- Proof that the √5 is irrational:
This will be a proof by contradiction.
- Assume that the √5 is rational (can be expressed in the form , where a and b are integers.
- Then √5 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 5.
- It follows that a2 / b2 = 5 and a2 = 5 b2.
- Therefore a2 is divisible by 5 because it is equal to 5 b2 which is obviously divisible by 5.
- It follows that a must be divisible by 5.
- Because a is divisible by 5, there exists an integer k that fulfills: a = 5k.
- We insert the last equation of (3) in (6): 5b2 = (5k)2 is equivalent to 5b2 = 25k2 is equivalent to b2 = 5k2.
- Because 5k2 is divisible by 5 it follows that b2 is also divisible by 5 which means that b is divisible by 5 because only numbers divisible by 5 have squares divisible by 5.
- By (5) and (8) a and b are both divisible by 5, which contradicts that a / b is irreducible as stated in (2).
Therefore, √5 is irrational.