Rational Numbers

A rational number is any number that can be written in the form

For example, the numbers 0, 2, 4.5, 2/3, −8/7, −7.2, 0.333, and √64 are included.

Key Points

• All natural numbers, whole numbers, and integers are rational numbers.
• Every rational number can be represented on a number line.
• Rational numbers can be positive, negative, or zero.
• 0 is neither positive nor negative, but it is a rational number because it can be written as 0/1.

Representation of Rational Numbers

1. Fraction Form: A rational number is written as p/q, where p is the numerator and q ≠ 0 is the denominator.

Examples: 3/5, −7/2

2. Decimal Form: A rational number can also be written as a terminating or repeating decimal.

Examples:

• 1/2 = 0.5 (terminating decimal)
• 1/3 = 0.333… (repeating decimal)

3. Representing Rational Numbers on Number Line

Rational numbers are a subset of real numbers therefore, they can be represented on the real line. The steps to represent a rational number are as follows:

• Step 1: First find if the number is positive or negative, if positive then it will be plotted on the RHS of zero and if negative, it will be on the LHS of zero.
• Step 2: Identify if the given rational number is proper or improper. If proper then it will lie between 0 and 1 in case of positive and 0 and -1 in case of negative rational number.
• Step 3: If improper then convert it into a mixed fraction. In this case, the rational number will lie just beyond the whole number part.
• Step 4: Now after steps 1, 2, and 3 we have to plot only the proper fraction part. To plot this cut the area between the two successive desired numbers by drawing lines n-1 times where n is the denominator of the proper fraction part.
• Step 5: Now count the lines equal to the value of the numerator. This will represent the desired rational number on a real line.

Example 1: Represent 2/5 on Real Line

Example 2: Represent -2/5 on Real Line

Types of Rational Numbers

Rational Numbers can be classified into following types:

1. Standard Form of Rational Numbers

The standard form of a rational number is defined as having no common factors other than one between the dividend and divisor, and hence the divisor is positive.

Example: 12/36 is a rational number. When simplified by dividing the numerator and denominator by 12: 12/36 = 1/3

Since 1 and 3 have no common factors other than 1, the rational number 1/3 is in standard form.

2. Positive Rational Numbers

Positive rational numbers are those in which both numerators and denominators are either positive or negative. In case both numerators and denominators are negative, -1 can be eliminated as common factor, which gives us ultimately a positive rational number.

Examples: 2/5, -3/-5, etc.

3. Negative Rational Numbers

Negative rational numbers are those in which either the numerator or the denominator is a negative integer.

Examples: -1/2, 3/-4

4. Terminating Rational Numbers

Terminating decimals are the rational numbers whose decimal representations end or terminate after a certain number of digits. A rational number has a terminating expansion if the denominator is in the form of 2^m × 5n, where either m or n can be zero.

Examples: 4/5 = 0.8, 3/4 = 0.75

5. Non-Terminating and Repeating Rational Numbers

Repeating decimals are the rational numbers whose decimal representations have a repeating pattern. The decimal expansion of a non-terminating rational number doesn't end. The same digit or group of digits is repeated after fixed interval

Examples: 1/3 = , 2/7 =

Properties of Rational Numbers

• Terminating or Repeating Decimals: When converted to decimal form, rational numbers either terminate after a finite number of digits (terminating decimals) or repeat a digit or a group of digits indefinitely (repeating decimals).
• Additive Identity: The additive identity for rational numbers is 0, which means adding 0 to any rational number gives the same number (a + 0 = a).
• Multiplicative Identity: The multiplicative identity for rational numbers is 1, because multiplying any rational number by 1 leaves it unchanged (a × 1 = a).
• Additive Inverse: Every rational number a has an additive inverse −a, such that (a + (−a) = 0).
• Multiplicative Inverse: Every non-zero rational number a has a multiplicative inverse 1/a, such that (a × 1/a = 1). However, 0 does not have a multiplicative inverse.
• Closure Property: Rational numbers are closed under addition, subtraction, and multiplication, which means the sum, difference, or product of any two rational numbers is also a rational number.
• Division Property: The quotient of two rational numbers is also a rational number, provided the divisor is not zero.
• Distributive Property: Rational numbers follow the distributive property of multiplication over addition: (a(b + c) = ab + ac).
• Ordering Property: Rational numbers can be ordered. For any two rational numbers, one number is either greater than, less than, or equal to the other.

