Can someone help with these problems

1- Prove the number 0.123456789101112.....is irrational.

2- Prove the number e/pi is irrational.

Thanks much!

Originally posted by smaia
1- Prove the number 0.123456789101112.....is irrational.

2- Prove the number e/pi is irrational.

Thanks much!

Question 1: 0.123456789101112... is not periodic, proof: suppose it is periodic with period length n. But in 0.123456789101112... there are 2n consecutive zeros an infinite number of times. So the period can only be n zeros, which is impossible.

Question 2 is an open problem. e/pi is not known to be irrational.

Originally posted by David113
Question 1: 0.123456789101112... is not periodic, proof: suppose it is periodic with period length n. But in 0.123456789101112... there are 2n consecutive zeros an infinite number of times. So the period can only be n zeros, which is impossible.

Question 2 is an open problem. e/pi is not known to be irrational.

Thanks!
are you aware of any existing research for problem 2?

Originally posted by smaia
Thanks!
are you aware of any existing research for problem 2?

http://mathworld.wolfram.com/e.html has some interesting tidbits about e, and about a quarter of the way down the page discusses briefly e/pi as well as (e + pi) which has also not been proven rational or irrational. interestingly, each of these seem likely to be transcendental (i.e. not satisfy any polynomial with integer coefficients) though this has not been proven for average integer coefficients larger than 10^9, or for polynomials of degree higher than 8.

in addition (i know it's slightly off topic from the OP) it's been proven that (e*pi) and (e+pi) are not both algebraic thanks to the Gelfond-Schneider theorem. this is not to say that they are "both not" algebraic, but rather that one or the other or both must be transcendental.

Originally posted by Aetherael
http://mathworld.wolfram.com/e.html has some interesting tidbits about e, and about a quarter of the way down the page discusses briefly e/pi as well as (e + pi) which has also not been proven rational or irrational. interestingly, each of these seem likely to be transcendental (i.e. not satisfy any polynomial with integer coefficients) though this has ...[text shortened]... y are "both not" algebraic, but rather that one or the other or both must be transcendental.

Thanks much!
This is really very interesting stuff.