Contents

Let \(\frac{a}{b}\) and \(\frac{c}{d}\), be any two unique rational numbers.

**Property 1:**

#### Closure Property:

The multiplication of any two rational numbers is a rational number.

Therefore, **( \(\frac{a}{b}\) X \(\frac{c}{d}\) )** is a rational number.

**Verification:**

We have,

1) \(\frac{1}{3}\) X \(\frac{5}{9}\) = \(\frac{(1 X 5)}{(3 X 9)}\)

= \(\frac{5}{27}\)

which is a rational number.

2) \(\frac{-7}{12}\) X \(\frac{5}{6}\) = \(\frac{(-7) X 5}{12 X 6}\)

= \(\frac{-35}{72}\)

which is a rational number.

3) \(\frac{-13}{5}\) X \(\frac{-7}{3}\) = \(\frac{(-13) X (-7)}{5 X 3}\)

= \(\frac{91}{15}\)

which is a rational number.

**Property 2:**

#### Commutativity:

Two rational numbers can be multiplied in any order.

Therefore, **( \(\frac{a}{b}\) X \(\frac{c}{d}\) ) = ( \(\frac{c}{d}\) X \(\frac{a}{b}\) ).**

**Verification:**

We have,

1) \(\frac{5}{2}\) X \(\frac{4}{7}\) = \(\frac{(5 X 4)}{(2 X 7)}\)

= \(\frac{20}{14}\) = \(\frac{10}{7}\),

\(\frac{4}{7}\) X \(\frac{5}{2}\) = \(\frac{(4 X 5)}{(7 X 2)}\)

= \(\frac{20}{14}\) = \(\frac{10}{7}\)

Therefore, \(\frac{5}{2}\) X \(\frac{4}{7}\) = \(\frac{4}{7}\) X \(\frac{5}{2}\)

2) \(\frac{-5}{8}\) X \(\frac{-9}{7}\) = \(\frac{(-5 X -9)}{(8 X 7)}\)

= \(\frac{45}{56}\),

\(\frac{-9}{7}\) X \(\frac{-5}{8}\) = \(\frac{(-9 X -5)}{7 X 8}\)

= \(\frac{45}{56}\)

Therefore, \(\frac{-5}{8}\) X \(\frac{-9}{7}\) = \(\frac{-9}{7}\) X \(\frac{-5}{8}\)

**Property 3:**

#### Associativity:

For any three rational numbers \(\frac{a}{b}\), \(\frac{c}{d}\), \(\frac{e}{f}\), we have **( \(\frac{a}{b}\) X \(\frac{c}{d}\) ) X \(\frac{e}{f}\) = \(\frac{a}{b}\) X ( \(\frac{c}{d}\) X \(\frac{e}{f}\) )** i.e., while multiplying three or more rational numbers, they can be grouped in any order.

**Verification:**

1) Let \(\frac{4}{7}\), \(\frac{-5}{12}\), \(\frac{9}{-11}\) be three rational numbers. Then,

( \(\frac{4}{7}\) X \(\frac{-5}{12}\) ) X \(\frac{9}{-11}\) = \(\frac{-20}{84}\) X \(\frac{9}{-11}\)

= \(\frac{-20}{84}\) X \(\frac{-9}{11}\) = \(\frac{180}{924}\),

\(\frac{4}{7}\) X ( \(\frac{-5}{12}\) X \(\frac{9}{-11}\) ) = \(\frac{4}{7}\) X ( \(\frac{-5}{12}\) X \(\frac{-9}{11}\) )

= \(\frac{4}{7}\) X \(\frac{45}{132}\) = \(\frac{180}{924}\)

Therefore, ( \(\frac{4}{7}\) X \(\frac{-5}{12}\) ) X \(\frac{9}{-11}\) = \(\frac{4}{7}\) X ( \(\frac{-5}{12}\) X \(\frac{9}{-11}\) ).

