I'm taking 'Introduction to Mathematical Thinking' on Coursera platform and following proof steps are given :
Proof of $\sqrt{2}$ is irrational.
Assume $\sqrt{2}$ is rational. $\sqrt{2}=p/q$
p and q have no common factors.
Why do p and q have no common factors? Is this a consequence of a property of the rational numbers? As p and q can be rational numbers we can set p = 6, q = 9 so p, q have common factors?
Answer
This is because every rational of the form $$\frac{a}{b}$$ can be simplified to the form $$\frac{p}{q}$$ where $p$ and $q$ are coprime. This follows from the fact that if $\gcd(a,b)=d$ then $a=pd, b=qd$ where $p$ and $q$ are copime as seen here from the property of the common divisor.
So we are trying to express $\sqrt{2}$ in the simplest way possible, which should always be possible if it is a rational.
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