Prove: For all a,b in R,

if a is rational,

a+b is irrational

then b is irrational

PF

Let a be rational, and b be irrational. Then a=c/d for some c,d and there is no e,f such that e/f=b.

Assume a+b is rational. Then there exists a j,k, such that j/k=a+b.

Then b=(a+b)-a = j/k-c/d = (jd - ck)/ dk.

Hence e= dj-ck and f= dk.

Contradiction