Rational numbers

Rational numbers

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Definition of Rational Numbers

A rational number is a number that is of the form \(\dfrac{p}{q}\) where:

  • \(p\) and \(q\) are integers
  • \(q \neq 0\)

Definition of rational numbers - the numerator is an integer and the denominator is a non-zero integer.

The set of rational numbers is denoted by \(Q\).

Examples of Rational Numbers

If a number can be expressed as a fraction where both the numerator and the denominator are integers, the number is a rational number.

Some examples of rational numbers are:

  • \( \dfrac{1}{2}\)
     
  • \( \dfrac{-3}{4}\)
     
  • \(0.3\) (or) \( \dfrac{3}{10}\)
     
  • \(-0.7\) (or) \( \dfrac{-7}{10}\)
     
  • \(0.141414...\) (or) \( \dfrac{14}{99}\)

Different Types of Rational Numbers

There are different types of rational numbers.

We shouldn't assume that only fractions with integers are rational numbers.

The different types of rational numbers are:

  • integers like \(-2, 0, 3,\) etc.
  • fractions whose numerator and denominators are integers like \( \dfrac{3}{7}\), \( \dfrac{-6}{5}\), etc.
  • terminating decimals like \(0.35, 0.7116, 0.9768\), etc.
  • non-terminating decimals with some repeating patterns (after the decimal point) such as \(0.333..., 0.141414...\), etc.

How to Identify Rational Numbers?

In each of the above cases, the number can be expressed as a fraction of integers. 

Hence, each of these numbers is a rational number.

To find whether a given number is a rational number, we can check whether it matches with any of these conditions.

Example:

Is \(0.923076923076923076923076923076...\) a rational number?

Solution:

The given number has a set of decimals \(923076\) which is repeating continuously.

To identify rational numbers, check if a number has repeating patterns of decimals. If it does, it is a rational number.

Thus, it is a rational number.

Is 0 a Rational Number?

Yes, \(0\) is a rational number as it can be written as a fraction of integers like \( \dfrac{0}{1},  \dfrac{0}{-2},...\) etc.

0 can be written as 0 over 1, or 0 over minus 2, or 0 over 3 etc. Thus, 0 is a rational number.

List of Rational Numbers

From the above information, it is clear that there are an infinite number of rational numbers.

Hence, it is not possible to determine the list of rational numbers.

Smallest Rational Number

Since we cannot determine the list of rational numbers, we cannot determine the smallest rational number.

 
important notes to remember
Important Notes
  1. Rational numbers are NOT only fractions but any number that can be expressed as fractions.
  2. Natural numbers, whole numbers, integers, fractions of integers and terminating decimals are rational numbers.
  3. Non-terminating decimals with repeating patterns of decimals are also rational numbers.
  4. If a fraction has a negative sign either to the numerator or to the denominator or in front of the fraction, the fraction is negative. i.e,
    \[ \dfrac{-a}{b} = \dfrac{a}{-b}= - \dfrac{a}{b} \]

Arithmetic Operations on Rational Numbers

Rational numbers can be added, subtracted, multiplied or divided just like fractions.

Adding and Subtracting Rational Numbers

The process of adding and subtracting rational numbers can be done in the same way as fractions.

To add or subtract any two rational numbers, we make their denominators the same and then add the numerators.

Example :

\[ \begin{aligned}\dfrac{1}{2}- \dfrac{-2}{3} &= \dfrac{1}{2}- \left(-\dfrac{2}{3}\right)\\[0.3cm]&=\dfrac{1}{2} \times \dfrac{3}{3}+ \dfrac{2}{3}\times \dfrac{2}{2}\\[0.3cm] &= \dfrac{2}{6}+ \dfrac{4}{6}\\[0.3cm] &= \dfrac{6}{6}\\[0.3cm] &= \color{blue}{\boxed{\mathbf{1}}} \end{aligned}\]

We can learn more about addition of fractions and subtraction of fractions.

Multiplying and Dividing Rational Numbers

The process of multiplying and dividing rational numbers can be done in the same way as fractions.

To multiply any two rational numbers, we multiply their numerators and their denominators separately and simplify the resultant fraction.

Example:

\[\dfrac{3}{5} \times \dfrac{-2}{7} = \dfrac{3 \times -2}{5 \times 7}= \color{blue}{\boxed{\mathbf{\dfrac{-6}{35}}} } \]

To divide any two fractions, we multiply the first fraction (which is dividend) by the reciprocal of the second fraction (which is the divisor).

Example:

\[\dfrac{3}{5} \div \dfrac{2}{7}=\dfrac{3}{5} \times \dfrac{7}{2}= \color{blue}{\boxed{\mathbf{\dfrac{21}{10}}}} \text{ (OR) } \color{blue}{\boxed{\mathbf{2 \dfrac{1}{10}}}}\]


Irrational and Rational Numbers

The numbers which are NOT rational numbers are called irrational numbers.

