Question

Which of the following is not true?
(a) Rational numbers are closed under addition.
(b) Rational numbers are closed under subtraction.
(c) Rational numbers are closed under multiplication.
(d) Rational numbers are closed under division.

Answer 1

Addition of two rational numbers:

Addition of any two rational numbers results in a rational number.
For example:
$\frac{1}{2}+\frac{2}{3}=\frac{1\times 3}{2\times 3}+\frac{2\times 2}{3\times 2}=\frac{3}{6}+\frac{4}{6}=\frac{7}{6}$

Hence, rational numbers are closed under addition.

Subtraction of two rational numbers:

Subtraction of any two rational numbers results in a rational number.
For example:

$\frac{3}{2}-\frac{1}{2}=\frac{3-1}{2}=\frac{2}{2}=1$

Hence, rational numbers are closed under subtraction.

Multiplication of two rational numbers:

Multiplication of any two rational numbers results in a rational number.
For example:
$\frac{3}{4}\times \frac{2}{5}=\frac{3\times 2}{4\times 5}=\frac{6}{20}=\frac{3}{10}$

Hence, rational numbers are closed under multiplication.

Division of two rational numbers:

Division of any rational number by 0 is undefined.

Hence, rational numbers are not closed under division.

The correct option is d.