Question

How do you prove that square root 15 is irrational?

Answer 1

See explanation...

Explanation:

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

Suppose `sqrt(15) = p/q` for some `p, q in NN`. and that `p` and `q` are the smallest such positive integers.

Then `p^2 = 15 q^2`

The right hand side has factors of `3` and `5`, so `p^2` 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 = 3 * 5 * k = 15k` for some `k in NN`.

Then we have:

`15 q^2 = p^2 = (15k)^2 = 15*(15 k^2)`

Divide both ends by `15` to find:

`q^2 = 15 k^2`

So `15 = q^2 / k^2` and `sqrt(15) = q/k`

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

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