How do you prove that square root 15 is irrational?

1 Answer
Sep 20, 2015

Answer:

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.