Rational Numbers
Rational numbers are in the form of p/q, where p and q can be any integer and q ≠ 0. This means that rational numbers include natural numbers, whole numbers, integers, fractions of integers, and decimals (terminating decimals and recurring decimals). Let us learn more about rational numbers and how to identify rational numbers and examples of rational numbers in this lesson.
1.  What are Rational Numbers? 
2.  Types of Rational Numbers 
3.  How to Identify Rational Numbers? 
4.  Rational vs Irrational Numbers 
5.  FAQs on Rational Numbers 
What are Rational Numbers?
The word 'rational' originated from the word 'ratio'. So, rational numbers are well related to the concept of fractions which represent ratios. In other words, If a number can be expressed as a fraction where both the numerator and the denominator are integers, the number is a rational number.
Rational Numbers Definition
A rational number is a number that is of the form p/q where p and q are integers and q is not equal to 0. The set of rational numbers is denoted by Q. Observe the following figure which defines a rational number.
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 as follows.
 56 (which can be written as 56/1)
 0 (which is another form of 0/1)
 1/2
 √16 which is equal to 4
 3/4
 0.3 or 3/10
 0.7 or 7/10
 0.141414... or 14/99
Types of Rational Numbers
The different types of rational numbers are given as follows.
 Integers like 2, 0, 3, etc., are rational numbers.
 Fractions whose numerators and denominators are integers like 3/7, 6/5, etc., are rational numbers.
 Terminating decimals like 0.35, 0.7116, 0.9768, etc., are rational numbers.
 Nonterminating decimals with some repeating patterns (after the decimal point) such as 0.333..., 0.141414..., etc., are rational numbers. These are popularly known as nonterminating repeating decimals.
How to Identify Rational Numbers?
Rational numbers can be easily identified with the help of the following characteristics.
 All integers, whole numbers, natural numbers, and fractions with integers are rational numbers.
 If the decimal form of the number is terminating or recurring as in the case of 5.6 or 2.141414, we know that they are rational numbers.
 In case, the decimals seem to be neverending or nonrecurring, then these are called irrational numbers. As in the case of √5 which is equal to 2.236067977499789696409173... which is an irrational number.
 Another way to identify rational numbers is to see if the number can be expressed in the form p/q where p and q are integers and q is not equal to 0.
Example: Is 0.923076923076923076923076923076... a rational number?
Solution: The given number has a set of decimals 923076 which is recurring and repeated continuously. Thus, it is a rational number.
Let us take another example.
Example: Is √2 a rational number?
Solution: If we write the decimal value of √2 we get √2 = 1.414213562....which is a nonterminating and nonrecurring decimal. Therefore, this is not a rational number. It is an irrational number.
Rational Numbers in Decimal Form
Rational numbers can also be expressed in decimal form. Do you know 1.1 is a rational number? Yes, it is because 1.1 can be written as 1.1= 11/10. Now let us talk about nonterminating decimals such as 0.333..... Since 0.333... can be written as 1/3, therefore it is a rational number. Therefore, nonterminating decimals having repeated numbers after the decimal point are also rational numbers.
Is 0 a Rational Number?
Yes, 0 is a rational number as it can be written as a fraction of integers like 0/1, 0/2,... etc. In other words, 0/5 = 0, 0/2 = 0, 0/1 = 0, and so on.
List of Rational Numbers
From the above information, it is clear that there is an infinite number of rational numbers. Hence, it is not possible to determine the whole list of rational numbers. However, a few rational numbers can be listed as 3, 4.57, 3/4, 0, 7, and so on. This shows that all natural numbers, whole numbers, integers, fractions, and decimals (terminating decimals and recurring decimal numbers) are considered to be rational numbers.
Adding and Subtracting Rational Numbers
For adding and subtracting rational numbers, we use the same rules of addition and subtraction of integers. Let us understand this with the help of an example.
Example: Solve 1/2  (2/3)
Solution: Let us solve this using the following steps:
 Step 1: As we simplify 1/2  (2/3), we will follow the rule of addition and subtraction of numbers which says that the subtraction fact can change to an addition fact and the sign of the subtrahend gets reversed. This will make it 1/2 + 2/3
 Step 2: Now, we need to add these fractions 1/2 + 2/3
 Step 3: Using the rules of addition of fractions, we will convert the given fractions to like fractions to get common denominators so that it becomes easier to add them. For this, we need to find the LCM of the denominators 2 and 3 which is 6. Then we will convert the fractions to their respective equivalent fractions which will make them 3/6 + 4/6. This will give the sum as 7/6 which can be written in the form of a mixed fraction \(1\dfrac{1}{6}\)
Multiplying and Dividing Rational Numbers
The multiplication and division of 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. Let us understand this with the help of an example.
Example: Multiply 3/5 × 2/7
Solution: Let us solve this using the following steps:
 Step 1: In order to multiply 3/5 × (2)/7, we will first multiply the numerators and then multiply the denominators.
 Step 2: In this case, when we multiply the numerators, it will be 3 × (2) = 6.
 Step 3: When we multiply the denominators, it will be 5 × 7 = 35. Therefore, the product will be 6/35.
When we need to divide any two fractions, we multiply the first fraction (which is the dividend) by the reciprocal of the second fraction (which is the divisor). Let us understand this with the help of an example.
Example: Divide 3/5 ÷ 2/7
Solution: Let us solve this using the following steps:
 Step 1: In order to divide 3/5 ÷ 2/7, we will first write the reciprocal of the second fraction. This will make it 3/5 × 7/2
 Step 2: Now, we will multiply the numerators This will be 3 × 7 = 21.
 Step 3: Then, we will multiply the denominators, it will be 5 × 2 = 10. Therefore, the product will be 21/10 or \(2\dfrac{1}{10}\)
Rational vs Irrational Numbers
The numbers which are NOT rational numbers are called irrational numbers. The set of irrational numbers is represented by Q´. The difference between rational and irrational numbers can be understood from the following figure and table given below.
Rational Numbers  Irrational Numbers 

