Operation on Real Numbers

Content Standards

In this lesson, students will learn about:

Closure property of real numbers under addition, subtraction, multiplication, and division (except division by zero).
Commutative and associative laws for addition and multiplication of real numbers.
Distributive law of multiplication over addition and subtraction.
Operations between rational and irrational numbers.
Identification of cases where operations yield rational or irrational results.

Performance Standards

Students will be able to:

Apply operations correctly on real numbers.
Verify and state properties of operations (closure, commutativity, associativity, distributivity).
Identify situations where operations between rational and irrational numbers produce rational or irrational results.
Rationalise denominators containing surds.
Use properties to simplify algebraic expressions involving real numbers.

Alignment Standards

Reference: NCERT Book Alignment

The lesson is aligned with the NCERT Grade 9 Mathematics Textbook, Chapter 1:Number Systems, Section – 4: Operations on Real Number.

Learning Objectives

By the end of the lesson, students will be able to:

Recognise that real numbers are closed under addition, subtraction, and multiplication (and under division except by zero).
State and verify commutative and associative properties of addition and multiplication.
Apply the distributive property of multiplication over addition and subtraction.
Use these identities to simplify numerical expressions.
Determine when results of operations between rational and irrational numbers are rational or irrational.
Rationalise simple denominators containing surds.
Appreciate that all these operations and properties together make the set of real numbers a well-structured number system.

Prerequisites (Prior Knowledge)

Representation of rational and irrational numbers on a number line.
Basic arithmetic operations with rational numbers.
Meaning of rational and irrational numbers.
Simple surds such as √2, √3, etc.

Introduction

In earlier grades, students have already worked with natural, whole, integers, and rational numbers. This lesson helps them extend that understanding to the set of real numbers, which includes both rational and irrational numbers, and study how these numbers behave under the four basic operations — addition, subtraction, multiplication, and division. Students will also discover the fundamental properties (identities) that hold true for real numbers, forming the basis for algebraic manipulation in higher classes.

Timeline (40 Minutes)

Title	Approximate Duration	Procedure	Reference Material
Engage	5	Begin with quick reasoning questions: “What do you get when you multiply √2 by itself?” “Can we divide any real number by 0?” Display number cards (rational & irrational) and ask: “What happens if I add these two?”	Slides
Explore	10	Exploring further through a virtual lab for real numbers. Introduction to identities through an area model example.	Slides + Virtual Lab
Explain	10	Teacher explains each identity in simple, intuitive language: Identity 1: Multiplication of Real Numbers √ab=√a √b Example: √2.3=√2 . √3=√6 Identity 2: Division of Real Numbers √a/b=√a / √b Example: √2/3=√2 / √3 Identity 3: Distributive Property-1 (√a +√b)(√a -√b)=a-b Example: (√3+√2)(√3-√2) =3-2=1 Identity 4: Distributive Property-2 (a+√b)(a-√b)=a² -b Example: (3+√2)(3-√2) =9-2=7 Identity 5: Distributive Property-3 (√a+√b)(√c+√d)= √ac + √ad + √bc + √bd Example: (√2+√3)(√4+√5) =√2.4 + √2.5 + √3.4 + √3.5 = √8 + √10 + √12 + √15 Identity 6 (√a+√b)²=a+2√ab+b Example: (√2+√3)² =2+2.√2.3+3 =2+ 2√6+3 =5+ 2√6 Identity 7: Rationalisation of Denominator Explain the process of multiplying and dividing the real number by its conjugate. Example: 1/√3 =1/√3 . √3 /√3 = √3/√3. √3 = √3/3	Slides
Evaluate	10	Students will attempt the Self Evaluation task on LMS.	Virtual Lab
Extend	5	Challenge: Verify using Identities learned. (5 + √3) + √2 = 5 + (√3 + √2)	Slides

Operation on Real Numbers

Introduction

In earlier classes, you learned about natural numbers, whole numbers, integers, and rational numbers. You also studied irrational numbers like √2 and √3, which cannot be expressed as fractions.

In this topic, you will learn how both rational and irrational numbers together form the set of real numbers and how they behave under the four basic operations — addition, subtraction, multiplication, and division. You will also understand some important properties (or laws) that help simplify calculations with real numbers.

These properties make real numbers a complete and consistent number system, meaning all arithmetic operations are possible (except division by zero).

Theory

Operations on Real Numbers

The four fundamental operations — addition, subtraction, multiplication, and division — can be performed on all real numbers.
Let’s explore how real numbers behave under each operation.

a) Closure Property

A set of numbers is said to be closed under an operation if performing that operation on any two numbers in the set gives a number that also belongs to the same set.

For real numbers:

Addition: a + b is a real number.
Subtraction: a – b is a real number.
Multiplication: a × b is a real number.
Division: a ÷ b is a real number (only when b ≠ 0).

