One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, also known as an indirect proof, in that the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, thereby implying that the proposition must be true.
Assume that is a rational number, meaning that there exists an integer and an integer in general such that .
Then can be written as an irreducible fraction such that and are coprime integers.
It follows that and . ( )
Therefore is even because it is equal to . ( is necessarily even because it is 2 times another whole number and even numbers are multiples of 2.)
It follows that a must be even (as squares of odd integers are never even).
Because a is even, there exists an integer k that fulfills: .
Substituting from step 6 for a in the second equation of step 3: is equivalent to , which is equivalent to .
Because is divisible by two and therefore even, and because , it follows that is also even which means that b is even.
By steps 5 and 8 a and b are both even, which contradicts that is irreducible as stated in step 2.
Q.E.D.
Because there is a contradiction, the assumption (1) that is a rational number must be false. This means that is not a rational number; i.e., is irrational.
This proof was hinted at by Aristotle, in his Analytica Priora, §I.23.[10] It appeared first as a full proof in Euclid's Elements, as proposition 117 of Book X. However, since the early 19th century historians agree that this proof is an interpolation and not attributable to Euclid.[11]