Question

can anyone tell me how to prove a sum of rational and irrational is equal to irrational number

Answer 1

Answer:

The proof is given below↓

Step-by-step explanation:

First 'a/b' be any rational number. 'x' be any irrational number.

To prove :- a/b + x = irrational

Assume that a/b + x is rational.

a/b + x = m/n  (where m/n is a rational number)

x = m/n - a/b

Take LCM = $\frac{mb + an}{nb}$

'nb' is an integer which is rational.

'mb' is an integer which is rational.

'an' is an integer which is rational.

∴ we can conclude that 'x' = rational.

But we took 'x' as irrational.

It was our contradiction.

∴ Our assumption was wrong and 'x' is an irrational number.

Hence proved !