Question

If n ' is a rational number and √y is irrational then prove that x+√y is rational?

Answer 1

Let us assume to the contrary that x+√y is rational
So x+√y can be written in the form a/b,where a and b are co-primes and b not equal to 0
x+√y=a/b
√y=(a/b)-x
√y=(a-bx)/b
Since x,a,b are all integers, therefore they are rational
But this contradicts the fact that√y is irrational
Hence our assumption is incorrect.
Therefore x+√y is irrational.