CBSE Class 9 Maths Ch1 Number Systems Notes PDF Download

Class 9 Chapter 1 Maths Notes are designed strictly according to the CBSE Syllabus Class 9 Maths. This chapter builds the foundation of real numbers, irrational numbers, decimal expansions and laws of exponents.

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If you understand this chapter clearly, higher class topics like polynomials and algebra will become much easier. Let’s break down the Number System in the simplest way possible–step by step.

Maths Class 9 Chapter 1 Number System Summary

In this chapter, you shall study the foundational concepts of mathematics – the number system. You will begin by understanding what numbers are and how they are categorized into different types.

The chapter explains how real numbers include both rational and irrational numbers and how they can be represented on a number line. 

You will also learn about the decimal expansion of rational and irrational numbers, how to convert recurring and terminating decimals into fractions and how to rationalise irrational denominators.

Introduction to Number Systems

The number system is the process of representing numbers on the number line using certain rules and symbols. A number line is a depiction of numbers on a straight line with a specified interval between them. 

The number system is used to do mathematical computations ranging from complex scientific calculations to counting the number of chocolates left in the box. In this chapter, we are going to understand the different types of numbers, their properties and how they interact with each other.

What are Numbers?

A number is an arithmetical value representing a particular quantity. The various types of numbers include natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers. 

These categories help us classify and work with numbers in mathematical operations more efficiently.

Natural Numbers

Natural numbers, denoted by N, are the basic counting numbers starting from 1. They include all positive integers like 1, 2, 3, 4 and so on. 

Natural numbers do not include zero, decimals or negative numbers. These are the first set of numbers we learn as children when we start counting.

Whole Numbers

Whole numbers, denoted by W, are natural numbers including zero. Hence, the set becomes 0, 1, 2, 3, 4 and so on. Whole numbers do not include negative values, fractions or decimals. Zero is the only number in this set that is not a natural number.

Integers

Integers are numbers that include all whole numbers and their negative counterparts. The set of integers, denoted by Z, includes {... –3, –2, –1, 0, 1, 2, 3...}. Integers do not include fractions or decimals. They are used to represent gains and losses, elevations, temperatures, etc.

Rational Numbers

A number is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0. Rational numbers include all integers, fractions and terminating or recurring decimals. For example, 4, –5/2, 0.75 and 1.333... are all rational numbers.

Irrational Numbers

Any number that cannot be expressed in the form of p/q (where p and q are integers and q ≠ 0) is an irrational number. These numbers have non-terminating and non-repeating decimal expansions. Common examples are √2, π and √5. Their values cannot be exactly expressed as fractions.

Properties of Irrational Numbers:

1. The negative of an irrational number is also irrational.
2. The sum or difference of a rational and an irrational number is always irrational.
3. The sum or product of two irrational numbers may be rational or irrational.
4. The product of a rational number and an irrational number is irrational unless the rational number is zero.

Identities Involving Irrational Numbers

Some useful identities to simplify expressions with surds and irrational numbers:

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

These are frequently used in algebraic manipulations and simplifications involving real numbers.

Proof that √2 is Irrational

Assume √2 is rational. So √2 = p/q (in lowest form)

Squaring both sides ⇒ 2 = p² / q² ⇒ p² = 2q²

This shows p² is even → p is even.

Let p = 2k

Substitute ⇒ (2k)² = 2q² ⇒ 4k² = 2q² ⇒ q² = 2k²

So q is also even. This contradicts the assumption that p/q is in lowest form. Therefore, √2 is irrational.

Real Numbers

The set of real numbers, denoted by R, includes both rational and irrational numbers. Every real number can be represented on the number line. This means every point on the number line corresponds to a real number. Real numbers include values like –3, 0, √2, 3.14, etc.

Euclid’s Division Lemma

For any two positive integers a and b: a = bq + r, where: 0 ≤ r < b. This is used to find HCF of numbers.

Euclid’s Division Lemma

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of prime numbers, and this factorisation is unique (except for order). For example, 24 = 2 × 2 × 2 × 3.

Prime factorisation tree of 24 Class 9 Maths

Representation of √x on Number Line

Steps:

1. Draw AB = x units
2. Extend to C such that BC = 1
3. Find midpoint of AC
4. Draw semicircle
5. Drop perpendicular from B

BD = √x

Construction of square root on number line Class 9

Decimal Representation of Rational Numbers

Rational numbers can be expressed in decimal form by dividing the numerator by the denominator. The decimal expansion can either terminate or become non-terminating but recurring.

Examples:

• 1/2 = 0.5 (terminating)
• 1/3 = 0.333… (non-terminating recurring)

If the prime factors of the denominator (after simplification) are only 2 and/or 5, the decimal is terminated. Otherwise, it is recurring.

Decimal Expansion of Rational and Irrational Numbers

If the denominator (in lowest form) has only prime factors 2 and/or 5 ⇒ decimal terminates. Otherwise ⇒ decimal is non-terminating, recurring. For example:

• 1/8 = 0.125 (terminating)
• 1/3 = 0.333… (recurring)

Expressing Decimals as Rational Numbers

Let x = 0.333… ⇒ 10x = 3.333…

Subtract ⇒ 10x – x = 3 ⇒ 9x = 3

⇒ x = 1/3

Rationalisation

Rationalisation is the process of eliminating irrational numbers from the denominator of a fraction. This helps in making the number easier to work with, especially in further calculations or representations on a number line. For example, to rationalise 1/√2

Multiply numerator & denominator by √2 ⇒ √2 / 2

Laws of Exponents for Real Numbers

For real numbers a and b and integers m and n:

• a^m × aⁿ = a^m+n
• (a^m)ⁿ = a^mn
• a^m / aⁿ = a^m–n
• (ab)^m = a^m × b^m
• a⁰ = 1 (a ≠ 0)
• a^–n = 1/aⁿ

Practice Questions on Number System

Solve the following questions for better practice and better marks in exams.

1. Prove that √5 is irrational.
2. Find HCF of 306 and 657 using Euclid’s Division Algorithm.
3. Determine whether 77/2100 will terminate or recur.
4. Represent √7 on a number line.
5. Convert 0.272727… into rational form.

FAQs

Q1. What is Euclid’s Division Lemma in Class 9 Maths?

Ans. Euclid’s Division Lemma states that for any two positive integers a and b ⇒ a = bq + r; where 0 ≤ r < b. It is used to find HCF of numbers.

Q2. How do you know if a rational number will terminate?

Ans. If the denominator (in lowest form) has only prime factors 2 and/or 5, the decimal will terminate. Otherwise, it will be recurring.

Q3. Is √2 a real number?

Ans. Yes. √2 is an irrational number and all irrational numbers are real numbers.

Q4. What is the difference between rational and irrational numbers?

Ans. Rational numbers can be written as p/q. Irrational numbers cannot be written as p/q and have non-repeating decimals.

Q5. Why is the Fundamental Theorem of Arithmetic important?

Ans. It ensures that every composite number has a unique prime factorisation, which is used to study decimal expansions and number properties.