BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon103 Learners

Last updated on July 9th, 2025

Math Whiteboard Illustration

Rational Numbers

Professor Greenline Explaining Math Concepts

Rational numbers can be written in the form of p/q where both p, q are integers and q≠0. All fractions, decimals, whole numbers, natural numbers belong to the set of rational numbers. In this article, we will learn about rational numbers.

Rational Numbers for US Students
Professor Greenline from BrightChamps

What Are Rational Numbers?

A rational number can be written as p/q, where p and q are integers and q  0. This means that rational numbers consist of natural numbers, whole numbers, integers, fractions of integers, and decimals (terminating or repeating). 

 

Rational numbers and fractions are interrelated concepts, as both can be represented as ratios. Rational numbers include all fractions (proper, improper, positive, or negative) with integer numerators and non-zero denominators. In a fraction, the numerator and denominator are integers, with the denominator not zero. However, in a rational number, the numerator and denominator can be any integers, as long as the denominator is not zero. A rational number is any number expressible as p/q, where p and q are integers and q ≠ 0, including natural numbers, whole numbers, integers, fractions, and terminating or repeating decimals.
 

Professor Greenline from BrightChamps

Properties of rational numbers:

  • Termination or Repeating Decimals: When expressed as decimals, Rational numbers as decimals are either terminating (end after a finite number of digits) or repeating (have a repeating pattern of digits).

 

  • Additive Identity: The additive identity of rational numbers is 0, meaning that adding 0 to any rational number does not change its value (a + 0 = a).

 

  • Multiplicative Identity: The multiplicative identity for rational numbers is 1 because multiplying any rational number by 1 results in the same number (a  1 = a). 
  • Additive Inverse: Every rational number a has an additive inverse –a, such that when they are added together, the result is 0 (a + (-a) = 0).

 

  • Multiplicative Inverse: Every nonzero rational number a has a multiplicative inverse 1/a, meaning their product is 1(a  1/a = 1). However, 0 has no multiplicative inverse. 

 

  • Closure under Addition, Subtraction, and Multiplication: Rational numbers are closed under addition, subtraction, and multiplication, meaning the result is always a rational number. 

 

  • Division Property: When dividing two rational numbers, the result is also a rational number, if the divisor is nonzero.

 

  • Distributive property: Rational numbers adhere to the distributive property, which states that a(b + c) = ab + ac. 

 

  • Ordering: Rational numbers can be ordered on a number line, where for any two rational numbers, one is greater than, less than, or equal to the other.
     
Professor Greenline from BrightChamps

Categories of Rational Numbers

Category

Description

Examples

Positive Rational Numbers

Rational numbers greater than 0

1/2, 3/4, 5, 7.2

Negative Rational Numbers

Rational numbers less than 0

-1/3, -4, -2.75

Proper Fractions

Here the numerator is always smaller than the denominator

1/2, 3/5, 7/9

Improper Fractions

The numerator is always greater than the denominator

5/3, 9/4, 7/7

Terminating Decimals

Decimal form ends after a few digits

0.5, 2.75, 4.125

Repeating Decimals

Decimal form has a repeating pattern

0.333..., 2.666..., 1.8181...

Whole Numbers

Rational numbers with denominator 1

0, 1, 2, 3, 4, ...

Integers

All positive and negative whole numbers

-3, -2, -1, 0, 1, 2, 3

Square Roots (Rational Only)

Square roots that result in a rational number

√4 = 2, √9 = 3, √16 = 4

Cube Roots (Rational Only)

Cube roots that result in a rational number

∛8 = 2, ∛27 = 3, ∛125 = 5

Professor Greenline from BrightChamps

Difference Between Rational and Irrational Numbers

Students often get confused between rational and irrational numbers, and they get stuck trying to differentiate them. Rational numbers can be expressed as p/q, Where p and q are integers and q  0, while irrational numbers cannot. Let’s understand this better using a table.