The set of irrational numbers is represented by \(I\).

The differences between rational and irrational numbers are as follows:

Rational Numbers Irrational Numbers

These are numbers that can be expressed as fractions of integers.

Examples: \(0.75, \dfrac{-31}{5}\), etc

These are numbers that CANNOT be expressed as fractions of integers.

Examples: \(\sqrt{5}, \sqrt[3]{-2}\), etc.

They can be terminating decimals. They are NEVER terminating decimals.

They can be non-terminating decimals with repetitive patterns of decimals.

Example: \(1.414 \, 414 \, 414 ...\) has repeating patterns of decimals where \(414\) is repeating.

They should be non-terminating decimals with NO repetitive patterns of decimals.

Example: \( \sqrt{5} = 2.236067977499789696409173....\) has no repeating patterns of decimals

The set of rational numbers contains all natural numbers, all whole numbers, and all integers. The set of irrational numbers is a separate set and it does NOT contain any of the other sets of numbers.

Examples of rational and irrational numbers are shown. Rational numbers include natural numbers, whole numbers and integers.

We can understand the formation of rational and irrational numbers using the following simulation:

Challenge your math skills
Challenging Questions
  1. Is \( \sqrt{4}\) rational or irrational?
  2. Is \( \dfrac{\sqrt{7}}{2} \) rational or irrational?
  3. Is \(\pi\) rational or irrational?
    Hint: Note that the exact value of \(\pi\) is NOT \(\dfrac{22}{7}\). Its value is a decimal \(3.141592653589793238...\), which has no repeating patterns of decimals.

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Solved Examples

Example 1

 

 

Identify the rational numbers among the following:

\[ \sqrt{4},\, \sqrt{3},\, \dfrac{\sqrt{5}}{2},\, \dfrac{-4}{5},\, \pi,\, 1.41421356237309504...\]

Solution:

A rational number when simplified should either be a terminating decimal or a non-terminating decimal with repeating patterns of decimals.

Thus, the rational numbers among the given numbers are:

\[\mathbf{ \sqrt{4} \text{ and } \dfrac{-4}{5} }\]
Example 2

 

 

Find a rational number that lies between the following rational numbers:

\[ \dfrac{1}{2} \text{ and } \dfrac{2}{3}\]

Solution:

We know that the average of any two numbers lies between the two numbers.

Let's find the average of the given two rational numbers.

\[ \begin{aligned} \dfrac{ \dfrac{1}{2}+ \dfrac{2}{3}}{2} &= \dfrac{\dfrac{3}{6}+ \dfrac{4}{6}}{2}\\[0.3cm] &= \dfrac{ \left(\dfrac{7}{6} \right)}{2}\\[0.3cm] &= \dfrac{ \left(\dfrac{7}{6} \right)}{ \left(\dfrac{2}{1} \right)}\\[0.3cm] &= \dfrac{7}{6} \times \dfrac{1}{2}\\[0.3cm] &= \dfrac{7}{12} \end{aligned} \]

Thus, the rational number is:

\[\mathbf{\dfrac{7}{12}}\]

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Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Important topics 

Given below are the list of topics that are closely connected to Rational Numbers. These topics will also give you a glimpse of how such concepts are covered in Cuemath.


Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. What is a rational number? Give an example.

Any number that is of the form \(\dfrac{p}{q}\) where both \(p\) and \(q\) are integers is a rational number.

Some examples of rational numbers are:

  • \( \dfrac{1}{2}\)
     
  • \( \dfrac{-3}{4}\)
     
  • \(0.3\) (or) \( \dfrac{3}{10}\)
     
  • \(-0.7\) (or) \( \dfrac{-7}{10}\)
     
  • \(0.141414...\) (or) \( \dfrac{14}{99}\)

We can learn more about this under the "Definition of Rational Numbers" section of this page.

2. Is 3.14 a rational number?

Yes, \(3.14\) is a rational number as it is a terminating decimal.

But note that \(\pi\) is NOT a rational number because the exact value of \(\pi\) is NOT \(\dfrac{22}{7}\)

Its value is a decimal \(3.141592653589793238...\) which has no repeating patterns of decimals.

Is 7 a rational number?

Yes, \(7\) is a rational number as it can be expressed as a fraction of integers \(\dfrac{7}{1}\)

3. How can you identify a rational number?

To identify whether a given number is rational, just convert it into decimal form.

If the decimal is either terminating or non-terminating with repeating patterns of decimal, the number is rational.

Otherwise, the number is irrational.

4. Is 0 an irrational number?

No, \(0\) is a rational number as it can be written as a fraction of integers like \( \dfrac{0}{1},  \dfrac{0}{-2},...\) etc.

 
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