Prove that and are irrational numbers.

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.

A man saves ₹ 5,000 every year and invests it at the end of the year at 10% p.a. compound interest. Calculate the total amount of his savings at the end of the third year.

View Answer Bookmark Now

Mathematics

Prove that and are irrational numbers.

Rational Irrational Nos

Related Questions

A man saves ₹ 5,000 every year and invests it at the end of the year at 10% p.a. compound interest. Calculate the total amount of his savings at the end of the third year.