# Irrationality proof

David113
Posers and Puzzles 31 Oct '07 13:21
1. 31 Oct '07 13:212 edits
n is a positive integer. Here is a proof that sqrt(n) can't be a non-integer rational number:
Assume this is not the case. Then there is a minimal positive integer k such that k*sqrt(n) is an integer. But k*sqrt(n)-k*floor(sqrt(n)) is a smaller positive integer having the same property.

My question: can you write a SIMILAR proof, that will show that for all positive integers m and n, the mth root of n can't be a non-integer rational? It is easy to prove this theorem if we are allowed to use the fundamental theorem of arithmetic (=every integer has a unique prime factorization). But I want a proof similar to the proof I gave for square roots.

Thanx ðŸ™‚
2. TheMaster37
Kupikupopo!
31 Oct '07 15:011 edit
Originally posted by David113
But k*sqrt(n)-k*floor(sqrt(n)) is a smaller positive integer having the same property.
What property do you mean?
3. 31 Oct '07 18:28
Originally posted by TheMaster37
What property do you mean?
The property of being an integer when multiplied by sqrt(n).
4. 31 Oct '07 20:56
Originally posted by David113
The property of being an integer when multiplied by sqrt(n).
But that's not a contradiction, as k*sqrt(n)-k*floor(sqrt(n)) is not a multiple of sqrt(n).

You haven't used any property of sqrt(n) in your proof.
5. 01 Nov '07 10:421 edit
Originally posted by mtthw
But that's not a contradiction, as k*sqrt(n)-k*floor(sqrt(n)) is not a multiple of sqrt(n).

You haven't used any property of sqrt(n) in your proof.
Why does k*sqrt(n)-k*floor(sqrt(n)) need to be a multiple of sqrt(n)?

Of course it is a contradiction:

k*sqrt(n)-k*floor(sqrt(n)) is a positive integer, since k*sqrt(n) and k*floor(sqrt(n)) are integers and k*floor(sqrt(n)) < k*sqrt(n).

It is smaller than k, since sqrt(n)-floor(sqrt(n)) < 1.

When multiplied by sqrt(n), the result is an integer:
[k*sqrt(n)-k*floor(sqrt(n))]sqrt(n)=kn-(k*sqrt(n))(floor(sqrt(n)))=difference between two integers=integer.

But k was supposed to be the SMALLEST positive integer that gives an integer when multiplied by sqrt(n).