Eight Standard >> Rational numbers | Introduction

Rational numbers: Introduction

Rational numbers are a subset of real numbers that can be expressed as the quotient or fraction of two integers (a/b), where b is not equal to zero. Put simply, rational numbers are those that can be depicted as fractions, with both the numerator and denominator being whole numbers. Rational numbers include whole numbers, integers, and fractions.

First, let's understand what Rational Numbers are. Rational numbers are a subset of real numbers that can be expressed as fractions, where both the numerator and the denominator are whole numbers. In this fraction, 'a' is the numerator, and 'b' is the denominator, which must not be equal to zero. Rational numbers include whole numbers, integers, and fractions. Even integers and whole umbers can be written as fractions by giving them a denominator of 1.

Nature of Different Sets of Numbers:

1. Natural Numbers (N): Natural numbers are the counting numbers and start from 1. They include all positive integers (1, 2, 3, 4, ...) but do not include zero or negative numbers.

2. Integers (Z): Integers include all the natural numbers along with their negatives and zero. They are represented as (..., -3, -2, -1, 0, 1, 2, 3, ...).

3. Rational Numbers (Q): Rational numbers, as mentioned earlier, are numbers that can be represented as fractions (a/b), where 'a' and 'b' are integers, and 'b' is not equal to zero. Rational numbers encompass both integers and fractions within their set.

Basic Difference between Fractions and Rational Numbers:

Fractions and rational numbers are closely related, but there is a slight difference between the two:

• A fraction is a way of representing a rational number. It consists of a numerator and a denominator, both of which are integers. For example, 3/5 is a fraction representing a rational number.
• Conversely, a rational number is any numeral that can be represented in the form of a fraction. It includes not only fractions but also integers since they can be written as fractions with a denominator of 1 (e.g., 5 can be written as 5/1).

Properties of Rational Numbers:

Rational numbers possess several important properties:

1. Closure Property: The sum, difference, product, or quotient of any two rational numbers is also a rational number. In other words, the result of any arithmetic operation between rational numbers remains within the set of rational numbers.

2. Dense Property: Between any two rational numbers, there exists an infinite number of other rational numbers. This property means that there are no "gaps" in the rational number line.

3. Additive Inverse: For every rational number 'a,' there exists another rational number '-a,' such that their sum is zero (a + (-a) = 0).

4. Multiplicative Inverse: For every non-zero rational number 'a,' there exists another rational number '1/a,' such that their product is one (a * 1/a = 1).

5. Associative Property: The addition and multiplication of rational numbers are associative, which means the grouping of numbers in a sum or product does not affect the result.

6. Commutative Property: The addition and multiplication of rational numbers are commutative, which means the order of numbers in a sum or product does not affect the result.