Rational Numbers (Definition, Properties, Examples) - BYJUS

# Rational Numbers

A rational number is a number that can be written in the $$\frac{p}{q}$$ form where p and q both are integers and q ≠ o. In the following article we will learn about the rational numbers, their properties and operations on them. ...Read MoreRead Less ## What is a Rational Number?

In mathematics, a number that can be written in the fraction form $$(\frac{p}{q})$$ where both the numerator p and denominator q are integers, and the denominator is a non-zero value, is called a rational number.

## A Number is a Rational Number if

• It can be written in $$\frac{p}{q}$$ form, where  p and q both are integers.
• And $$q~\neq~o$$

For example: $$\frac{5}{2}$$, $$\frac{3}{7}$$, $$\frac{4}{1}$$, $$\frac{1}{3}$$ and  $$\frac{0}{1}$$ are rational numbers, but for $$\frac{5}{0}$$ or $$\frac{1}{0}$$  the denominator is equal to zero and the value of $$\frac{5}{0}$$ or $$\frac{1}{0}$$ is not defined, so they are not rational numbers.

A rational number in fractional form can also be expressed in decimal form. The decimal form of rational number is either,

• Terminating, for example 0.125 and 2.75, or,
• Non-terminating and repeating, for example 1.66666666…………

Let us understand this with a few examples:

$$\frac{3}{25}$$ = 0.12, which is a terminating decimal.

$$\frac{2}{8}$$ = 0.25, which is a terminating decimal.

$$\frac{1}{3}$$ = 0.33333…………., which is a non-terminating repeating decimal.

$$\frac{2}{7}$$ = 0.285714285714…………….., which is a non-terminating repeating decimal.

## Properties of Rational Numbers

• When we add, subtract or multiply two rational numbers, the result is a rational number.
• If we multiply or divide the numerator and denominator of a rational number by the same number, the rational number remains the same.
• Addition or subtraction of zero to a rational number does not change the rational number.

## How to write Repeating Decimals as Fractions?

• Apply the following steps to convert a repeating decimal into fraction.

Step 1: Equate the repeating decimal to a variable.

Step 2: Multiply both sides by 10$$^n$$ where n is the number of repeating digits.

Step 3: Subtract the original equation from the equation obtained in step 2.

Step 4: Solve for the variable.

Let us understand this with an example

Coverer 1.$$\overline{3}$$ into a fraction.

The stepwise procedure will be as follows:

Let x = 1.$$\overline{3}$$

x = 1.$$\overline{3}$$                                 [Write the equation]

10 x  $$x$$ = 10 x 1.$$\overline{3}$$                 [There is one repeating digit, so multiply each side by 10$$^1$$ = 10.]

10x = 13.$$\overline{3}$$                            [Simplify]

10x – x = 13.$$\overline{3}$$  – 1.$$\overline{3}$$            [Subtract the original equation, that is, x = 1.$$\overline{3}$$]

9x = 12                               [Simplify]

x = $$\frac{12}{9}$$                                [Solve for x]

x = $$\frac{4}{3}$$                                 [Simplify]

So the repeating decimal 1.$$\overline{3}$$  can be represented as a fraction $$\frac{4}{3}$$.

## Additive Inverse and Multiplicative Inverse of Rational Numbers

Additive inverse: A number, which when added to a given rational number results in zero, is called additive inverse of the rational number.

Additive inverse of $$\frac{p}{q}$$ is $$-\frac{p}{q}$$.

Multiplicative inverse of a rational number: A number, which when multiplied by a given rational number results in one, is called multiplicative inverse of the rational number.

Multiplicative inverse of $$\frac{p}{q}$$ is $$\frac{q}{p}$$.

## Solved Examples

Example 1: Find the value of $$\frac{2}{3}$$ + $$\frac{5}{6}$$  .

Solution:

$$\frac{2}{3}$$  + $$\frac{5}{6}$$       [Write the expression]

$$\frac{4}{6}$$ + $$\frac{5}{6}$$        [Rewrite $$\frac{2}{3}$$ as $$\frac{4}{3}$$]

$$\frac{4~+~5}{6}$$          [Simplify]

$$\frac{9}{6}$$               [Add]

Therefore the value of $$\frac{2}{3}$$ + $$\frac{5}{6}$$  is $$\frac{9}{6}$$.

Example 2: Coververt 0.$$\overline{23}$$ into a fraction.

Solution:

Let x = 0.$$\overline{23}$$

x = 0.$$\overline{23}$$                                              [Write the equation]

100 x $$x$$ = 100 x 0.$$\overline{23}$$                          [There are two repeating digits, so multiply each side by 10$$^2$$ = 100.]

100x = 23.$$\overline{23}$$                                      [Simplify]

100x – x = 23.$$\overline{23}$$ – 0.$$\overline{23}$$                    [Subtract the original equation, that is, x = 0.$$\overline{23}$$]

99x = 23                                           [Simplify]

x = $$\frac{23}{99}$$                                               [Solve for x]

So, 0.$$\overline{23}$$ can be written as $$\frac{23}{99}$$ .

Example 3: Compare the rational numbers $$-\frac{3}{5}$$  and $$-\frac{1}{5}$$.

Solution:

Let us use a number line to compare the given rational numbers.

Graph  $$-\frac{3}{5}$$ and $$-\frac{1}{5}$$ on a number line. $$-\frac{3}{5}$$ is on the left of $$-\frac{1}{5}$$

So, $$-\frac{3}{5}$$ < $$-\frac{1}{5}$$.

Example 4: What is the multiplicative inverse and additive inverse of $$\frac{3}{4}$$.

Solution:

Multiplicative inverse of $$\frac{3}{4}$$ is $$\frac{4}{3}$$.

Additive inverse of $$\frac{3}{4}$$ is $$-\frac{3}{4}$$.