proof that $\sqrt(2)$ is irrational / why is the fraction irreducible

Question

The proof that shows the square root of 2 is irrational starts by assuming, for a contradiction, that it is rational. It starts with the assumption that it can be written as p/q where p and q have no factors in common.

Why do we start by assuming p and q have no factors in common? If we went on to find that they have factors in common, we could cancel them later on?

In an actual fraction, can you really keep cancelling indefinitely? — Deepak, Nov 29 '17 at 12:38

score 2 · Answer 1 · answered Nov 29 '17 at 12:42

2

Every natural number can be factored uniquely into a finite product of prime numbers. Hence every fraction of natural numbers will have a simplest form.

answered Nov 29 '17 at 12:42

neptun

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proof that $\sqrt(2)$ is irrational / why is the fraction irreducible

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