Prove that each of the following numbers is irrational: (i) (ii) 3 -

Question

KnowledgeBoat · Accepted Answer

(i) Let us assume is a rational number. Let = x Squaring both sides, we get; Here, x is rational, ∴ x2 is rational ..................(1) ⇒ x2 - 5 is rational So, is rational. But is irrational, as it is square root of non-perfect square. ⇒ is irrational i.e. x2 - 5 is irrational and so x2 is irrational ....................(2) From (1), x2 is rational, and From (2), x2 is irrational ∴ We arrive at a contradiction. So, our assumption that is a rational number is wrong. Hence, is an irrational number. (ii) Let us assume 3 - is a rational number. Let, 3 - = x Squaring both sides, we get; Here, x is rational, ∴ x2 is rational ..................(1) ⇒ 11 - x2 is rational So, is rational. But is irrational, as it is a square root of non-perfect square. ⇒ is irrational i.e. 11 - x2 is irrational and so x2 is irrational ....................(2) From (1), x2 is rational, and From (2), x2 is irrational ∴ We arrive at a contradiction. So, our assumption that 3 - is a rational number is wrong. Hence, 3 - is an irrational number.

Mathematics

Prove that each of the following numbers is irrational: (i) (ii) 3 -

Rational Irrational Nos

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