2) Let \(\frac{-4}{5}\), \(\frac{3}{7}\), \(\frac{-8}{11}\) be three rational numbers. Then,

( \(\frac{-4}{5}\) X \(\frac{3}{7}\) ) X \(\frac{-8}{11}\) = \(\frac{-12}{35}\) X \(\frac{-8}{11}\)

= \(\frac{96}{385}\),

\(\frac{-4}{5}\) X ( \(\frac{3}{7}\) X \(\frac{-8}{11}\) ) = \(\frac{-4}{5}\) X \(\frac{-24}{77}\)

= \(\frac{96}{385}\)

Therefore, ( \(\frac{-4}{5}\) X \(\frac{3}{7}\) ) X \(\frac{-8}{11}\) = \(\frac{-4}{5}\) X ( \(\frac{3}{7}\) X \(\frac{-8}{11}\) ).

**Property 4:**

#### Property of 1:

If \(\frac{a}{b}\) is a rational number, then **( \(\frac{a}{b}\) X 1 ) = ( 1 X \(\frac{a}{b}\) ) = \(\frac{a}{b}\).**

**Verification:**

1) \(\frac{45}{127}\) X 1 = \(\frac{45}{127}\) X \(\frac{1}{1}\)

= \(\frac{45 X 1}{127 X 1}\) = \(\frac{45}{127}\)

2) 1 X \(\frac{-251}{69}\) = \(\frac{1}{1}\) X \(\frac{-251}{69}\)

= \(\frac{1 X (-251)}{1 X 69}\) = \(\frac{-251}{69}\)

1 is called the multiplicative identity for rational numbers.

**Property 5:**

#### Multiplication by 0:

Every rational number multiplied with 0 gives 0. Thus, if \(\frac{a}{b}\) is any rational number, then

**\(\frac{a}{b}\) X 0 = 0 = 0 X \(\frac{a}{b}\).**

**Verification:**

1) \(\frac{75}{13}\) X 0 = \(\frac{75}{13}\) X \(\frac{0}{1}\)

= \(\frac{75 X 0}{13 X 1}\) = \(\frac{0}{13}\) = 0,

0 X \(\frac{75}{13}\) = \(\frac{0}{1}\) X \(\frac{75}{13}\)

= \(\frac{0 X 75}{1 X 13}\) = \(\frac{0}{13}\) = 0.

Therefore, \(\frac{75}{13}\) X 0 = 0 = 0 X \(\frac{75}{13}\).

2) \(\frac{43}{-123}\) X 0 = \(\frac{-43}{123}\) X 0 = \(\frac{-43}{123}\) X \(\frac{0}{1}\)

= \(\frac{-43 X 0}{123 X 1}\) = \(\frac{0}{13}\) = 0,

0 X \(\frac{43}{-123}\) = 0 X \(\frac{-43}{123}\) = \(\frac{0}{1}\) X \(\frac{-43}{123}\)

= \(\frac{0 X -43}{1 X 123}\) = \(\frac{0}{123}\) = 0.

Therefore, \(\frac{43}{-123}\) X 0 = 0 = 0 X \(\frac{43}{-123}\).

**Property 6:**

#### (Distributivity of multiplication over addition):

The multiplication of rational numbers is distributive over addition. Thus, if \(\frac{a}{b}\), \(\frac{c}{d}\) and \(\frac{e}{f}\) are rational numbers, then **\(\frac{a}{b}\) X ( \(\frac{c}{d}\) + \(\frac{e}{f}\) ) = \(\frac{a}{b}\) X \(\frac{c}{d}\) + \(\frac{a}{b}\) X \(\frac{e}{f}\).**

**Verification:**

1) Let \(\frac{2}{3}\), \(\frac{-1}{2}\), \(\frac{5}{-2}\) be three rational numbers. Then,

