multiplication to exponentials won’t save you: there are irrational numbers to irrational powers that are raational.
In other words: any irrational at all
Otherwise, supposing for instance that (n/m)x is rational for integers n, m, both non-zero, and irrational x, we can express (n/m)x as a ratio of two integers p, q, q non-zero: (n/m)x = p/q if and only if x = (mp)/(qn). Since integers are closed under multiplication, x is rational, against supposition; thus by contradiction (n/m)x is irrational for any rational r = (n/m), with integers n, m both non-zero. Similarly for the case of addition.
irrational number + irrational number could be rational or irrational.
5 - sqrt(2) is irrational
sqrt(2) is irrational
Add them up you get 5, which is rational
[1] If it were rational, you will be able to construct a rational representation of the irrational number using this equation.
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1. For p,q∊ℚ, let p=a/b, q=c/d, a,b,c,d∊ℤ, then p+q=(ad+bc)/bd, but the products and sums of integers are integers, so p+q∊ℚ
when x is an irrational number > 1:
"x - floor(x)" is just the fractional part of x, so it's an irrational number which is not equal to x.
Subtracting the fractional part from the original leaves only the integer part, which is obviously rational.
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
∫(−∞ 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.
And for a given variance, gaussian distributions are exactly the maximal entropy distribution.
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.
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.