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Mathematics > Chapter 1 Rational Numbers > Introduction

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Rational Numbers

Properties of rational number

Introduction to rational numbers,

Rational Numbers

A number is called Rational if it can be expressed in the form p/ q where p and q are integers (q> 0). It includes all natural, whole number, and integers.

Case1/2, 4/3, 5/7, 1 etc.

Natural Numbers - All the positive integers from 1, 2, 3,, ∞.

Whole Numbers - All the natural numbers including zero are called Whole Numbers.

Integers - All negative and positive numbers including zero are called Integers.

Closure property,

Closure- Rational numbers are closed under addition, subtraction and multiplication. For eg.- If p and q are any two rational numbers, then and the sum, difference and product of these rational numbers is also a rational number. This is known as the closure law

Commutative property,

Commutativity- Rational numbers are commutative under addition and multiplication. If p and q are two rational numbers, then:

Commutative law under addition says- p + q = q + p.

Commutative law under multiplication says p x q = q x p.

Note- Rational numbers, integers and whole numbers are commutative under addition and multiplication. Rational numbers, integers and whole numbers are non commutative under subtraction and division.

Associativity property,

▪  Associativity- Rational numbers are associative under addition and multiplication. If a, b, c are rational numbers, then:

Associative property under addition: p + (q + r) = (p + q) + r

Associative property under multiplication: p(qr) = (pq)r

The role of 0 and 1

Role of zero and one- 0 is the additive identity for rational numbers. 1 is the multiplicative identity for rational numbers.

Zero is the additive identity for whole numbers, integers and rational numbers.

▪  Multiplicative inverse- If the product of two rational numbers is 1, then they are called multiplicative inverse of each other.

Eg. 4/9 * 9/4 = 1

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