[FOM] Re: The Search for Mathematical Roots
Timothy Y. Chow
tchow at alum.mit.edu
Sat Mar 6 21:17:14 EST 2004
Marcus Schaefer wrote:
> I recently purchased Ivor Grattan-Guinness' "The Search for Mathematical
> Roots 1870-1940".
Your quotations do sound hair-raising. This is a bit of a tangent, but
one of the quotations raised a question in my mind.
> "He also proved here the existence of irrational numbers by a lovely
> reductio argument that has never gained the attention that it deserves:
> assume that the equation t^2 = Du^2 in integers (D not a square) has a
> solution, and let u be the smallest integer involved; then exhibit a
> smaller integer also to satisfy the equation, a contradiction which
> establishes sqrt(D) as irrational."
This sounds like a garbled version of the following proof that if D is not
the square of an integer, then sqrt(D) is irrational. Suppose to the
contrary that sqrt(D) is rational but not an integer. Then there exist
positive integers n such that n*sqrt(D) is an integer; pick the least
such n. But now let m = n*(sqrt(D) - floor(sqrt(D))); then m is a
positive integer less than n and m*sqrt(D) is an integer, which is a
contradiction.
In particular there is no appeal made to unique factorization.
Is this argument due to Dedekind?
Tim
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