These are numbers that can be expressed as fractions of integers. Examples: 1/2, 0.75, 31/5, etc 
These are numbers that cannot be expressed as fractions of integers. Examples: √5, π, etc. 
They are terminating decimals.  They are NEVER terminating decimals that do not have an accurate value. 
They can be nonterminating decimals with repetitive patterns of decimals or recurring decimals. Example: 1.414, 414, 414 ... has repeating patterns of decimals where 414 is repeating. 
They should be nonterminating decimals with NO repetitive patterns of decimals. Example: √5 = 2.236067977499789696409173.... has no repeating patterns of decimals 
The set of rational numbers contains allnatural 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. 
Rational Numbers Tips and Tricks
 Rational numbers include fractions and any number that can be expressed as fractions.
 Natural numbers, whole numbers, integers, fractions of integers, and terminating decimals are rational numbers.
 Nonterminating decimals with repeating patterns of decimals, that is recurring decimals are also rational numbers.
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Rational Numbers Examples

Example 1: Identify the rational numbers among the following: √4, √3, √5/2, 4/5, π, 1.41421356237309504.....
Solution:
A rational number when simplified should either be a terminating decimal or a nonterminating decimal with a repeating pattern of decimals. Therefore, the rational numbers among the given numbers are √4 (which results in 2) and 4/5.

Example 2: State true or false with respect to rational numbers.
a.) Every integer is a rational number.
b.) Every rational number is a whole number.
Solution:
a.) True, every integer is a rational number.
b.) False, every rational number is not a whole number. It can be a fraction of integers or a decimal. Rational numbers include natural numbers, whole numbers, integers, and fractions of integers.

Example 3:
Write the following rational number in decimal form: 1/2
Solution:
The rational number 1/2 can be converted to a decimal number by dividing the numerator by the denominator. We need to divide 1 by 2 and we get 0.5
FAQs on Rational Numbers
What is a Rational Number?
Any number in the form of p/q where p and q are integers and q is not equal to 0 is a rational number. Examples of rational numbers are 1/2, 3/4, 0.3, or 3/10.
How to Identify a Rational Number?
To identify whether a given number is rational or irrational, we need to convert it into its decimal form. If the decimal is terminating or nonterminating with repeating patterns of decimal, then the number is rational. Otherwise, it is irrational. For example, 1.414, 414, 414 ... has repeating patterns of decimals where 414 is repeating. So, it is a rational number. On the other hand, if we take the example of √5 = 2.236067977499789696409173...., we can see that it is nonterminating and has no repeating patterns of decimals. So, it is an irrational number. Another way to identify a rational number is to check whether it is a natural number, a whole number, an integer, or a fraction of integers. If it is one of these, then it is a rational number.
What are Terminating Rational Numbers?
Terminating rational numbers are those decimal numbers that end after a specific number of decimal places. For example, 1.5, 3.4, 0.25, etc are terminating numbers. All terminating numbers are rational numbers as they can be written in the form of p/q easily.
What is the Difference Between Rational and Irrational Numbers?
A rational number is a number whose decimal form is finite or recurring in nature. For example, 2.67 and 5.666...Whereas, irrational numbers are those numbers whose decimal form neither terminates nor repeats after a specific number of decimal places. For example, √5 = 2.236067977499789696409173.... has no repeating patterns of decimals and it does not end, so it is an irrational number.
What are Irrational Numbers?
Irrational numbers are those that cannot be represented using integers in the p/q form. The set of irrational numbers is denoted by Q´. A few examples of irrational numbers are √2, √5, and so on. Their decimal forms are nonterminating and nonrecurring.
Is 0 a Rational Number?
Yes, 0 is a rational number as we can write it as 0/1 where 0 and 1 are integers and the denominator is not equal to 0.
What Number is Added to Pi to Get a Rational Number?
If we add  π to π, we get,  π + π = 0. This sum is a rational number. Therefore, by adding  π to π we get a rational number.
How to Add Rational Numbers?
For adding rational numbers, we use the same rules of addition of integers. Let us understand this with the help of an example. If we need to add 4.53 + 2.31, we will start adding from the right side. After arranging the numbers in columns according to their place value, we will start from the hundredth place. So, 3 + 1 = 4. Moving on to the tenths column, we add 5 + 3 = 8. Then, we add the digits in the ones column, that is, 4 + 2 = 6. So, we get the sum as 6.84
How to Divide Rational Numbers?
In order to divide rational numbers, we use the usual rules of division of integers, fractions or decimals, whatever the case may be. For example, let us divide the fractions 7/5 ÷ 1/5. In order to divide 7/5 ÷ 1/5, we will first write the reciprocal of the second fraction. This will make it 7/5 × 5/1. Now, we will multiply the numerators This will be 7 × 5 = 35. Then, we will multiply the denominators, it will be 5 × 1 = 5. Therefore, the product will be 35/5 which is equal to 7.
How to Multiply Rational Numbers?
Rational numbers can be multiplied using the rules of multiplication. For example, let us multiply 43 × 10, we get the product as 430.
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