Rational Numbers: Definitions, Types, and Properties

In math, a rational number is any number that can be expressed as a fraction p/q where p and q are integers and q≠0. Examples include fractions like 1/2, 3/4​, and even whole numbers like 5 (which can be written as 5/1). 

In this blog, we will cover the definition, properties, and types of rational numbers, highlight the key differences between rational and irrational numbers, and give examples to help you understand how to identify and work with them. 

What are Rational Numbers?

Rational numbers are numbers that can be expressed in the form of a fraction p/q, where p and q are integers, and q is not equal to zero. This means any number that can be written as a ratio of two integers is considered a rational number. 

Rational numbers include integers, terminating decimals, and repeating decimals, as they can all be expressed as fractions. 

Examples of Rational Numbers

Let us consider this with some examples.

• Integers as rational numbers: 5 (5/1), -3 (-3/1), 0 (0/1)

• Terminating decimals as rational numbers: 1.5 (3/2), 2.5 (5/2), 0.8 (4/5)

• Repeating decimals as rational numbers: 0.333... (1/3), 0.666... (2/3)

Since all these numbers can be represented in fractional form, they are classified as rational numbers.

Types of Rational Numbers

Rational numbers can be classified into four types based on their properties and representation. Understanding these types will help in identifying different forms of rational numbers.

1. Positive Rational Numbers

These are rational numbers where both the numerator and denominator are positive, resulting in a value greater than zero. Since the division of two positive integers always yields a positive result, these numbers represent positive fractions.

Examples

• 7/10

• 15/22

• 3/4

Here, 3/4​ can be written in decimal form as 0.75 and remains a positive value.

2. Negative Rational Numbers

These rational numbers have either the numerator or the denominator (but not both) as negative, making the overall value negative.

Examples

• −5/9​

• 8/−11

• −13/6

Here, −5/9 can also be written as -5 ÷ 9 = -0.555... (a repeating decimal).

3. Integer Form of Rational Numbers

All integers are rational numbers because they can be expressed in the form of p/q, where q = 1. Since any integer divided by 1 remains unchanged, integers naturally fit within the category of rational numbers.

Examples: 

• 5 = 5/1

• −12 = −12/1

• 0 = 0/1​

4. Decimal Form of Rational Numbers

Rational numbers can also be represented as decimals. These decimals are either terminating (ending after a finite number of digits) or non-terminating but recurring (a repeating pattern).

Examples

• Terminating Decimal

• 0.5 = 1/2 

• 2.75 = 11/4

• Non-Terminating Recurring Decimal

• 0.333...= 1/3 

• 0.142857...= 1/7​

Each of these types follows the basic property of rational numbers, which is expressed in the form p/q, where q ≠ 0.

Properties of Rational Numbers

A rational number is a subset of the real number. Rational numbers exhibit several key properties, as follows.

• Rational numbers are closed under addition, subtraction, and multiplication.

• Adding 0 to a rational number doesn’t change it, and multiplying by 1 doesn’t change it either.

• Every rational number has an additive inverse, and every non-zero rational number has a multiplicative inverse.

• Multiplying or dividing both the numerator and denominator by the same non-zero integer keeps the value unchanged.

• Multiplication distributes over addition and subtraction.

• Between any two rational numbers, there is always another rational number.

• You can change the order or grouping of rational numbers when adding or multiplying, and it won’t change the result.

Learn more about the Properties of Rational numbers.

Standard Form of Rational Numbers

The standard form of a rational number is when the numerator and denominator have no common factors other than 1. This means the fraction is in its simplest or most reduced form. In other words, the Highest Common Factor (HCF) of the numerator and denominator is 1.

Example:

Let's simplify 24/36 

• Here, the common factors of 24 and 36 are as follows.

• 24 = 1x2x3x4x6x8x12x24

• 36 = 1x2x3x4x6x9x12x18x36

So, the HCF of 24 and 36 is 12.

• Now, divide both the numerator and the denominator by 12:
  24/36 = 24÷12/36÷12 = 2/3

Thus, 24/36​ in standard form is 2/3​.

Also Read: Rational Numbers in Standard Form - Conversion and Solved Examples

Now that we understand how to express rational numbers in their standard form, let's explore how these numbers operate in various arithmetic operations such as addition, subtraction, multiplication, and division.

Arithmetic Operations with Rational Numbers

When working with rational numbers (fractions), there are specific rules to follow for addition, subtraction, multiplication, and division. Here’s how each operation works.

1. Addition/Subtraction

To add or subtract fractions, you need to ensure that the denominators are the same. If the denominators are different, you first need to find a common denominator.

If the denominators are the same, you can add or subtract the numerators as it is. 

Example (Addition): With different denominator

Question: 1/4 + 1/2  

• Here, they both have a different denominator, we need to find a common denominator. The lowest common denominator of 4 and 2 is 4.

• 1/4​ is already in the form with a denominator of 4, so it remains 1/4 only. 

To convert 1/2 into a fraction with a denominator of 4, we multiply both the numerator and denominator by 2 (because 2 x 2 = 4) 

Therefore 1/2 = 1 x 2 / 2 x 2 = 2/4

• Now that both fractions have the same denominator, we can add them. Simply add the numerators and keep the denominator the same: 1/4 + 2/4 = 3/4

Answer: 3/4

Example (Subtraction): With the same denominator

Question: 3/5 − 2/5 

Since they both have the same denominator, we can directly do the subtraction of the numerator while the denominator remains as is.
3 - 2 = 1

Answer: 1/5

2. Multiplication

To multiply two fractions, simply multiply the numerators and multiply the denominators directly.

Example:

2/3 x 4/5 = 2 x 4 / 3 x 5 = 8/15

Answer = 8/15

3. Division

To divide one fraction by another, you multiply the first fraction by the reciprocal (flip) of the second fraction.

The reciprocal of a fraction ab\ba​ is ba\ab​

Example:

3/4 ÷ 2/5 = 3/4 x 5/2 (reciprocal of 2/5) = 3 x 5/4 x 2 = 15/8

​Answer = 15/8

Rational Vs Irrational Numbers

Rational numbers are numbers that can be written as fractions. Where the numerator is an integer, and the denominator is a non-zero integer. In other words, a rational number can be written as p/q, where p and q are integers, and q ≠ 0.

Examples:

• 1/2 (a fraction)

• 3 (can be written as 3/1)

• -5/8

• 0.75 (which is equal to 3/4)

• 0.333... (which is equal to 1/3)

Read more about: Rational and Irrational Numbers

Irrational numbers are numbers that cannot be written as fractions. They are numbers that have non-terminating, non-repeating decimal expansions. 

Example:

• π (Pi): This is the number we use when working with circles, and its decimal form starts like 3.14159... and goes on forever, without repeating.

Conclusion

Understanding rational numbers helps students solve math problems more easily. Here are 5 key takeaways from the blog.

• Rational numbers are numbers that can be written as fractions, including whole numbers and decimals that end or repeat.

• You can add, subtract, multiply, and divide rational numbers by following simple steps.

• Standard form means simplifying fractions to their simplest form.

• Rational numbers have important properties, such as being closed under addition, subtraction, and multiplication, having an additive identity (0) and multiplicative identity (1).

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