a) Interpreting a rational m/n as m • n^–1 ∈ R, use Postulate 1 to prove that for m, n, p, q,  ℓ ∈ Z and n, q, ℓ ≠ 0.

b) Using Remark 1.1, Prove that Postulate 1 holds with Q in place of R.

c) Prove that the sum of a rational and an irrational is always irrational. What can you say about the product of a rational and an irrational?

d) Let m/n, p/q ∈ R with n, q > 0. Prove that

(Restricting this observation to Q gives a definition of "<" on Q.)

Transcribed Image Text:

m P mq +np mp mp m q ng m=-m , and n q nq I G 0 m < 2 9 if and only if mq < np .