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That's a great question, and the answer is by direct inspection that repeating digits cause the number to be rational. For instance, 0.123123123... is checked

That's a great question, and the answer is by direct inspection that repeating digits cause the number to be rational. For instance, 0.123123123... is checked to be the same as 123/999, a fraction -- hence rational. Similarly, 0.abcdabcdabcd... is the same as abcd/9999. This works for repeating blocks of digits of any length.

coldcache 2y ago
To add on to this, the question then can become why can’t the number start repeating after a certain point (e.g., 3.14133333333…). But then we can represent it as a sum of 3.141 and 0.000333333…, i.e., two rational numbers. Then we can construct a fraction that represents the number.
- empath75 2y ago
  and it can also be shown that all rational numbers repeat because there's only so many remainders possible for a given denominator (all the numbers from 0 to n-1), and as soon as you repeat a remainder, you necessarily have to repeat everything from the last occurrence of that remainder.
BobaFloutist 2y ago
Ok, a different question: How do we know that several orders of magnitude past the digits we've calculated so far, Pi (or e, or 2^1/2, or any irrational number) doesn't start repeating (or end), and turn out to be rational. If an irrational number has to have infinite digits without repeating (or stopping), and we can't calculate infinite digits, how can we ever know that a number is actually irrational? What if it's just an absurdly specific rational number, and we just haven't gotten to the end?
- pfdietz 2y ago
  There are proofs that pi and e are irrational. These proofs aren't entirely trivial (especially for pi), but they're not very long. There are longer proofs that neither number is algebraic (solution to a polynomial equation with rational coefficients.) https://en.wikipedia.org/wiki/Proof_that_e_is_irrational https://en.wikipedia.org/wiki/Proof_that_e_is_irrational https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrationa... The "third" fundamental constant of mathematics, Euler's constant, is thought to be irrational but there is as yet no proof (so it's conceivable it could be rational!)
- xigoi 2y ago
  You don’t need to know the digits of a number to prove that it’s irrational. You just need to demonstrate that it can’t be expressed as the ratio of two integers.
  - IngoBlechschmid 2y ago
    Indeed! For instance, Wikipedia presents a couple of proofs of the irrationality of √2 (including a geometric one). None of these require knowledge of its digits. https://en.wikipedia.org/wiki/Square_root_of_2 https://en.wikipedia.org/wiki/Square_root_of_2
  - pfdietz 2y ago
    I think the person you were responding to doesn't understand proof by contradiction.
- coldcache 2y ago
  You may enjoy reading through this: https://mathoverflow.net/questions/32967/have-any-long-suspected-irrational-numbers-turned-out-to-be-rational https://mathoverflow.net/questions/32967/have-any-long-suspe...

coldcache 2y ago

To add on to this, the question then can become why can’t the number start repeating after a certain point (e.g., 3.14133333333…). But then we can represent it as a sum of 3.141 and 0.000333333…, i.e., two rational numbers. Then we can construct a fraction that represents the number.

empath75 2y ago
and it can also be shown that all rational numbers repeat because there's only so many remainders possible for a given denominator (all the numbers from 0 to n-1), and as soon as you repeat a remainder, you necessarily have to repeat everything from the last occurrence of that remainder.

BobaFloutist 2y ago

Ok, a different question: How do we know that several orders of magnitude past the digits we've calculated so far, Pi (or e, or 2^1/2, or any irrational number) doesn't start repeating (or end), and turn out to be rational. If an irrational number has to have infinite digits without repeating (or stopping), and we can't calculate infinite digits, how can we ever know that a number is actually irrational? What if it's just an absurdly specific rational number, and we just haven't gotten to the end?

pfdietz 2y ago
There are proofs that pi and e are irrational. These proofs aren't entirely trivial (especially for pi), but they're not very long. There are longer proofs that neither number is algebraic (solution to a polynomial equation with rational coefficients.) https://en.wikipedia.org/wiki/Proof_that_e_is_irrational https://en.wikipedia.org/wiki/Proof_that_e_is_irrational https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrationa... The "third" fundamental constant of mathematics, Euler's constant, is thought to be irrational but there is as yet no proof (so it's conceivable it could be rational!)
xigoi 2y ago
You don’t need to know the digits of a number to prove that it’s irrational. You just need to demonstrate that it can’t be expressed as the ratio of two integers.
- IngoBlechschmid 2y ago
  Indeed! For instance, Wikipedia presents a couple of proofs of the irrationality of √2 (including a geometric one). None of these require knowledge of its digits. https://en.wikipedia.org/wiki/Square_root_of_2 https://en.wikipedia.org/wiki/Square_root_of_2
- pfdietz 2y ago
  I think the person you were responding to doesn't understand proof by contradiction.
coldcache 2y ago
You may enjoy reading through this: https://mathoverflow.net/questions/32967/have-any-long-suspected-irrational-numbers-turned-out-to-be-rational https://mathoverflow.net/questions/32967/have-any-long-suspe...