Prove that and are irrational numbers.

Question

KnowledgeBoat · Accepted Answer

Let be a rational number. ∴ where, p,q are integers and q 0. ⇒ (squaring both the sides) ⇒ ⇒ As 3 divides 75q2 , so 3 divides p2 ans so p is also divisible by 3 ...................(1) So, let p = 3m for some integer m. ⇒ (squaring both the sides) ⇒ ∵ ⇒ Since 3m2 is divisible by 3, the right-hand side 25q2 must also be divisible by 3. But 25q2 = 52q2 is not divisible by 3 unless q itself is divisible by 3. Thus, 3 divides q. ...................(2) From 1 and 2, we get p and q both are divisible by 3 i.e., p and q have 3 as their common factor. This contradicts our assumption that is rational i.e. p and q do not have any common factor other than unity (1). ⇒ is not rational. ⇒ is not rational i.e., is irrational. Let be a rational number. ∴ where, p,q are integers and q 0. ⇒ (squaring both the sides) ⇒ ⇒ As 5 divides 45q2 , so 5 divides p2 ans so p is also divisible by 5 ...................(1) So, let p = 5m for some integer m. ⇒ (squaring both the sides) ⇒ ∵ ⇒ Since 5m2 is divisible by 5, the right-hand side 9q2 must also be divisible by 5. But 9q2 = 32q2 is not divisible by 5 unless q itself is divisible by 5. Thus, 5 divides q. ...................(2) From 1 and 2, we get p and q both are divisible by 5 i.e., p and q have 5 as their common factor. This contradicts our assumption that is rational i.e. p and q do not have any common factor other than unity (1). ⇒ is not rational. ⇒ is not rational i.e., is irrational. Hence, and are irrational numbers.

Mathematics

Prove that and are irrational numbers.

Rational Irrational Nos

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