(Solution) - A Interpreting a rational m n as m n 1 -(2025 Original AI-Free Solution)

Paper Details

a) Interpreting a rational m/n as m ? n-1 ? R, use Postulate 1 to prove that

for m, n, p, q, ? ? Z and n, q, ? ? 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.)