 

Feature

Rational Numbers

Irrational Numbers

Definition

Can be represented as p/q, where p and q are integers and q ≠ 0.

Cannot be represented as a simple fraction.

Decimal Form

Either terminating (e.g., 0.75) or repeating (e.g., 0.3333...).

Non-terminating and non-repeating (e.g., 3.141592...).

Examples

1/2, -3/4, 5, 0.25, 1.333...

√2, π (pi), e (Euler’s number), √3, 0.10110111011110...

Square & Cube Roots

The square root of a perfect square number and the cube root of a perfect cube number are always rational numbers. (e.g., √4 = 2, ∛8 = 2).

The square and cube roots of non-perfect squares and cubes are irrational (e.g., √2, ∛5).

Existence on Number Line

Can be located precisely on the number line.

Can also be located, but are not expressed exactly in fraction form.

Professor Greenline from BrightChamps

What Are the Types of Rational Numbers?

Rational numbers have different types depending on their form and properties:

  • Standard Form: The standard form of a rational number means that the numerator and the denominator have no common factor other than 1. 

 

  • For example, 18/24 is a rational number, but it can be simplified to 3/4, where the numerator and denominator share only 1 as a common factor. Therefore, the rational number 3/4 is in its standard form. 

 

  • Positive Rational Numbers: Positive rational numbers have a positive value, e.g., 4/6 or -6/-9 (where negative signs cancel). If both are negative, dividing by –1 simplifies the fraction into a positive rational number. Examples of positive rational numbers include 4/6, -6/-9, etc. For example, -6/-9 = 6/9, as the negative signs cancel out.

 

  • Negative Rational Numbers: Negative rational numbers are those where either the numerator or the denominator is a negative integer. Examples of negative rational numbers include -4/6, 6/-7, etc.

 

  • Terminating Rational Numbers: Rational numbers with a decimal representation that ends after a certain number of digits are known as terminating decimals.A rational number has a terminating decimal of its denominator, after simplification, is of the form 2^m × 5^n, where m and n are non-negative integers.

 

  • Non Terminating and Repeating Rational Numbers:  Repeating decimals are rational numbers whose decimal representation follows a repeating pattern. The decimal expansion of a non-terminating rational number does not end: instead, a single digit or a group of digits repeats at a fixed interval. For example, 5/6 = 0.8333 … , 7/9 = 0.777 … 
Professor Greenline from BrightChamps

How to Represent Rational Numbers on a Number Line?

Since rational numbers are a subset of real numbers, they can be represented on a number line. The following steps explain how to place a rational number on the number line.

 

Steps to Represent a Rational Number on a Number Line

 

Step 1: Determine the sign:
    
If the number is a positive, it will be plotted to the right of zero. 
If the number is negative, it will be plotted to the left of zero.

For example,  -3 (negative number) and +5 (positive number) on a number line can be represented like this on the number line
 

 

 

Step 2: Identify the type of fraction

If the rational number is a proper fraction (numerator < denominator), it lies between 0 and 1 (for positive numbers) or 0 and -1 (for negative numbers).
If the rational number is an improper fraction (numerator ≥ denominator), convert it into a mixed fraction. The number will be located beyond its whole number part.

Step 3: Divide the number line

Identify the two consecutive whole numbers between which the fraction lies.
Divide the section into equal parts based on the denominator of the fraction.

Step 4: Locate the Desired Value

Count the required number of divisions as indicated by the numerator and mark the point. 

Example: Represent 5/4 on the number line.

Solution: The number 5/4 is positive, so it will be placed on the right side of zero.

    5/4 is an improper fraction. Converting it into a mixed fraction, we get 114. 
    
    The number lies between 1 and 2

    Divide the segment between 1 and 2 into 4 equal parts (since the denominator is 4).

    Count 1 division beyond 1 (since 5/4 = 11/4) to mark the point. 

Professor Greenline from BrightChamps

How to Identify Rational Numbers?

