Question

Let us assume on the contrary that ( 3+2 sqrt{5} ) is rational. Then there exist co-prime positive integers ( a ) and ( b ) such that
( 3+2 sqrt{5}=frac{a}{b} )
( Rightarrow quad 2 sqrt{5}=frac{a}{b}-3 quad Rightarrow quad sqrt{5}=frac{a-3 b}{2 b} )
( Rightarrow sqrt{5} ) is rational
( therefore a, b ) are integers ( therefore frac{a-3 b}{2 b} ) is a rational
This contradicts the fact that ( sqrt{5} ) is irrational. So, our
supposition is incorrect.
Hence, ( 3+2 sqrt{5} ) is an irrational number.

# Prove that 3+2V5 is irrational.

Solution