Question

Which one of the following is correct and which one is not correct? Give reasons.

1) If 'a' is a rational number and 'b' is irrational, then a+b is irrational.

2) The product of a non-profit rational number with an irrational number is always irrational.

3) Addition of any two irrational numbers can be rational.

4) Division of any two integers is an integer.

Answer 1

(1) true.
If 'a' is a rational number and 'b' is irrational, then a+b is irrational.
proof:

Let p+q=z and suppose z is rational

Then p=a/b and z=c/d for some integers a,b,c,d. Then p-z=a/b - c/d = (ad-bc)/bd.

Since (ad-bc) and bd are both integers, then (ad-bc)/bd is of the form g/h, where g and h are also both integers so (ad-bc)/bd is rational as well.

But (ad-bc)/bd=p-z=q

Therefore q is rational

Contradiction! We’ve previously stated that q is irrational so z(=p+q) must be irrational

(2)true:The product of a non-profit rational number with an irrational number is always irrational.
proof:

assume that a is a rational number and r is an irrational number.

As a is rational we can express it as a=p/q .

We prove it by contradiction.

Let a*r=a' ; a' is rational. As a' is rational we can find p',q' such that a'=p'/q'. So

a*r=a'

(p/q )*r= (p'/q')

r=(p'/q')*(q/p)

r=(p'*q)/(q'*p)

Which implies that r is irrational , a contradiction , which occurred due to our assumption that a' i.e product of rational and irrational is rational .

(3)False:Addition of any two irrational numbers can be rational.
reason : 2sqrt(3) + 5sqrt(3) = 7sqrt(3); which is irrational hence false

(4)False:Division of any two integers is an integer.
reason:7 ÷ 4 =1.75 , the result we get is not an integer.