Example:
√2 + 3 = 4.414… → real number
5 ÷ 2 = 2.5 → real number

So, real numbers are closed under all operations except division by zero.

b) Commutative Property

Changing the order of numbers does not change the result for addition or multiplication.

a+b = b+a and a×b =b×a

Example:
2 + √3 = √3 + 2
5 × √2 = √2 × 5

c) Associative Property

The way numbers are grouped does not affect the result for addition or multiplication.

(a+b)+c = a+(b+c) and (a×b)×c = a×(b×c)

Example:
(2 + 3) + 4 = 2 + (3 + 4) = 9
(√2 × 3) × 5 = √2 × (3 × 5) = 15√2

d) Distributive Property

Multiplication distributes over both addition and subtraction.

a×(b+c) = a×b + a×c

Example:
2(√3 + 5) = 2√3 + 10
√2(√2 + 3) = (√2 × √2) + 3√2 = 2 + 3√2

e) Identities Involving Real Numbers

Some useful algebraic identities frequently used while simplifying expressions:

√ab= √a. √b
√a/b= √a./√b
(√a+√b)(√a−√b)=a − b
(a+√b)(a−√b)=a² − b
(√a+√b)(√c+√d)= √ac + √ad + √bc + √bd
(√a+√b)² =a+ 2√ab + b

Example:
(√5 + 2)(√5 – 2) = (√5)² – (2)² = 5 – 4 = 1

f) Rationalisation of Denominator

If the denominator of a fraction has a surd (square root), we make it rational by multiplying numerator and denominator by the conjugate of the denominator.

Example:

1/√3
=1/√3 . √3 /√3
= √3/√3. √3
= √3/3

g) Operations Between Rational and Irrational Numbers

Operation	Result
Rational + Rational	Rational
Irrational + Irrational	Sometimes Rational (e.g., √2 + (–√2) = 0)
Rational + Irrational	Irrational
Rational × Irrational	Irrational
Irrational × Irrational	Sometimes Rational (e.g., √2 × √2 = 2)

Vocabulary

This is the list of vocabulary terms used throughout the lesson.

Real Numbers: Numbers that include both rational and irrational numbers.
Rational Numbers: Numbers that can be expressed as p/q, where p and q are integers and q ≠ 0.
Irrational Numbers: Numbers that cannot be expressed as p/q; their decimal expansion is non-terminating and non-repeating.
Closure Property: When an operation on two numbers gives a number of the same set.
Commutative Property: Changing the order of numbers does not change the result.
Associative Property: Grouping of numbers does not affect the result.
Distributive Property: Multiplication distributes over addition or subtraction.
Identity Element: A number that leaves another number unchanged when operated upon (0 for addition, 1 for multiplication).
Inverse Element: A number which when operated with another gives the identity element (–a for addition, 1/a for multiplication).
Rationalization: The process of removing a surd from the denominator.
Conjugate: The expression obtained by changing the sign between two terms; e.g., conjugate of (√3 + 2) is (√3 – 2).
Surd: A number containing an irrational root, like √2 or √5.
Difference of Squares: The identity (a+b)(a−b)= a²−b²
Real Number System: The collection of all rational and irrational numbers forming a single set.

Operation on Real Numbers

Introduction

In this Virtual Reality (VR) Lab, you will explore how real numbers behave under basic arithmetic operations — addition, subtraction, multiplication, and division — and understand important algebraic identities to solve and simplify numericals containing real numbers. The lab uses interactive 3D visuals and games to make abstract mathematical concepts easier to understand.
You will learn to:

Understand how operations combine and distribute in real situations.
Apply mathematical properties while solving fun interactive challenges.

By the end of this lab, you’ll not only recall how real numbers work but also see these operations come to life in an engaging way.

Key Features

Dynamic Area Model: Observe how operations such as multiplication and distributive property are represented using rectangular area models.
Step-by-Step Concept Exploration: The lab progresses from basic understanding of real numbers to applying identities.
Interactive Game: Help John climb the ladder by solving questions correctly — each correct answer lifts him closer to the apple!
Built-in Quiz: Reinforce your learning through a quick self-assessment at the end.

Step-by-Step Procedure for VR Experience

Step 1: Introduction to Real Numbers

Understand what real numbers are and how the operation on real numbers is different from the natural and whole numbers.

Step 2: Explore Operations using the Area Model

Understand the multiplicative operation on a real number using an area model of the rectangle.

Step 3: Understanding the Distributive Property

Use the concept of multiplication of real numbers and the distributive property to solve complex numerical problems having real numbers.

Step 4: The Ladder Game — “Help John Grab the Apple!”

Use the logic of the operations on real numbers to help John reach the top of the ladder to get an apple.

Step 5: Evaluation

After interaction, students proceed to the quiz:
- 2 MCQs