Rational Numbers Definition (with examples)
A rational number is of the form \(\dfrac{p}{q}\) where:
 \(p\) and \(q\) are integers, and
 \(q \neq 0\)
The set of rational numbers is denoted by \(Q\) or \(\mathbb{Q}\)
Examples:
 \( \dfrac{1}{4}\)
 \( \dfrac{2}{5}\)
 \(0.3\) (or) \( \dfrac{3}{10}\)
 \(0.7\) (or) \( \dfrac{7}{10}\)
 \(0.151515...\) (or) \( \dfrac{15}{99}\)
Types of Rational Numbers
The different types of rational numbers are:
 Integers like \(1, 0, 5,\) etc.
 Fractions like \( \dfrac{2}{5}\), \( \dfrac{1}{3}\), etc.
 Terminating decimals like \(0.12, 0.625, 1.325\), etc.
 Nonterminating decimals with repeating patterns (after the decimal point) such as \(0.666..., 1.151515...\), etc.
Decimal Representation of Rational Numbers
A rational number can have two types of decimal representations (expansions):
 Terminating
 Nonterminating but repeating
Let's try to understand them better.
Consider \(\begin{align}\frac{a}{b}\end{align}\)
While dividing a number \(a \div b \), if we get zero as the remainder, the decimal expansion of such a number is called terminating.
Example: \(\dfrac{1}{2}\)
While dividing a number, if the decimal expansion continues and the remainder does not become zero, it is called nonterminating.
Example:
\(\begin{align}\frac{1}{3} = 0.33333....\end{align}\) is a recurring, nonterminating decimal.
You can notice that the digits in the quotient keep repeating.
If the decimal expansion is nonterminating and nonrecurring, it is an irrational number.
NonTerminating Decimal and Terminating Decimal Representation
Terminating Decimal Expansion
The terminating decimal expansion means that the decimal representation or expansion terminates after a certain number of digits.
For example:
Here, the decimal expansion of \(\begin{align}\frac{1}{{16}}\end{align}\) terminates after 4 digits.
Note that in terminating decimal expansion, you will find that the prime factorization of the denominator has no other factors other than 2 and 5.
Nonterminating but Repeating Decimal Expansion
The nonterminating but repeating decimal expansion means that although the decimal representation has an infinite number of digits, there is a repetitive pattern to it.
For example:
In nonterminating but repeating decimal expansion, you will find that the prime factorization of denominator has factors other than 2 and 5.
Here is a small activity for you . In the simulation below check if the given rational number has a terminating or non terminating decimal expansion.
How Decimal Expansions Correspond to a Rational Number
It is easy to see why a terminating decimal representation corresponds to a rational number.
Consider a number \({x}\) which has a terminating decimal representation with a certain number of digits (say \({n}\)) after the decimal point.
We take a particular example, with \({n}\) equal to 5:
\[x = 1.23867\]
We can convert this into a rational form easily.
Remove the decimal point, and divide by 10 raised to the power \({n}\) (or 1 followed by \({n}\) zeroes):
\[x = \frac{{123867}}{{{{10}^5}}} = \frac{{123867}}{{100000}}\]
However, it is not so easy to see why a nonterminating but repeating decimal representation is also rational.
To understand this, check the Solved examples section.
 In terminating decimal expansion, the prime factorization of the denominator has no other factors other than 2 and 5
 In nonterminating but repeating decimal expansion, you will find that the prime factorization of the denominator has factors other than 2 and 5
 Think: What type of numbers will represent nonterminating, nonrepeating decimal expansions?
Solved Examples
Example 1 
Express \(\begin{align}\frac{1}{13}\end{align}\) in decimal form.
Solution:
Let's divide 1 by 13
We see that the quotient is 0.0769230769...which is a recurring decimal quotient.
\(\begin{align}\therefore \frac{1}{13} =0.\overline{076923} \end{align}\) 
Example 2 
Determine if \(\begin{align}\frac{11}{25}\end{align}\) is a terminating or a nonterminating number.
Solution:
A rational number is terminating if it can be expressed in the form \(\begin{align} \frac{p}{2^n \times 5^m}\end{align}\)
The prime factorisation of 25 is \(5 \times 5 \)
\( \begin{align}\frac{11}{25} = \frac{11}{2^0 \times 5^2} \end{align}\)
Thus, \( \begin{align}\frac{11}{25}\end{align}\) is a terminating rational number.
\( \begin{align}\therefore \frac{11}{25}\end{align}\) is a terminating number. 