\(\frac{2}{3}\) X ( \(\frac{-1}{2}\) + \(\frac{5}{-2}\) ) = \(\frac{2}{3}\) X ( \(\frac{-1}{2}\) + \(\frac{-5}{2}\) )

= \(\frac{2}{3}\) X ( \(\frac{-1 + (-5)}{2}\) ) = \(\frac{2}{3}\) X ( \(\frac{-6}{2}\) )

= \(\frac{(2 X -6)}{(3 X 2)}\) = \(\frac{-12}{6}\) = \(\frac{-2}{1}\),

\(\frac{2}{3}\) X \(\frac{-1}{2}\) + \(\frac{2}{3}\) X \(\frac{5}{-2}\) = \(\frac{2}{3}\) X \(\frac{-1}{2}\) + \(\frac{2}{3}\) X \(\frac{-5}{2}\)

= \(\frac{(2 X -1)}{(3 X 2)}\) + \(\frac{(2 X -5)}{(3 X 2)}\)

= \(\frac{-2}{6}\) + \(\frac{-10}{6}\) = \(\frac{-2 + (-10)}{6}\)

= \(\frac{-12}{6}\) = \(\frac{-2}{1}\)

Therefore, \(\frac{2}{3}\) X ( \(\frac{-1}{2}\) + \(\frac{5}{-2}\) ) = ( \(\frac{2}{3}\) X \(\frac{-1}{2}\) ) + ( \(\frac{2}{3}\) X \(\frac{5}{-2}\) ).

2) Let \(\frac{4}{5}\), \(\frac{-3}{4}\), \(\frac{5}{6}\) be three rational numbers. Then,

\(\frac{4}{5}\) X ( \(\frac{-3}{4}\) + \(\frac{5}{6}\) ) = \(\frac{4}{5}\) X ( \(\frac{-3 X 3 + 5 X 2}{12}\) )

= \(\frac{4}{5}\) X ( \(\frac{-9 + 10}{12}\) ) = \(\frac{4}{5}\) X ( \(\frac{1}{12}\) )

= \(\frac{(4 X 1)}{(5 X 12)}\) = \(\frac{1}{15}\),

\(\frac{4}{5}\) X \(\frac{-3}{4}\) + \(\frac{4}{5}\) X \(\frac{5}{6}\) = \(\frac{(4 X -3)}{(5 X 4)}\) + \(\frac{(4 X 5)}{(5 X 6)}\)

= \(\frac{(1 X -3)}{(5 X 1)}\) + \(\frac{(2 X 1)}{(1 X 3)}\) = \(\frac{-3}{5}\) + \(\frac{2}{3}\)

= \(\frac{-3 X 3 + 2 X 5}{15}\) = \(\frac{1}{15}\)

Therefore, \(\frac{4}{5}\) X ( \(\frac{-3}{4}\) + \(\frac{5}{6}\) ) = ( \(\frac{4}{5}\) X \(\frac{-3}{4}\) ) + ( \(\frac{4}{5}\) X \(\frac{5}{6}\) ).

**Property 7:**

#### (Existence of Multiplicative Inverse):

Every non-zero rational number \(\frac{a}{b}\) has its multiplicative inverse \(\frac{b}{a}\). Thus, **( \(\frac{a}{b}\) X \(\frac{b}{a}\) ) = ( \(\frac{b}{a}\) X \(\frac{a}{b}\) ) = 1.**

**Verification:**

Let \(\frac{7}{4}\) be a rational number. Then its multiplicative inverse is \(\frac{4}{7}\).

Therefore, \(\frac{7}{4}\) X \(\frac{4}{7}\) = \(\frac{(7 X 4)}{(4 X 7)}\)

= \(\frac{28}{28}\) = 1.

**Example:**

Reciprocal of 17 is \(\frac{1}{17}\)

Reciprocal of -8 is \(\frac{-1}{8}\)

Reciprocal of \(\frac{-15}{16}\) is \(\frac{-16}{15}\)