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CBSE Notes
Class 9
Maths
Chapter 1 Number Systems

CBSE Notes Class 9 Maths Chapter 1 Number System

The number system forms the core of Mathematics and is vital for understanding advanced topics. From counting objects to solving complex problems in Physics, engineering, and computer science, numbers are everywhere. This blog will provide a comprehensive overview of the CBSE Class 9 Maths chapter 1 number system: exploring number system, its types, properties, and significance in Mathematics.

1.0Download CBSE Notes for Class 9 Maths Chapter 1 Number System - Free PDF!!

Students can now access and download free PDF for CBSE Class 9 Maths Notes Chapter 1 Number System, designed for clear understanding and flexible study anytime, anywhere.

Class 9 Maths Chapter 1 Revision Notes:

2.0Types of Numbers

Number systems are fundamental structures in mathematics that provide a way to represent and manipulate numbers. Here are the key number systems:

  1. Natural Numbers (N): Natural numbers are positive integers beginning from 1 and extending infinitely (1, 2, 3, 4, ...)
  2. Whole Numbers (W):  Whole numbers are similar to natural numbers but include zero (0, 1, 2, 3, ...).
  3. Integers (Z): Integers are whole numbers, including positive numbers, negative numbers, and zero. They are denoted by the symbol "Z" in mathematics. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.
  4. Rational Numbers (Q): Rational numbers are those that can be expressed as fractions of two integers where the denominator is not equal to zero (e.g., 1/2, -3/4, 5, -7, 0).
  5. Real Numbers (R): Real numbers encompass both irrational and rational numbers. They are numbers that can be located on the number line and include integers, fractions, decimals, and square roots of non-negative numbers. Examples of real numbers are -3, 0, 1.5, 2​, and π.
  6. Complex Numbers (C): Complex numbers are mathematical entities of the form a + bi, where "a" and "b" are real numbers and "i" represents the imaginary unit ((−1)​). Complex numbers can be represented on a complex plane. It may be noted that N ⊂ Z ⊂ Q ⊂ R ⊂ C.

types of numbers

  1. Irrational Number: An irrational number is a real number that cannot be expressed as a fraction of two integers or in qp​ form. Its decimal expansion is non-terminating and non-repeating, Examples of irrational numbers are √2 , e, or π.

3.0Properties of Irrational Numbers

  1. Non-Rational: Irrational numbers cannot be expressed as a ratio of two integers.
  2. Non-Terminating, Non-Repeating Decimals: Their decimal expansion neither terminates nor repeats (e.g., π = 3.14159...).
  3. Closure: Irrational numbers are not closed under addition, subtraction, multiplication, or division. (e.g., √2 + √2 = 2√2, which is irrational, but √2 × √2 = 2, which is rational).
  4. Subset of Real Numbers: Irrational numbers are part of the set of real numbers (ℝ).

4.0Representation of Rational Number on the Number Line  

Irrational numbers can be challenging to represent due to their non-terminating, non-repeating decimal expansions, but they still have precise locations on the number line. Here's how you can visualize and represent them:

Steps to Represent an Irrational Number (e.g., √2) on the Number Line:

  • Draw a Number Line: Start with a simple number line with integers like 0, 1, –1, 2, –2, and so on.
  • To locate √2, draw a right-angled triangle where both legs are of length 1 unit (based on the Pythagorean theorem: a2 + b2 = c2).
  • For this draw a base OA measuring 1 unit. And on ‘A’ draw a height of 1 unit long. Name it as AB.
  • Now join line OB>
  • So by using Pythagoras Theorem 

OA = 1, AB = 1 

OB2 = OA2 + OC2

OB2 = 12 + 12

OB2 = 2

Taking square root,

OB = 2​ 

The hypotenuse of this triangle will have a length of √2 units.

  • Place one end of the hypotenuse at O on the number line. Using a compass or ruler, mark the point where the hypotenuse intersects the number line and call them C. This point represents √2.

This process can be used for other square roots, like √3 or √5, and even for more complex irrational numbers by refining their decimal approximation.

5.0Operations on Real Numbers

  1. The addition or subtraction of a rational number with an irrational number always leads to an irrational number.
  2. The product or division of a non-zero rational number and an irrational number always leads to an irrational number.
  3. When adding, subtracting, multiplying, or dividing two irrational numbers, the result can be either rational or irrational.

6.0Identities involving square roots for Real Numbers

  1. mn​=m​⋅n​
  2. (m​+n​)(m​−n​)=m−n
  3. (p+n​)(p−n​)=p2−n
  4. (m​+n​)(q​+r​)=mq​+mr​+nq​+nr​
  5. (m​+n​)(q​−r​)=mq​−mr​+nq​−nr​
  6. (m​+n​)2=m+2mn​+n

7.0Solved Examples on Number System

Example 1: Determine whether the following numbers are rational or irrational:

1. 16​

2. 7​

3. 35​

  1. 07​

Solutions:

  1. 16​: 16​=4 is a rational number because it can be represented as 14​.
  2. 7​: Since 7 is not a perfect square, 7​ is irrational.
  3. 35​: This is a ratio of two integers where the denominator is not zero, so 5/3 is a rational number.
  4. 07​: Division by zero is undefined, so 07​ is not a valid number.

