Prove that the following are irrationals : (i) 1/√2(ii)7√5 (iii) 6+√2

Question

KnowledgeBoat · Accepted Answer

(i) Let us assume, to the contrary, that is a rational number. Then, , where a and b have no common factors other than 1. Solving above equation, Since b and a are integers, is a rational number and so, is rational. We know that is irrational. So, our assumption was wrong. Hence, proved that is an irrational number. (ii) Let us assume, to the contrary, that is a rational number. Then, , where a and b have no common factors other than 1. Since, a, 7, and b are integers, so, is a rational number. This means is rational but this contradicts the fact that is irrational. So, our assumption was wrong. Hence, proved that is an irrational number. (iii) Let us assume, to the contrary, that is rational. Then, , where a and b have no common factors other than 1. Since, a, b, and 6 are integers, so, is a rational number. This means is also a rational number. This contradicts the fact that is irrational. So, our assumption was wrong. Hence, proved that is an irrational number.

Mathematics

Prove that the following are irrationals : (i) 1/√2(ii)7√5 (iii) 6+√2

Irrational Numbers

Related Questions

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.