Last updated on July 9th, 2025
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.
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.
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 |
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. |
Rational numbers have different types depending on their form and properties:
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.
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.
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).
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.
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:
Is -8/5 a rational number?
Yes, -8/5 is a rational number.
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.
Express 0.375 as a rational number.
0.375 = 3/8.
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.
Add 2/7 and 3/4.
2/7 + 3/4 = 29/28.
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).
Express 0.666… (repeating) as a rational number.
0.666… = 2/3.
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.
Subtract 5/6 from 3/2.
3/2 – 5/6 = 2/3.
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.
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.