Prove that √5 is an irrational number. Hence, show that -3 + 2√5 is an irrational number.

Question

KnowledgeBoat · Accepted Answer

Let be a rational number, then where p, q are integers, q ≠ 0 and p, q have no common factors (except 1) As 5 divides 5q², so 5 divides p² but 5 is prime Let p = 5m, where m is an integer. Substituting this value of p in (i), we get As 5 divides 5m², so 5 divides q² but 5 is prime Thus, p and q have a common factor 5. This contradicts that p and q have no common factors (except 1). Hence, is not a rational number. So, we conclude that is an irrational number. Suppose that is a rational number, say r. Then, = r (note that r ≠ 0) As r is rational and r ≠ 0, so is rational is rational But this contradicts that is irrational. Hence, our supposition is wrong. is an irrational number.

Mathematics

Prove that √5 is an irrational number. Hence, show that -3 + 2√5 is an irrational number.

Rational Irrational Nos

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