1) let A be rational and B be irrational number prove that A+B is irrational number 2)prove that product of rational and irrational number is irrational number. 3)if a is rational and b is irratonal then a/b is irrational number.

If a is rational and b is irrational, to prove that (a+b) is irrational. 

We prove this result by contradiction,

let if possible (a+b) is rational.

Since (a+b) is rational and 'a' is rational and the difference of two rational number is rational,

so (a+b)–a is also a rational number.

This gives 'b' is rational.

Which contradict that "b" is irrational.

Hence our supposition is wrong and (a+b) is irrational. 

Similar proof can be done for others.