With reference to the above image, why is it for this proof that sqrt(2) is irrational, after making the first assumption that sqrt(2) is rational, we can also make what seem like an additional assumption - that the fraction (m/n) is in its lowest form / irreducible (which is later used to produce a contradiction).
It seems like we are making more than one assumption for this proof by contradiction which begs the following 2 questions:
- How do we know when such additional assumptions can be made? - My classmate have shared that this assumption is fairly made due to a "Proof by infinite descent" - I will like to still hear this community's thought on this / any resources which will point me to a similar direction etc. for a deeper understanding will be greatly appreciated !
- How can we know which assumption is responsible for the contradiction?
Actually we implicitly use here another statement: if $q$ is rational number, then there exist integer numbers $m$ and $n$ s.t. $q = m / n$ and $m / n$ is irreducible.
So, better formulation will be: assume $\sqrt{2}$ is rational, then there exist $m$ and $n$ s.t. $m / n$ is irreducible and $\sqrt{2} = m/n$. Then we use this numbers in the rest of proof.
Or, in other words, we don't need to assume that $m / n$ is irreducible, we can choose it to be irreducible.