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.

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.

Set of Rational Numbers

The set of rational numbers is denoted by Q. It is to be noted that rational numbers include natural numbers, whole numbers, integers, and decimals.

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

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.
• Non-terminating decimals with some repeating patterns (after the decimal point) such as 0.333..., 0.141414..., etc., are rational numbers. These are popularly known as non-terminating repeating decimals.

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 never-ending or non-recurring, 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 non-terminating and non-recurring 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 non-terminating decimals such as 0.333..... Since 0.333... can be written as 1/3, therefore it is a rational number. Therefore, non-terminating 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 \(2\dfrac{1}{10}\)

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 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.
• Non-terminating decimals with repeating patterns of decimals, that is recurring decimals are also rational numbers.

☛ Related Articles

• Prime Numbers
• Composite Numbers
• Even Numbers
• Odd Numbers
• Real Numbers
• Natural Numbers
• Whole numbers
• Irrational Numbers
• Counting Numbers
• Cardinal Numbers
• Even and Odd Numbers

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What is a Rational Number in Math?

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 non-terminating 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 non-terminating 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 non-terminating and non-recurring.

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.

Where do Rational Numbers Start from?

Rational numbers do not start from any specific number. They include all natural numbers, whole numbers, and integers, along with terminating decimals and fractions that can be represented in the form of p/q where q is not equal to 0.

What is the Difference Between Fractions and Rational Numbers?

There is a slight difference between fractions and rational numbers which is given below:

• Any number which can be represented in the form of p/q where 'p' and 'q' are integers and q should not be equal to 0. For example, 1/2, -3/4, 7/9, -100/21, and so on.
• Fractions can be written in the form of p/q, but 'p' and 'q' should be whole numbers. For example, 3/4, 1/9, and so on. This means the value of p and q cannot be negative.

Are all Integers Rational Numbers?

Yes, all integers are rational numbers because rational numbers include all natural numbers, whole numbers, integers, terminating decimals, and fractions in the form of p/q where 'p' and 'q' are integers and q should not be equal to 0.

Are all Whole Numbers Rational Numbers?

Yes, all whole numbers are rational numbers because rational numbers include all natural numbers, whole numbers, integers, terminating decimals, and fractions in the form of p/q where 'p' and 'q' are integers and q should not be equal to 0.