Thread: Pure Mathematics problems

Hi my names jacob and im having some problems with these questions, just wondering if anyone can help???
(fraction in lowest terms). Let r 2 Q.
(a) Prove that
r has a least possible denominator n, i.e., r = m=n for some m 2 Z
and
n 2 N and that for no n′ < n is it possible to write r = m′=n′. We say that
such
n is denominator of r in lowest terms and the fraction m=n is the lowest terms
representation of
r as a fraction.
Hint
. Introduce the set
S
:= fl 2 N : 9 k 2 Z : r = k=lg : (3.3.1)
Show that
S is non-empty and apply the well ordering property of N.

(b) Show that if
m=n is the lowest term fraction of r ̸= 0 then m and n are co-prime.

(c) Show that there is a unique pair (
m; n) 2 Z N for which r = m=n in lowest
terms.

When presenting a problem, I would suggest that you should either learn how to write it out in LaTex, or write the it out in the English language. I, and I'd bet many others, are not going to want to suffer through trying to decipher your improvised notation.