Rational numbers can be recognized using the following characteristics:

 

Type of numbers: All integers, whole numbers, natural numbers, and fractions with integer numerators and denominators are rational numbers.

 

  • Decimal Representation: A number is rational if its decimal form is either terminating or repeating (e.g., 5.6 or 2.141414…).

 

  • Irrational Numbers: If the decimal form is non-terminating and non-repeating, the number is irrational. For example, 5 = 2.236067977… is an irrational number.

 

  • Fraction Form: Any number that can be written as p/q is considered rational if p and q are integers and q  0.
Professor Greenline from BrightChamps

Operations of Rational Numbers

There are four common arithmetic operations of rational numbers. Let’s learn more about it below.

Addition of Rational Numbers: The addition of two rational numbers can be performed using a step-by-step method. Below, the sum 5/8 and 2/5 is explained as an example.

    Step 1: Find the least common denominator (LCD)

        The least common denominator of 8 and 5 is 40.

    Step 2: We now convert the fractions to have the same denominator.
            
        5/8 = 5 × 5/8 × 5 = 2540

        2/5 = 2 × 8/5 × 8  = 1640

    Step 3: Add the numerators

        25/40 + 16/40 = (25 + 16)/40 = 41/40

    Step 4: Simplify the result if possible

        4140 is an improper fraction and can be written as a mixed number.

        1140

    Thus, the sum of 5/8 and 2/5 is 41/40 or 11/40.

Subtraction of Rational Numbers: The subtraction of two rational numbers can be performed using a step-by-step method. Below, the subtraction of 7/9 and 1/4 is explained. 

    Step 1: Find the Least Common Denominator (LCD)

        The least common denominator of 9 and 4 is 36.

    Step 2: We now convert the fractions to have the same denominator.

        79 = 7 × 4/9 × 4 = 28/36

        14 = 1 × 9/4 × 9 = 9/36

    Step 3: Subtract the numerators

        28/36 – 9/36 = (28 -9)/36 = 19/36

    Step 4: Simplify the result if possible

        1936 is already in its simplest form

    Therefore, 7/9 – 1/4 = 19/36

Multiplication of Rational Numbers: Multiplication of two rational numbers is done by simply multiplying their numerators and denominators. Below is a step-by-step method using -7/2 and 3/8 as an example. 

    Step 1: Write the rational numbers with a multiplication sign

        -7/2 × 3/8 

    Step 2: Multiply the numerators and denominators individually 

        (-7) × 3/2 × 8 = -21/16

    Step 3: Simplify the result if possible

        -2116 is already in its simplest form

    Thus, -7/2 × 3/8 = – 21/16.

Division of Rational Numbers: Division of two rational numbers is done by multiplying the first number by the reciprocal of the second number. Below is a step-by-step method using 5/6  2/9 as an example.

    Step 1: Write the rational numbers with the division sign

        5/6 × 2/9

    Step 2: Change “” to “” and take the reciprocal of the second rational number

5/6   9/2

Step 3: Multiply the numerator and denominators individually

    5 × 9/6 × 2 = 45/12

Step 4: Simplify the result if possible

    The greatest common factor (GCF) of 45 and 12 is 3.

    45  3/12  3 = 15/4

Thus, 5/6  2/9 = 15/4 or 3 3/4 (as a mixed fraction).

Max Pointing Out Common Math Mistakes

Common Mistakes of Rational Numbers and How to Avoid Them

Understanding rational numbers is an essential part of mathematics, but students often make common mistakes while learning about them. Here are some of the common mistakes students might encounter and how to avoid them.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusion between Rational and Irrational Numbers

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Some students assume that all decimals are rational, even if they are non-terminating and non-repeating. For example, assuming that numbers like 2 or π are rational. Remember that a number is rational only if it can be written as p/q (where p and q are integers and q  0).

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrect Addition or Subtraction of Rational Numbers

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Adding or subtracting fractions without finding the common denominator first. For example, incorrectly adding 1/2 + 1/3 as 2/5 is wrong. Instead, use the LCD (6): 1/2 = 3/6, 1/3 = 2/6, so 3/6 + 2/6 = 5/6. 

