Question

4. [5 Pts] Prove that the product of a non-zero rational number and an irrational number is irrational. Can you use a direct proof? Why or why not?

Answer 1

The proof is by contradiction, i.e., indirect proof.

Let p/q be a non-zero rational number and r be an irrational number, then their product is r*(p/q).

Let us assume that their product is a rational number a/b.

therefore, r*(p/q) = a/b

this implies that r = (a/b) *(q/p)

this implies that r = aq/bp

but aq/bp is a rational number and since r is an irrational number, hence this cannot be true.

Therefore r*(p/q) is an irrational number.

This result cannot be proved using a direct proof as for that , we will need to verify all the products obtained while multiplying a rational and an irrational number exhaustively. Since the products are infinite, hence they all cannot be verified that each product produces an irrational number. Hence we prove it by negating the possibility of the product being a rational number which is the only choice left.