Show that the square root 15 is irrationalSolutionThis proof

Show that the square root 15 is irrational.

Solution

This proof uses the unique prime factorisation theorem that every positive integer has a unique factorisation as a product of positive prime numbers.

Suppose 15=p/q for some p,qN. and that p and q are the smallest such positive integers.

Then p²=15q²

The right hand side has factors of 3 and 5, so p2 must be divisible by 3 and by 5. By the unique prime factorisation theorem, p must also be divisible by 3 and 5.

So p=35k=15k for some kN.

Then we have:

15q²=p²=(15k)²=15(15k²)

Divide both ends by 15 to find:

q²=15k²

So 15=q²/k² and 15=q/k

Now k<q<p contradicting our assertion that p,q is the smallest pair of values such that 15=p/q.

So our initial assertion was false and there is no such pair of integers.

And therefore, 15 is proven to be an irrational number.