Question

prove √2+✓3 is irrational by using contradiction method​

Answer 1

Answer:

Let us assume that

√2 + √3 is not an irrational

So it becomes rational

√2 + √3 is a rational

√2 + √3 = p/q where p, q belongs to integers and q ≠ 0 .

Squaring on both sides

( √2 + √3 ) 2 = ( p/q ) 2

(√2)2 + (√3)2 + 2 × √2 × √3 = p2 / q2

2 + 3 + 2√6 = p2 / q2

5 + 2√6 = p2 / q2

2√6 = p2 / q2 - 5 /1

2√6 = p2 - 5q2 / q2

√6 = p2 - 5q2 / 2q2

LHS = √6 is an irrational because "6" is not a perfect square

RHS = p2 - 5q2 / 2q2 becomes rational

where p , q belongs to integers and q ≠ 0

But ,

LHS ≠ RHS

It is contradiction to our Assumption

Therefore our assumption is wrong

√2 + √3 is an irrational .