Question

Prove that the sum of rational and irrational number is always irrational

Answer 1

Let rational number + irrational number = rational number

And we know " rational number can be expressed in the form of pq , where p , q are any integers And q 0 ,
So, we can expressed our assumption As :

pq + x = ab ( Here x is a irrational number )

x = ab - pq
So,

x is a rational number , but that contradict our starting assumption .
Hence

rational number + irrational number = irrational number ( hence proved )

Answer 2

Answer: Let a be a rational number and b be an irrational

number.

Let us assume that a b + is rational, say r .

Then a b + = r

b = r a −

As r and a are both rational numbers, so r a − is a

rational number

b is a rational number.

But this contradicts that b is irrational

Hence, our assumption is wrong. Therefore, a b + is

an irrational number

i e. . the sum of a rational and an irrational number is

always an irrational number.