Consider the assertion: If r is a non-zero rational and s is irrational then rs is irrational.?

Question

Prove, stating the proof technique that you are using.
Direct proof
Contrapositive proof or
Contradiction

Answer 1

Let´s use contradiction. Suppose rs is rational. Then, there exist integers m and n<>0 (non-zero) such that r*s = m/n. In addition, since r is a non-zewro rational, there exist integers p<>0 and q<> such that r = p/q. It follows that

m/n = p/q * s , which implies that s = (m*q)/(p*n), since p*n <>0. But since m, n, p and q are integers, it follows so are m*q and p*n, which implies s can be given by the ratio of 2 integers. Since this contradicts the assumption that s is a an irrational, the theorem is proved.

Answer 2

it is a simple logic.It is defenitely a irrational.B'coz the intergral the uncertainity in the irrational number is not eliminated by an y mathematical operation.Suppose 10/3=3.333333333333333333333333333333333333333333333333333333333333333333333333333333333............>U try to make a definite form after decimal part by any mathematical form.It gives u a proof.