For instance, is there a number like 2.3838383838383… and we have written this out to trillions of decimals and we still find the …38383838… pattern, but yet we haven’t proven it’s rational? (i.e. there could be a .38938 somewhere)
Is there any number that has a repeating decimal expansion for every finite sequence we have checked, but that we have not proven is rational?
To expand on this a bit for the OP, what you can do is take some finite length sequence s of digits and chop up a real x between 0 and 1 into blocks Bₖ of this length, call it ℓ. Then on the kth block you say that x[Bₖ]=s if <universally quantified open problem on ℕ> is true at the kth possible instance of the problem, and x[Bₖ]=0 otherwise, where 0 is the length ℓ zero sequence. Since the problem is unsolved, it must be true at every instance checked so far and so is periodic on some long initial segment of digits. It just repeats s over and over until you find a counterexample to the problem statement, x=s⌢s⌢…?… .
Good point
An easier one would be to use the halting problem. For digit n after the decimal, if the Turing machine hasn't halted on step n, digit n=1, else n=0. Any machine that doesn't halt produces the number 0.111... and any that does truncates at some point.
In both cases the result would be rational, though. You could modify your idea to say, if the machine hasn't halted on step n, the digit is 1, else if n is a prime it's 2, otherwise 3 (or some other permutation of digits).
or just swap to digits of pi or e or any other transcendental number.
Note that the opposite happens quite a lot: there are lots of constants for which we have computed thousands or millions of decimals places, found that it does not repeat, but we still don't know for sure that they're irrational.
One reason that what you asked for is likely much less common (so much less common that I can't think of a single instance of it happening) is that if you have a conjectured repeating decimal, then you actually have a conjectured exact form in mind (e.g., if I came up with a number like 2.38383838... in my research, I wouldn't just conjecture that it's rational, I'd conjecture that it's 236/99). And once you have a specific value in mind like that, it's a lot easier to prove it correct.
There's a story that almost fits.
Let π(n) be the number of primes less than or equal to n. Legendre conjectured that as n tends to infinity the quantity log(n) - n/π(n) converges to a limit, and estimated this limit as 1.08366. It was eventually proved to converge to 1, which can therefore be called 'Legendre's Constant'.
Interesting!
Using pi as the function is so cursed
Edit: my bad guys I just wasn’t used to that I’m sorry 😭😭
Pi is used as a function all the time in upper level math. Apart from prime number counting fucntion, it is quite common to use it for projections and for permutations.
Greek letters are a scarce resource in maths - you have to reuse them!
I haven't seen omicron or upsilon used to be honest.
Omicron is used, we just call it "oh", because it's identical to the letter o. Latex doesn't even have code for it, since it's identical to o.
Upsilon yeah, not sure I've ever seen that used in math, and looking at the list I've seen everything else used.
There is no TeX macro for capital alpha, beta, epsilon, zeta, eta, iota, kappa, mu, rho, tau or chi for similar reasons, although this can be slightly problematic for copying or searching via unicode strings.
That being said, I'm fairly convinced that we use O (Latin) rather than Ο (omicron) or О (Cyrillic) or whatever similar characters other alphabets have.
Knuth thinks of big O notation as using a capital Greek omicron. Everyone else thinks of it as a capital Latin O. They are both right, because those are the same letter.
That's cool, hadn't heard that before
Popular in category theory as the name of the projection maps from products. And it shows up in capital form in dependent type theory as the "pi-type" (dependent function)
It’s the standard symbol for that function. If you called it anything else that would be unusual. Pi is also commonly used to represent functions in other contexts, for example permutations are often represented with pi.
😭😭 don't cry 😭😭 🥺🥺 hug 🤗 hug 🤗🤗🫂
In bond percolation on the square lattice at the critical point where the prob ability of putting a bond on an edge is 1/2, it has been conjectured that the probability that two next nearest neighbor vertices belong to the same cluster equals 11/16. So, you could consider a sequence of larger and larger lattices and for each lattice this probability is, of course, a rational number, But the limit to in infinity is then conjectured to be exactly 11/16.
This conjecture follows directly from work I did a long time ago: Equation 44 on page 13 of this article.
R(L,2) = (11 L^2 + 4)/[16 (L^2 -1)]
That this implies that next nearest neigbors have the limiting probability of 11/16 for being connected to each other, is explained here. And you'll find a lot more examples of correlations in bond percolation conjectured to be rational numbers in these papers.
It's usually the other way around. We find a number, it looks irrational, but we can't prove it. For example:
- Feigenbaum's constant is 4.66920160910299067185320382046620161725818557747576863274565134300413433021131473713868974402394801381716...
- Euler's constant is 0.577215664901532860606512090082402431042159335939923598805767234884867726777664670936947063291746749...
We expect both of these numbers to be irrational, and probably transcendental, but we have yet to prove it. In the case of Euler's constant, it's been extremely well-studied and we have computed over a trillion digits, and it hasn't started repeating, but we still haven't proven that it's irrational.
That’s cool! Thanks
This probably goes beyond my understanding of the field, but what prevents us from determining it to be irrational?
Simply knowing lots of digits doesn't prove it, it could be that the period is just really long and we haven't discovered it yet. Unless we find some way in which it being rational leads to a contradiction, then how are we supposed to know?
You can always cheese this.
Pick your favorite open problem P in a system L
Consider the sequence given by s(n)=0 if there is no proof for Pnof length less than or equal to n, and s(n)=a_n if there is a proof for P of length less than n where a_n is the nth digit of e. Now consider x=0.s(1)...s(n)...
Then, so long as there is no proof of P, x will be 0 up to the nth decimal. But it could be the case that P is provable and then x will be irrational (will be e(10n)) for for n the length of the minimal proofs for P.
But this is cheating
1/3 + (e + pi)/g, where g is Graham’s number. Good luck finding the nonrepeating part.
underrated comment
If you multiply by the suspected base it's like asking do we know any numbers which is as far as we know an integer but could be not. And then if we subtract the suspect integer it's asking if we have some construction of a number for all we calculate is zero but could be not. By this equivalence we can substitute in any two numbers constructed in different ways that we suspect are the same and have checked digits but don't know for sure
If you know a number has a repeating decimal expansion, then it must be rational. The number 0.abcdabcdabcd... is abcd/(104 - 1), and that pattern works for repeating decimal sequences of any length, not just 4. (Also, this shows that every integer divides some number of the form 2x5y(10k - 1).
The tricky part is establishing that the number in question has a repeating decimal expansion.
I believe what OP means is a number defined through some external property which is shown to have repeating decimal expansion on computation but hasn’t been shown to be rational
this has nothing to do with the question that was asked.
Every repeating decimal is rational. If you take the repeating digits and divide them by as many 9s as digits, you get the ratio. You can also adjust for a finite number of digits before the sequence begins to repeat.
That isn't what OP asked
No, it can be proven that any repeating decimal expansion can be written rationally, and in my Calculus II course I show how to do it using infinite series.
I think the OP knows that already. The point is that we don't know whether it is repeating or not; just that there is a lot of numerical evidence that it is.
Thank you - I misunderstood their question.
Yeah I’m basically talking about a number that looks rational but hasn’t been proven to be
Aah, I misunderstood what you were asking - you're looking for numbers that are "close" to being rational. We could always create these (as you sort of have), so you must be looking for ones that have arisen in other situations.
I doubt any number like this has naturally shown up, but one can be constructed, for instance by encoding something like the Goldbach conjecture into a number like this.