Question

prove that 2-3 root 5 is an irrational number

Answer 1

let 2-3√5 be a rarional no.
rational number are those which can be written in the form of p/q and P and Q do not have any other factor other than 1.

so, 2-3√5=p/q
2-p/q = 3√5. ( since rational - rational = rational)
(2q-p)/q = 3√5( since rational/rational = rational)
(2q-p)/3p = √5
hence,
(2q-p)/3p is rational if 2-3√5 is also rational
but we know that √5 is irrational and rational no. can never be equal to irrational.
therefore, our assumption is wrong and 2-3√5 is irrational