Question

1) x and y be rational and irtational numbers , respectively. Is x+ y necessarily a rational number? Give an example in support of your answer.

2) x be rational and y be irrational. Is xy necessarily irrational ? justify your answer and give an example.

Answer 1

Heya

1. The sum of rational and an irrational number is always an irrational number.

For example 2+√2

Let us assume sum of x (rational) and √y (irrational) is a rational number z

x+√y = z

√y = z-x

Difference between rational and a rational number is a rational number.

So we get irrational=rational.

But this is contradiction.

So we can conclude that sum of ray and irrational is irrational.

2. Product of a rational and irrational number is always an irrational number.

Example 4√2 = √16×2 = √32

Let us suppose it is rational.

x√y=z

√y = z/x

Irrational= rational

This is contradiction.

Hope it helps! ^_^