Identification of Rational Number

To identify a rational number, check if it can be written as a fraction where both the top number (numerator) and the bottom number (denominator) are whole numbers, and the bottom number isn't zero. Rational numbers also have decimal forms that either end after a few digits or repeat a specific pattern.

For example: Which of the following numbers are rational numbers?

a) -1.75
b) 2/3
c) √5
d) π

a) -1.75 is a rational number as it has a terminating decimal expansion.

b) 2/3 is also a rational number as it can be expressed in the form of a ratio of two integers.

c) √5 is an irrational number because  it has a decimal expansion with infinitely many digits without any repetition.

d) π is also an irrational number as it has a decimal expansion with infinitely many digits without any repetition.

Thus, only (a) and (b) are the rational numbers out of all the given numbers.

Arithmetic Operations on Rational Numbers

There are four most common operations for Rational Numbers, which includes the following:

• Addition
• Subtraction
• Multiplication
• Division

Addition of Rational Numbers

To add rational numbers, first make the denominators the same and then add the numerators.

Example:Add 3/4 and 1/6
Step 1: Find the LCM of 4 and 6 = 12
Step 2: Convert to equivalent fractions
3/4 = 9/12, 1/6 = 2/12
Step 3: Add the numerators
9/12 + 2/12 = 11/12

Subtraction of Rational Numbers

Subtraction is similar to addition. First make the denominators the same, and then subtract the numerators.

Example:Subtract 2/5 from 1/3
Step 1: LCM of 3 and 5 = 15
Step 2: Convert to equivalent fractions
1/3 = 5/15, 2/5 = 6/15
Step 3: Subtract the numerators
5/15 − 6/15 = −1/15

Multiplication of Rational Numbers

Multiply the numerators together and the denominators together.

Example:−11/3 × 4/5

= (−11 × 4) / (3 × 5)
= −44/15

Division of Rational Numbers

To divide rational numbers, multiply the first fraction by the reciprocal of the second fraction.

Example:3/5 ÷ 4/7

= 3/5 × 7/4
= (3 × 7) / (5 × 4)
= 21/20

Related Articles

• Real Numbers
• Operations on Real Numbers
• Difference between rational and irrational numbers
• Imaginary Numbers

Solved Examples

Example 1: Check which of the following is irrational or rational: 1/2, 13, -4, √3, and π.

Rational numbers are numbers that can be expressed in the form of p/q, where q is not equal to 0.

1/2, 13, and -4 are rational numbers as they can be expressed as p/q.

√3, and π are irrational numbers as they can not be expressed as p/q.

Example 2: Check if a mixed fraction, 3(5/6), is a rational number or an irrational number.

Simplest form of 3(5/6) is 23/6

Numerator = 23, which is an integer

Denominator = 6, is an integer and not equal to zero.

So, 23/6 is a rational number.

Example 3: Determine whether the given numbers are rational or irrational.

(a) 1.33  (b) 0.1  (c) 0  (d) √5

a) 1.33 is a rational number as it can be represented as 133/100.

b) 0.1 is a rational number as it can be represented as 1/10.

c) 0 is a rational number as it can be represented as 0/1.

d) √5 is an irrational number as it can not be represented as p/q.

Example 4: Simplify (2/3) × (6/8) ÷ (5/3).

(2/3) × (6/8) ÷ (5/3) = (2/3) x (6/8) × (3/5)

= (2 × 6 × 3)/(3 × 8 × 5) 

= 36/120 = 3/10

Example 5: Arrange the following rational numbers in ascending order: 1/3, -1/2, 2/5, and -3/4.

Common denominator for 3, 2, 5, and 4 is 60. Thus

1/3 = 20/60

-1/2 = -30/60

2/5 = 24/60

-3/4 = -45/60

For negative numbers, the one with the more negative value is smaller:

−45/60 < −30/60 < 20/60 < 24/60

Thus, ascending order of given rational numbers is: −3/4​ < −1/2 ​< 1/3 ​< 2/5​

Practice Questions

Q1. Find two rational number between 2/3 and 3/4

Q2. Find the sum of -3/5 and 6/7

Q3. Find the first five equivalent rational numbers of -7/8

Q4. Represent 4/3 on Real Line

Q5. Find the Product of -19/3 and 2/57