Rational Numbers - Definition, Properties & Types

Rational numbers is a type of real numbers that can be expressed in the form a/b, where a and b are integers and b ≠ 0. The set of rational numbers, denoted by letter 'Q' and includes natural numbers, whole numbers, integers, fractions and terminating or recurring decimals. Examples of rational numbers include 3/4, -5/2, 8/3, 0.71, 0.333, etc. Rational numbers play an important role in mathematics because they help represent parts of a whole, ratios, and quantities that cannot always be expressed as whole numbers. In this article, you will learn the definition of rational numbers, their properties, types, examples, and operations with rational numbers in a simple and easy-to-understand way.

Table of Contents

• What is a Rational Number?
• Rational Numbers Examples
• Rational Numbers Definition
• Positive and Negative Rational Numbers
• Types of Rational Numbers
• How to Identify a Rational Number
• Rational Numbers on the Number Line
• Difference Between Rational and Irrational Numbers
• The Real Number System: Where Do Rational Numbers Fit?
• Arithmetic Operations on Rational Numbers
• Properties of Rational Numbers
  
  

What is a Rational Number?

The word 'rational' originated from the word ratio, and in mathematics, rational number is represented in the form of a ratio of integers. That is why, it is expressed in the form of 'p/q'. For example, 1/2, 5/3 , etc. A rational number can be positive, negative, or zero. 

The collection of rational numbers is infinite and is denoted by set 'Q' that consist of whole numbers, integers, proper and improper fractions, as well as terminating and recurring decimals. So when you see ℚ in maths, it means the complete collection of all numbers expressible as p/q where q ≠ 0. On the other hand, the numbers that cannot be represented in the form of fraction are called irrational numbers. We will learn more in detail about the difference between rational numbers and irrational numbers on this page. Before that let's look at some examples of rational numbers.

Rational Numbers Examples

If a number can be expressed as a fraction of two integers (with a non-zero denominator), it's rational.

Rational Numbers Definition

A rational number can be defined as any number that can be expressed in the form p/q, where:

• p and q are integers

• q ≠ 0

Standard Form:

A rational number is said to be in standard form (or simplest form / lowest terms) when:

• The numerator (p) and denominator (q) have no common factors other than 1 (i.e., they are co-prime).

• The denominator is positive.
  
  

Positive and Negative Rational Numbers 

Positive rational numbers have the same sign in numerator and denominator (both positive or both negative): 3/5, -2/-7 (which simplifies to 2/7)

Negative rational numbers have opposite signs in numerator and denominator: -2/7, 5/-9. 

The negative sign can be placed in front of the fraction, with the numerator, or with the denominator, all three are equivalent: -(p/q) = (-p)/q = p/(-q)

Types of Rational Numbers

Rational numbers aren't just fractions. They come in several forms.

A. Integers as Rational Numbers

Every integer, positive, negative, or zero is a rational number, because any integer can be written with a denominator of 1. Examples: 7 = 7/1 , -12 = -12/1, 0 = 0/1 

This is why integers are a subset of rational numbers.

B. Fractions as Rational Numbers

Any fraction where both the numerator and denominator are integers (and the denominator is non-zero) is rational. 3/7, -6/5, 11/4, 22/7 are all rational.

C. Terminating Decimals

A decimal that ends (terminates) after a finite number of digits is always rational.

For example: 0.5 = 1/2 , -1.625 = -13/8

D. Non-Terminating Repeating Decimals

A decimal that goes on forever but repeats the same pattern is also rational.

Consider the following examples:  0.333... = 1/3, 0.141414... = 14/99, 2.0909... = 23/11 are all rational numbers.

How to Identify a Rational Number

The following checklist can be used to determine whether any number is rational:

Step 1: Can it be written as p/q where p and q are integers and q ≠ 0? If yes: it's rational.

Step 2: Is it a decimal?

If it terminates (ends) →  rational

If it repeats (pattern repeats forever) → rational

If it's non-terminating and non-repeating → irrational

Step 3: Is it a square root or other radical?

If the root gives a whole number (perfect square/cube) then it is a rational number

√25 = 5 = 5/1 → rational

∛27 = 3 = 3/1 → rational

√2 = 1.41421356... → irrational

√3 = 1.73205... → irrational

Rational Numbers on the Number Line 

Like whole numbers and integers, rational numbers also have precise positions on the number line, where 0 is the origin, positives lie to the right and negatives to the left.

To plot a proper fraction (numerator < denominator), divide the segment between the nearest integers (e.g., 0 and 1) into equal parts and count the numerator parts from the smaller integer; do the same to the left for negatives. For an improper fraction, convert to a mixed number, locate the whole part, then partition the next segment and move the fractional part.

Rational numbers are dense on the number line. Between any two rationals there are infinitely many others.

Difference Between Rational and Irrational Numbers

Irrational numbers are numbers that cannot be expressed as p/q where p and q are integers. The difference between rational and irrational numbers can be understood from the table given below.

The Real Number System: Where Do Rational Numbers Fit?

• Every natural number is a whole number, integer, and rational number.

• Every integer is a rational number (but not every rational is an integer).

• Every rational number is a real number.

• Rational and irrational numbers together make up all real number.
  
  

Arithmetic Operations on Rational Numbers 

A. Addition of Rational Numbers

Case 1: Same denominators: (a/b) + (c/b) = (a + c)/b 

Case 2: Different denominators: (a/b) + (c/d) = (ad + bc)/(bd)

Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12

B. Subtraction of Rational Numbers

Like addition, find common denominator, then subtract numerators.

(a/b) − (c/d) = (ad − bc)/(bd)

Example: 5/6 − 1/4 = 10/12 − 3/12 = 7/12

C. Multiplication of Rational Numbers

Multiply numerators together and denominators together.

(a/b) × (c/d) = (a × c)/(b × d)

Example: 3/5 × 2/7 = (3 × 2)/(5 × 7) = 6/35 

D. Division of Rational Numbers

Multiply the first fraction by the reciprocal of the second fraction.

(a/b) ÷ (c/d) = (a/b) × (d/c) = (ad)/(bc)

Example: 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2⅒

Properties of Rational Numbers

Rational numbers are closed under addition, subtraction, multiplication, and division (except by 0).

• Associative Property: (a + b) + c = a + (b + c)
• Commutative Property: a + b = b + a and a × b = b × a
• Distributive Property: a × (b + c) = a × b + a × c
• Identity Element: 0 is the identity for addition: a + 0 = a and 1 is the identity for multiplication: a × 1 = a
• Inverse Element: Additive inverse: a + (-a) = 0 and Multiplicative inverse (reciprocal): a × (1/a) = 1 (a ≠ 0)

 

Rational numbers form the basis of arithmetic and problem-solving in real life. They are applicable in performing basic operations like addition, subtraction, multiplication, and division (excluding division by zero). They are widely used everywhere in our lives, from kitchen measurements and monetary calculations to building work, sales, and sporting statistics. Being familiar with rational numbers is crucial not just to master math but also to understand and handle much of daily life with reason and precision. Learn how to identify the difference between rational and irrational numbers and confidently work with them by practicing with our fun exercises and examples.

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