Example 2: Simplify 72​ and express it in the form ab​, where a and b are integers and b is not a perfect square.

Solution:

Factorize 72:

72=36×2=62×2

Simplify:

72​=36×2​=36​×2​=62​

So, 72​ simplifies to 62​.

Example 3: Rationalize the denominator of 2​1​.

Solution:

Multiply by 2​2​​:

2​1​×2​2​​=22​​

So, the rationalized form of 2​1​ is 22​​.

Example 4: Simplify (35​+23​)−(5​−3​)

Solution:

Distribute and combine like terms:

(35​+23​)−(5​−3​)

=35​+23​−5​+3​

=25​+33​

So, the simplified form is 25​+33​.

Example 5: Find five rational numbers between 75​ and 76​.

Solution:

Multiply and divide 75​ and 76​ by 6.

7×65×6​=4230​ and 7×66×6​=4236​

So, now 5 rational number between 75​ and 76​ are equivalent to 4230​ and 4236​ are:

4231​,4232​,4233​,4234​,4235​

Example 5: Express 0.257 in the form qp​, where p and q are integers and q ≠ 0.

Solution:

Let x=0.257= 0.2575757….

Multiply x by 100

100x = 25.7575…

100x = 25.2 + 0.25757…

100x = 25.2 + x

100x – x = 25.2

99x = 25.2

99 x = 10252​

Which gives, x=990252​

8.0Practice Questions on Number System

  1. Find five rational numbers between 83​ and 85​.
  2. Express 37 in the form p/q, where p and q are integers and q ≠ 0.
  3. Determine whether the following numbers are rational or irrational:
  • 25​
  • 50​
  • 47​
  • 2​+3​
  1. Explain why 3​8​ is an irrational number. Rationalize the denominator of 3​8​.
  2. Express 27​+12​ in the simplest form.
  3. Simplify the expression after rationalizing the denominator: 2​−123​​
  4. Simplify the following expressions:
  • (7​+25​)−(7​−5​)
  • (6​×3​)+(12​−2​)

9.0Key Features of CBSE Maths Notes for Class 9 Chapter 1 Number System

  • Comprehensive Coverage: Includes all important topics such as rational and irrational numbers, number line representation, laws of exponents, and real numbers.
  • Aligned with Latest CBSE Syllabus: Notes are fully updated as per the latest NCERT and CBSE curriculum guidelines.
  • Concept Clarity: Topics are explained in simple language with step-by-step methods, helping students understand even complex concepts easily.
  • Prepared by Experts: Curated by experienced teachers and subject experts to ensure accuracy and depth of content.
  • Easy PDF Download: Instantly downloadable in PDF format, so students can access and study anytime, even without internet.

Chapter-wise CBSE Notes for Class 9 Maths:

Class 9 Maths Chapter 1 - Number Systems Notes

Class 9 Maths Chapter 2 - Polynomial Notes

Class 9 Maths Chapter 3 - Coordinate Geometry Notes

Class 9 Maths Chapter 4 - Linear Equation In Two Variables Notes

Class 9 Maths Chapter 5 - Introduction To Euclids Geometry Notes

Class 9 Maths Chapter 6 - Lines and Angles Notes

Class 9 Maths Chapter 7 - Triangles Notes

Class 9 Maths Chapter 8 - Quadrilaterals Notes

Class 9 Maths Chapter 9 - Circles Notes

Class 9 Maths Chapter 10 - Herons Formula Notes

Class 9 Maths Chapter 11 - Surface Areas and Volumes Notes

Class 9 Maths Chapter 12 - Statistics Notes


Chapter-wise NCERT Solutions for Class 9 Maths:

Chapter 1: Number Systems

Chapter 2: Polynomials

Chapter 3: Coordinate Geometry

Chapter 4: Linear Equations in Two Variables

Chapter 5: Introduction to Euclid’s Geometry

Chapter 6: Lines and Angles

Chapter 7: Triangles

Chapter 8: Quadrilaterals

Chapter 9: Circles

Chapter 10: Heron’s Formula

Chapter 11: Surface Areas and Volumes

Chapter 12: Statistics

Frequently Asked Questions

A number system is a way to represent numbers using symbols and rules. It helps to perform arithmetic operations and calculations. Common number systems include the decimal system (base 10), binary system (base 2), and others.

A rational number is any number that can be expressed in the form p/q, where p and q are integers, and q ≠ 0. Examples include 1/2, 3, -5 etc.

An irrational number cannot be expressed as a fraction p/q. It has a non-terminating, non-repeating decimal form. Examples include π, e, etc.

Real numbers include both rational and irrational numbers. They represent all the numbers on the number line, such as 1, -3, π, and many more.

Rational numbers either have terminating decimals (e.g., 1/4 = 0.25 ) or repeating decimals (e.g., 1/3 = 0.3333 ). Irrational numbers have non-terminating and non-repeating decimal expansions (e.g., π = 3.14159… ).

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