4 i If a is rational and b is irrational is ab necessarily i

4) (i) If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational?

(ii) If a is rational and b is irrational, is ab necessarily irrational? (Careful)

(iii) Is there a number a such that a² is irrational, but a⁴ is rational?

(iv) Is there two irrational numbers whose sum and products are both rational?

Solution

i)

Yes

Assume not true and there exist som rational a and irrational b so that

a+b is rational

But sum of two rational numbers is rational

Hence, a+b+(-a)=b is rational which is a contradiction

Hence no such a,b exist

If both a and b are irrational then a+b can be rational

Just choose :b=-a which a is irrational

Then, a+b=0

ii)

No.

Let a=0 and b any irrational number

ab=0 which is rational

iii)

Yes

a=2^{1/4} is irrational

a^2=2^{1/2} is irrational

a^4=2 which is rational

iv)

Yes

Let, a=1-sqrt{2}, b=1+sqrt{2}

a+b=2

ab=(1-sqrt{2})(1+sqrt{2})=1-2=-1