Last updated on July 4th, 2025
Operations on rational numbers are the arithmetic operations (addition, subtraction, multiplication, and division). Rational numbers are real numbers that can be expressed as p/q where p and q are integers and q ≠ 0. In this topic, we are going to talk about the operations on rational numbers and their properties.
A number that can be written as a fraction is a rational number. Any fraction where the denominator is not zero qualifies as a rational number. Rational numbers are expressed in the form p/q where q ≠ 0. There are various types of numbers that we can represent as rational numbers:
Fraction form of a rational number: A rational number consists of two integers and can be written in the form p/q.
Decimal form of a rational number: A rational number can be written in the form of a decimal number if the value is terminated or has recurring digits after the decimal point. For example, 0.33333 is a rational number.
Standard form of a rational number: A rational number is expressed as p/q, where p and q are integers with no common factor other than 1. For example, 3/9 is a rational number, but is not in the standard form as both the numerator and denominator have a common factor of 3, and can be further simplified to 1/3. Therefore, the standard form is 1/3.
To identify whether a number is rational or not, here are a few properties that we can use to identify the rational numbers:
Rational Numbers | Rational Numbers |
Any integer or number that we can write as a fraction, like 6, 5/6, or 7/8. | Numbers that cannot be written as fractions, such as 5 |
Here, Rational numbers either have terminating decimals or repeating decimals, such as 0.33333 | In irrational numbers, the decimal goes on forever without repeating, such as 3.141592… |
Can be added, subtracted, divided, or multiplied to produce rational numbers. | When added or multiplied with rational numbers, the result is irrational. |
Example: 0, -5, 0.75, 0.3333 | Example: 𝝅, 2.. |
When we solve for rational numbers, the operations we use are usually addition, subtraction, division, and multiplication. We know that a rational number is expressed in the form p/q. Here, we will explain each operation of rational numbers in detail:
We add rational numbers similar to how we add fractions. When adding rational numbers, there are usually two cases:
When we add two rational numbers with common denominators, we simply need to add the numerators and keep the same denominator. When rational numbers have different denominators, certain steps need to be followed:
Step 1: Since the denominators are different, we need to find the least common denominator.
Step 2: Find the rational number equivalent with the common denominator.
Step 3: Since the denominators are the same, we just need to add the numerators and use the same denominator.
The method of subtracting rational numbers is similar to the method of addition of rational numbers.
Step 1: Find the LCM of the denominators.
Step 2: Rewrite the numbers using a common denominator.
Step 3: Subtract the numbers.
Multiplying rational numbers is similar to multiplying fractions. Here are a few steps to multiply any two rational numbers:
Step 1: We first multiply the numerators
Step 2: Multiply the denominators
Step 3: Simplify the resulting number to its lowest form.
When we want to divide any two rational numbers, we see how many parts of the divisors are in the dividend. Below are the steps used to divide rational numbers:
Step 1: The reciprocal of the divisor should be taken (the second rational number)
Step 2: Multiply it by the dividend.
Step 3: The solution is the product of these two numbers.
Here are some properties that we can apply to operations on rational numbers:
Property | Explanation | Example |
Closure Property | This property states that when two rational numbers are added, subtracted, multiplied, or divided, the result will also be a rational number. | x/y × m/n = xm/yn |
Associative Property | When adding or multiplying three rational numbers, we can arrange the numbers internally without affecting the final answer. This property does not hold for subtraction and division of rational numbers. | x/y + (m/n + p/q) = (x/y + m/n) + p/q |
Commutative Property | This property states that when two rational numbers are added or multiplied, irrespective of their order. | x/y + m/n= m/n + x/y |
Additive/Multiplicative Identity |
0 is the number for the additive identity of any rational number. Here, the result is the number itself. 1 is the multiplicative identity for any rational number. When we multiply 1 with any rational number, the resultant will be the number itself. |
x/y + 0 = x/y x/y × 1 = x/y |
Additive/Multiplicative Inverse |
For any rational number x/y, there exists a negative equivalent of it such that the addition of both numbers gives 0. -x/y is the additive inverse of x/y. Similarly, for any rational number x/y, there exists a reciprocal such that the product of both numbers is equal to 1. y/x is the multiplicative inverse of x/y |
x/y + (-x/y) = 0
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We use rational numbers in various fields by researchers and engineers. Here are a few real-life applications of operations on Rational Numbers:
When learning about operations on rational numbers, students often make mistakes. Here are a few common mistakes that students make on operations on rational numbers and ways to avoid them:
Add 3/4 + 5/6
19/12
Find the LCM of 4 and 6, which is 12.
Convert fractions: 3/4 = 9/12, 5/6 = 10/12.
Add: 9/12 + 10/12 = 19/12.
Subtract 7/8 - 1/6
17/24
The LCM of 8 and 6 is 24
Convert fractions: 7/8 = 21/24, 1/6 = 4/24
Subtract: 21/24 - 4/24 =17/24
Multiply 2/5 × 3/7
6/35
Multiply the numerators: 2 × 3 = 6
Multiply denominators: 5 × 7 = 35
The final result is: 6/35
Divide 4/9 / 2/3
2/3
Take the reciprocal of 2/3, which is 3/2.
Multiply: 4/9 × 3/2 = 4 × 3/9 × 2 = 12/18. (Multiply the numerators with each other and the denominators with each other).
Simplify: 12/18 = 2/3
Add -5/6 + 1/3.
-1/2
The LCM of 6 and 3 is 6
Convert: 1/3 = 2/6
Add: -5/6 + 2/6 = -3/6
= -1/2
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.