Question

prove that 3+ root 5is an irrational number​

Answer 1

GIVEN :- 3 + √5

Let us suppose that √5 is irrational.

➡ √5 = p/q (q ≠ 0, p and q are coprime)

➡ √5q = p

Squaring both sides,

➡ (√5q)² = (p)²

➡ 5q² = p² ----------(i)

Therefore 5 divides p ---------(ii)

➡ 5 × r = p

Again squaring both sides,

➡ (5 × r)² = (p)²

➡ 25 × r² = p²

➡ 25 × r² = 5q² (from equation (i)

➡ 5 × r² = q²

5 divides q², therefore 5 divides q ---------(iii)

From (ii) and (iii), we can say that p and q have common factor 5 which contradicts our assumption that p and q are coprime.

Hence, our assumption is wrong and it's proved √5 is irrational.

Now, we know that any number added to irrational number also becomes irrational.

Hence proved that 3 + √5 is an irrational number.