Question

Given an example where the quotient of two irrational numbers is a rational number.

Answer 1

Rational Number: Any number which can be expressed in the form of $\frac{p}{q}$, where p and q are integers, $q\ne 0$.

Ex. $2,\frac{1}{3},\frac{6}{11},...$

Irrational Number: Any number which cannot be expressed in the form of $\frac{p}{q}$, where p and q are integers, $q\ne 0$.

Ex. $\sqrt{2},\sqrt{5},\sqrt{11},...$

We have to show that the quotient of two irrational numbers is a rational number.

Step 1: Consider any two irrational numbers.

Let two irrational numbers be $6+9\sqrt{3}\mathrm{and}2+3\sqrt{3}$.

Step 2: Find the quotient of those two irrational numbers.

$\frac{6+9\sqrt{3}}{2+3\sqrt{3}}=\frac{3\left(2+3\sqrt{3}\right)}{2+3\sqrt{3}}=3$

which is a rational number.

Hence, examples of two irrational numbers are $6+9\sqrt{3}\mathrm{and}2+3\sqrt{3}$.