Rational numbers are one of the most important parts of mathematics. This guide will help you learn their definition, types, properties, and real-life examples.
By the end of this blog, you will know exactly what rational numbers are, how to work with them, and how they differ from irrational numbers.
Let’s start with the meaning of rational numbers and then move through their types, operations, and examples step by step.
Definition of Rational Numbers
A rational number is any number that can be expressed in the form p/q, where:
- p and q are integers
- q ≠ 0
This means rational numbers include fractions, terminating decimals, and repeating decimals.
Examples:
- 3/4, -5/2, 0.75, -7, 8
"Get success in maths with our 1-on-1 online maths tutors."
Types of Rational Numbers
Rational numbers can be divided into categories:
- Positive Rational Numbers – Both numerator and denominator are positive.
Example: 5/3, 7/8 - Negative Rational Numbers – Either numerator or denominator (but not both) is negative.
Example: -4/9, -3/7 - Zero – 0 is also a rational number because it can be written as 0/1.
Standard Form of Rational Numbers
A rational number is in standard form if:
- The denominator is positive.
- The numerator and denominator have no common factors other than 1.
Example:
- -6/8 simplifies to -3/4 (standard form).
“Book a free assessment for maths tutoring.”
Positive and Negative Rational Numbers
Positive Rational Numbers
- Have both numerator and denominator positive.
- Represent values greater than zero.
Examples: 3/5, 10/7, 4
Negative Rational Numbers
- Either numerator or denominator is negative.
- Represent values less than zero.
Examples: -2/3, -8, -15/4
Arithmetic Operations on Rational Numbers
Rational numbers can be added, subtracted, multiplied, and divided using basic rules.
1. Addition
- Make denominators the same.
- Add numerators.
Example:
3/5 + 2/5 = (3+2)/5 = 5/5 = 1
2. Subtraction
- Make denominators the same.
- Subtract numerators.
Example:
7/8 – 3/8 = (7-3)/8 = 4/8 = 1/2
3. Multiplication
- Multiply numerators and denominators directly.
Example:
2/3 × 3/4 = 6/12 = 1/2
4. Division
- Multiply the first number by the reciprocal of the second.
Example:
(3/5) ÷ (6/7) = (3/5) × (7/6) = 21/30 = 7/10
Multiplicative Inverse of Rational Number
The multiplicative inverse (or reciprocal) of a rational number p/q is q/p.
Examples:
- Reciprocal of 4/5 = 5/4
- Reciprocal of -7 = -1/7
Properties of Rational Numbers
Rational numbers follow these key properties:
- Closure Property
- Addition, subtraction, multiplication of two rational numbers gives another rational number.
- Commutative Property
- p/q + r/s = r/s + p/q
- p/q × r/s = r/s × p/q
- Associative Property
- (p/q + r/s) + t/u = p/q + (r/s + t/u)
- Distributive Property
- p/q × (r/s + t/u) = (p/q × r/s) + (p/q × t/u)
Difference Between Rational and Irrational Numbers
| Feature | Rational Numbers | Irrational Numbers |
|---|---|---|
| Representation | p/q form | Cannot be written as p/q |
| Decimal Form | Terminating/Repeating | Non-terminating, Non-repeating |
| Examples | 1/2, 3.75, -7 | √2, π, √5 |
Finding Rational Numbers Between Two Rational Numbers
To find rational numbers between two numbers:
- Write both numbers with the same denominator.
- Choose fractions between them.
Example:
Find a rational number between 1/4 and 1/2.
- 1/4 = 2/8 and 1/2 = 4/8
- Number between them = 3/8
Examples of Rational Numbers
- 4/5
- -9/3
- 0.25
- 1.333… (Repeating decimal)
- 12 (written as 12/1)
Importance of Rational Numbers
- Used in measurements and sharing quantities.
- Help in calculations of percentages and ratios.
- Essential for algebra and higher mathematics.
Conclusion
Rational numbers are an essential part of mathematics and daily life. They include all numbers that can be written in the form p/q, where q is not zero.
From fractions and decimals to positive and negative values, rational numbers help us count, measure, and calculate with accuracy.
Understanding their properties, operations, and standard form builds a strong base for solving more complex problems in algebra and arithmetic.
By practicing examples and finding rational numbers between two values, students can develop confidence in maths. Keep exploring rational numbers to improve problem-solving skills and prepare for advanced mathematical concepts.
Read More What is Whole Number? Definition, Properties & Examples
FAQs
Is zero a rational number?
Yes, 0 can be written as 0/1, so it is rational.
Are all fractions rational numbers?
Yes, if the denominator is not zero.
Can decimals be rational numbers?
Yes, terminating and repeating decimals are rational numbers.
Is √2 rational?
No, √2 cannot be written as p/q, so it is irrational.
What is the standard form of rational numbers?
When numerator and denominator have no common factor other than 1, and denominator is positive.
Are integers rational numbers?
Yes, any integer can be written as p/1.
Can a rational number be negative?
Yes, if either numerator or denominator is negative.
Are all repeating decimals rational numbers?
Yes, repeating decimals can be written as fractions.
Are percentages rational numbers?
Yes, as they can be converted into fractions.
How many rational numbers exist between 0 and 1?
Infinitely many, like 1/2, 1/3, 2/3, etc.