The square root of 2 is irrational

The square root of 2 is irrational is the claim in number theory that there is no rational number that when multiplied by itself equals the number 2.

One proof by reductio ad absurdum consists of first assuming that the square root of 2 can be written as a rational number. Thus, there is a pair of coprime integers [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] such that their ratio [math]\displaystyle{ \frac{p}q = \sqrt 2 }[/math] (Every rational number has an irreducible representation). And so, [math]\displaystyle{ \frac{p^2}{q^2} = 2 }[/math], and [math]\displaystyle{ p^2=2q^2 }[/math]. As [math]\displaystyle{ p^2 }[/math] has a factor of [math]\displaystyle{ 2 }[/math], it is even, and thus [math]\displaystyle{ p }[/math] is also even. Thus [math]\displaystyle{ p }[/math] can be written as [math]\displaystyle{ 2k }[/math], where [math]\displaystyle{ k }[/math] is an integer. It then follows that [math]\displaystyle{ (2k)^2=2q^2 }[/math], [math]\displaystyle{ 4k^2=2q^2 }[/math], [math]\displaystyle{ 2k^2=q^2 }[/math], and thus [math]\displaystyle{ q^2 }[/math] also has a factor of [math]\displaystyle{ 2 }[/math], and thus [math]\displaystyle{ q }[/math] is also even. If [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] are both even, then they share a common factor of [math]\displaystyle{ 2 }[/math], and thus are not coprime, leading to a contradiction as we have already established that [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] are coprime. And so, no such [math]\displaystyle{ p,q }[/math] can exist, and the square root of 2 is irrational.

Claim

Statement of the claim	The square root of 2 is irrational
Level of certainty	Proven
Nature	Theoretical
Counterclaim
Dependent on	Every rational number has an irreducible representation
Dependency of