Question

let a and b be rational and irrational number respectively.is a+b an irrational number? justify your answer

Answer 1

Hi ,

Given ,

a is rational number,

b is irrational number take it as ( sqrt b )

We have to show the number a + ( sqrt b ) is an irrational

Let us assume to the contrary , that ( sqrt b ) is rational.

Then ,

a + sqrt b is rational . [ since sum of two rational is rational ]

So, we can find coprime integers p and q ( q is not equals to zero )

such that

a + sqrt b = p / q

Sqrt b = p / q - a

Since , p and q are integers , we get p / q is rational

and so , ( p / q - a ) is rational and so , sqrt b is rational.

But this contradicts the fact that sqrt b is irrational.

So , we conclude that a + sqrt b is irrational number.


I hope this will useful to you.

****