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

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

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) \( \dfrac{14}{99}\)

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.

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.

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.

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:

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

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.

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

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:

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

CLUEless in Math? Check out how CUEMATH Teachers will explain Rational numbers to your kid using interactive simulations & worksheets so they never have to memorise anything in Math again!

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

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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.

• Operations on Rational Number
• The decimal representation of Rational Numbers

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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:

• IMO Sample Paper Class 1
• IMO Sample Paper Class 2
• IMO Sample Paper Class 3
• IMO Sample Paper Class 4
• IMO Sample Paper Class 5
• IMO Sample Paper Class 6
• IMO Sample Paper Class 7
• IMO Sample Paper Class 8
• IMO Sample Paper Class 9
• IMO Sample Paper Class 10

To know more about the Maths Olympiad you can click here

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