Question

Use the well-ordering principle of natural numbers to show that for any positive rational number x ∈ Q, there exists a pair of integers a, b ∈ N such that x = a/b and the only common divisor of a and b is 1.

Answer 1

ANSWER:

Given that proof is needlessly the complex. we claim that every Fraction f con be written irredo cibly icf=alb where as b are coprime indeed by well ordering there enisting a representation f=alb with least de nominator b. if arb. were not coprime then we could canced a common factor dal from all yielding an equal fraction Cuith steallor denominator bld cb contra leastness of b.

nEOT { positive ratind homber) To prove - alb EN n=alb ged Caib) = 1 ic common and divider of a and bins Proof AS IKGQ+ the form of Pla S son con be written in ie x =P/2 Ret ged (819) #1 a dividov other then I thas dividor then there B both panda ged (019) K p=ak and gea Calb)=1 and q=bk Now x= Pla - all la vou x = alb and ged (alb)=1