Example 3 
Express \(\begin{align}\frac{1}{27}\end{align}\) using the recurring decimal form of \(\begin{align}\frac{1}{3} =0.33\overline{3} \end{align}\)
Find the value of \( \begin{align}\dfrac{83}{27}\end{align}\)
Solution:
\(\begin{align}\frac{1}{27} = \frac{1}{9} \times \frac{1}{3}\end{align}\)
Given,
\(\begin{align}\frac{1}{3} =0.33\overline{3} \end{align}\)
\(\begin{align}\frac{1}{27} = \frac{1}{9} \times 0.33\overline{3} \end{align}\)
Dividing \(\dfrac{0.333\overline{3}}{9}\) we get, a recurring decimal \(0.\overline{037}\)
\(\begin{align} \frac{83}{27} = 3 \frac{2}{27}\end{align}\)
We can rewrite it as:
\(\begin{align}3 + \frac{2}{27} = 3 +2 \times \frac{1}{27}\end{align}\)
Simplifying further,
\[ \begin{align}
&=3+2 \left( \frac{1}{27} \right)\\\\
&=3 + 2 \times 0.\overline{037} \\\\
&=3 +0. \overline{074} \\\\
&=3.\overline{074}
\end{align}\]
\(\therefore \dfrac{83}{27} =3.\overline{074} \) 
Example 4 
Convert the following into a rational form:
\[x = 0.343434...\]
Solution:
We have,
\[\begin{align} x &= 0.343434 \ldots \\ \Rightarrow 100x &= 34.343434 \ldots \end{align}\]
Subtracting these two, we have,
\[99x = 34\]
Or,
\(\begin{align}x = \frac{34}{99}\end{align}\)
\( \begin{align}\therefore x = \frac{34}{99} \end{align}\) 
Example 5 
Convert the following into a rational form:
\[y = 1.721873873873...\]
Solution:
We have,
\[y = 1.721873873873...\]
Multiplying by 1000 and 1000000, we get,
\[ \Rightarrow \left\{ {\begin{array}{*{20}{l}}
{1000y = 1721.873873873...} \\
{1000000y = 1721873.873873873...}
\end{array}} \right.\]
Subtracting these two, we have,
\[\begin{align}
\left( {1000000  1000} \right)y &= 1721873  1721 \hfill \\
\Rightarrow 999000y &= 1720152 \hfill \\
\end{align} \]
\[ \Rightarrow y = \frac{1720152}{999000}\]
\( \begin{align}\therefore y = \frac{{1720152}}{{999000}}\end{align}\) 
 Without performing long division, state whether the following rational numbers will have a terminating decimal expansion or a nonterminating but repeating decimal expansion
\[\frac{{129}}{{{2^2}{5^7}{7^5}}},\frac{6}{{15}},\frac{{77}}{{210}}\]

Convert \(0.9999...\) into a rational number. Are you surprised? (Tip: Let \(x = 0.9999...\) and then multiply \(10\) on both sides)
Practice Questions
Here are a few activities for you to practice.
Select/Type your answer and click the "Check Answer" button to see the result.
 If a number can be expressed in the form \(\begin{align} \frac{p}{2^n \times 5^m}\end{align}\) where \(p \in Z \) and \(m,n \in W\) then rational number will be a terminating decimal
 Terminating decimal expansion means that the decimal representation or expansion terminates after a certain number of digits
 Every nonterminating but repeating decimal representation corresponds to a rational number even if the repetition starts after a certain number of digits
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 gradewise 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
Frequently Asked Questions(FAQs)
1. Is a decimal a rational number?
Terminating and repeating decimal numbers are rational numbers.
2. How do you know if a decimal is rational?
If a decimal number can be expresed in the form \(\frac{p}{q}\) and \(q \neq 0 \), it is a rational number.
Example: \( 0.25 = \dfrac{25}{100} \) is a rational number.
3. What are the characteristics of a rational number when written as a decimal?
When expressing a rational number in the decimal form, it can be terminating or non terminating and the digits can recur in a pattern.
Example: \(\begin{align}\frac{1}{2} = 0.5 \end{align}\) is a terminating decimal number.
\(\begin{align}\frac{1}{3} = 0.33333...\end{align}\) is a nonterminating decimal number with the digit 3 repeating.
If it is nonterminating and nonrecurring, it is not a rational number.
Example: \(\pi\)
4. Is every decimal number represented as a rational number?
Nonterminating and nonrepeating digits to the right of the decimal point cannot be expressed in the form \(\frac{p}{q}\) hence they are not rational numbers.
5. Is 3.14 a rational number?
A rational number is a number that can be written as a fraction, \(\frac{a}{b}\) where a and b are integers.
Hence, the number 3.14 is a rational number.
6. How to know if a number is irrational?
Irrational number cannot be expressed in the form \(\frac{p}{q}\)
It has endless nonrepeating digits after the decimal point.
Examples: \(\pi = 3.141592…\) , \(\sqrt{2}= 1.414213…\)
7. How do you convert a rational number to a decimal?
To convert fractions to decimals, just divide the numerator by the denominator.
If the division doesn't end evenly, we can stop after a certain number of decimal places and round it off.
8. Is a decimal an integer?
No, they are both different.
Decimals refer to a number system in base of 10, which means it is written using the digits between 0 to 9.
Integers on the other hand are a set of numbers that include natural numbers, their negatives and 0.