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

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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.

 

img

 

\(-\frac{3}{5}\) 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 \(-\frac{3}{4}\).

Frequently Asked Questions

Yes 0 is a rational number.

An integer consists of all positive and negative numbers including zero. Every integer can be written in fraction form where the denominator is not equal to zero. Hence, all integers are rational numbers.

There are infinite rational numbers between two rational numbers.

If the numerator and denominator of a rational number are of the same sign then the number is known as a positive rational number.

If the numerator and denominator of a rational number are of opposite signs then the number is known as a negative rational number.