Prove: For all a,b in R,
if a is rational,
a+b is irrational
then b is irrational
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.