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Old 2021-01-04, 01:15   #75
Dr Sardonicus
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Feb 2017

32·5·101 Posts

(1) OP seems to be confounding "fractions" (rational numbers) and "decimal fractions," i.e. fractions that can be expressed with a power-of-ten denominator. Not all rational numbers are decimal fractions.

(2) OP also seems to think that invalidating a proof of A automatically proves ~A (not-A). It doesn't. (Here, A is "The square root of 2 is irrational.")

OP, of course, did not invalidate the proof. What he actually did was (1).

Expressing the statement that the (positive) square root of 2 is rational as an equation in positive integers p and q,

(*) p2 = 2*q2

invites a Euclidean proof that the square root of 2 is not rational, because the equation is impossible.

Euclid also proved a result now known as the Fundamental Theorem of Arithmetic, AKA unique factorization.

The equation (*) violates the Fundamental Theorem, because the left side is divisible by 2 evenly many times, while the right side is divisible by 2 oddly many times.
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