Question

Give an example of each, of two irrational numbers whose: 

(i) Difference is a rational number.

(ii) Difference is an irrational number.

(iii) Sum is a rational number.

(iv) Sum is an irrational number.

(v) Product is a rational number.

(vi) Product is an irrational number.

(vii) Quotient is a rational number.

(viii) Quotient is an irrational number.

Answer 1

(i) \(\sqrt3\) is an irrational number.

Now,\((\sqrt3)-(\sqrt3)\) = 0

0 is the rational number. 

(ii) Let two irrational numbers are \(5\sqrt{2}\) and \(\sqrt2\)

Now, \((5\sqrt{2})-(\sqrt2)\) = \(4\sqrt{2}\)

\(4\sqrt{2}\) is an irrational number.

(iii) Let two irrational numbers be \(\sqrt{11}\) and \(-\sqrt{11}\)

Now,\((\sqrt11)+(\sqrt11)\) = 0

0 is a rational number 

(iv) Let two irrational numbers are \(4\sqrt6\) and \(\sqrt6\)

Now,\((4\sqrt6)+(\sqrt6)\) = \(5\sqrt6\)

\(5\sqrt6\) is an irrational number. 

(v) Let two irrational numbers are \(2\sqrt3\) and \(\sqrt3\)

Now, \(2\sqrt3\times\sqrt3\) = \(2\times3\)

= 6 

6 is a rational number. 

(vi) Let two irrational numbers are \(\sqrt2\) and \(\sqrt5\)

Now, \(\sqrt2\times\sqrt5\) = \(\sqrt10\)

\(\sqrt10\) is a irrational number.

(vii) Let two irrational numbers are \(3\sqrt6\) and \(\sqrt6\)

Now, \(\frac{3\sqrt6}{\sqrt6}\) = 3

3 is a rational number.

(viii) Let two irrational numbers are \(\sqrt6\) and \(\sqrt2\)

Now, \(\frac{\sqrt6}{\sqrt2}\) = \(\frac{\sqrt3\times\sqrt2}{\sqrt{2}}\)

\(= \sqrt3\)

\(\sqrt3\) is an irrational number.