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Misplacing Negative Signs
 

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often misplace negative signs, especially when working with improper fractions or reciprocals. For example, thinking that -3/5 is different from 3/-5, when in reality both are the same. Remember that a fraction is negative if either the numerator or denominator is negative, but not both. Since -3/5 = 3/-5 (both equal -0.6), the negative sign can be in either the numerator or denominator.
 
 

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Multiplying instead of Dividing When Finding Reciprocals

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often forget to flip the second fraction when dividing rational numbers. For example, incorrectly computing 3/4 ÷ 2/5 as 3/4 × 2/5 is wrong. Instead, use the reciprocal: 3/4 × 5/2 = 15/8.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not simplifying Fractions to Their Lowest Terms

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Leaving fractions in simplified form instead of reducing them to the simplest fraction. For example, writing 12/18 instead of 2/3. Always find the greatest common factor and then simplify the fraction by dividing both the numerator and denominator by it. 

arrow-right
Professor Greenline from BrightChamps

Real Life Applications of Rational Numbers

Rational numbers play a crucial role in various real-world situations. From managing money to measuring ingredients in cooking, they help us make accurate calculations and decisions.

These applications demonstrate the practicality of rational numbers. Let’s go through a few examples:

 

  • Money and Finance: We commonly use rational numbers in tasks related to money and finance such as banking, shopping, calculating expenses, etc.

 

  • Cooking: When cooking, we often measure ingredients using fractions. For example, doubling or halving a recipe can be done by multiplying or dividing rational numbers.

 

  • Construction and Carpentry: Builders use fractions and decimals to measure wood, tiles, and bricks accurately. 

 

  • Sports and Statistics: Players’ batting averages, shooting percentages, and game scores involve rational numbers.  
Max from BrightChamps Saying "Hey"

Solved Examples for Rational Numbers

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Is -8/5 a rational number?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

Yes, -8/5 is a rational number.
 

Explanation

A rational number is any number that can be expressed in the form p/q where p and q are integers and q  0. Since -8 and 5 are both integers and the denominator is not zero, -8/5 is rational.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

Express 0.375 as a rational number.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

0.375 = 3/8.

Explanation

Since 0.375 is a terminating decimal, we can write it as 375/1000 and simplify it by dividing both the numerator and denominator by 125, resulting in 3/8.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

Add 2/7 and 3/4.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

2/7 + 3/4 = 29/28.
 

Explanation

The LCM of 7 and 4 is 28.
Convert fractions: 2/7 = 8/28, 3/4 = 21/28.
Add (8 + 21)/28 = 1 1/28 (an improper fraction). 

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

Express 0.666… (repeating) as a rational number.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

0.666… = 2/3.

Explanation

Since 0.666… is a repeating decimal, we can express it as a fraction. Let x = 0.666…. Multiply by 10 to get 10x = 6.666…. Subtract the original x from this: 10x – x = 6.666… – 0.666…, giving 9x = 6. Solve for x: x = 6/9 = 2/3. Since 2 and 3 are integers and the denominator is not zero, 0.666… = 2/3 is a rational number.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

Subtract 5/6 from 3/2.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

3/2 – 5/6 = 2/3.

Explanation

To subtract 5/6 from 3/2, find the least common multiple (LCM) of the denominators 2 and 6, which is 6. Convert the fractions: 3/2 = 9/6 (multiply numerator and denominator by 3), and 5/6 = 5/6. Now subtract: (9 – 5)/6 = 4/6. Simplify 4/6 by dividing numerator and denominator by 2, giving 2/3. So, the result is 2/3.

Max from BrightChamps Praising Clear Math Explanations
Math Teacher Background Image
Math Teacher Image

Hiralee Lalitkumar Makwana

About the Author

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.

Max, the Girl Character from BrightChamps

Fun Fact

: She loves to read number jokes and games.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom