Homework Sourselet ab be rationals and x irrational show that if xaxb is ra
let a,b be rationals and x irrational. show that if (x+a)/(x+b) is rational then a=b.
let a,b be rationals and x irrational. show that if (x+a)/(x+b) is rational then a=b.
let a,b be rationals and x irrational. show that if (x+a)/(x+b) is rational then a=b.
Solution
(x+a)/(x+b) is rational
=>
(x+a)/(x+b) = p/q where p,q are integers such that (p,q) = 1
=>
q(x+a) = p(x+b)
=>
(q-p)x = (bp-aq)
but x is irrational =>
q = p
=>
(x+a)/(x+b) = 1
=>
x+a = x+b
=>
a= b
thus proved
