Number Systems Class 9 Notes CBSE Maths Chapter 1(Free PDF Download)

• Real numbers and imaginary numbers together form number systems.

• We will discuss imaginary numbers in higher classes, let us restrict our discussion to real numbers

• Real numbers are the set of natural numbers, whole numbers, integers, rational and irrational numbers. Denoted by R

• Natural numbers:

• These are counting numbers starting from $1$.

•  The set $\left\{ 1,2,3,4,5,6,7.... \right\}$ is called natural numbers.

•  Denoted by N

• Whole numbers:

• These are the set of natural numbers including $0$. 

• The set $\left\{ 0,1,2,3,4,5,6.... \right\}$ is called whole numbers.

• Denoted by W

• Integers: 

• These are the set of negative numbers, positive numbers and $0$ excluding fractions. 

• The set $\left\{ ....-3,-2,-1,0,1,2,3.... \right\}$ is called integers.

• Denoted by Z

• Rational numbers:

• These are those numbers which can be expressed in the form of fraction i.e., $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$. 

• For example: $\dfrac{3}{5},\dfrac{-2}{9},\dfrac{-3}{4},$ etc. 

• Denoted by Q

• There are infinitely many rational numbers between any two rational numbers.

• Irrational numbers:

• These are those which are not rational i.e., which cannot be expressed in the form of  $\dfrac{p}{q}$ where $p$ and $q$ are integers and$q\ne 0$.

• For example: $\sqrt{2},\sqrt{3},\sqrt{5},$ etc.

Real Numbers and Their Decimal Expansions

There are two cases of decimal expansions

1. Remainder becomes zero

• Decimal expansion of numbers whose remainder becomes zero after some step is called terminating.

• For example: $\dfrac{7}{8}=0.875$ , the remainder becomes zero after some steps

2. Remainder never become zero

• Decimal expansion of numbers whose remainder never becomes zero after some step is called non-terminating.

• It is further divided into non-terminating recurring and non-terminating non-recurring.

• Non-terminating recurring means numbers which keep on repeating the same value after decimal point.

• For example: $\dfrac{9}{11}=0.818181....$ 

• Non-terminating non-recurring means numbers which do not keep on repeating the same value after decimal point but remainder never become zero.

• For example: value of $\pi =3.141592653589793283....$ 

• Decimal expansion of rational numbers is either terminating or non-terminating.

• Decimal expansion of irrational numbers is non-terminating non-recurring.

Representing Real Number on Number Line

• Representation of real numbers on the number line can be done by the process of successive magnification.

• For example: If we want to locate $4.377$ on the number line we proceed by successive magnification i.e., $4.37$ lies between $4$ and $5$ then locate $4.37$ between $4.36$ and $4.38$further divide this portion into ten equal parts then $4.377$ will lie between $4.376$ and $4.378$. The number line is shown below

Operations on Real Numbers

• Real numbers can be added, subtracted, multiplied and divided.

• For example: 

Add $2+\sqrt{3}$ and $2-2\sqrt{3}$

$2+\sqrt{3}+2-2\sqrt{3}$ 

$=4-\sqrt{3}$ 

Subtract $2+\sqrt{3}$ and $2-2\sqrt{3}$ 

$\left( 2+\sqrt{3} \right)-\left( 2-2\sqrt{3} \right)$ 

$=2+\sqrt{3}-2+2\sqrt{3}$ 

$=3\sqrt{3}$ 

Multiply $2\sqrt{2}$ and $3\sqrt{3}$ 

$2\sqrt{2}\times 3\sqrt{3}$

$=2\times 3\times \sqrt{2}\times \sqrt{3}$ 

$=6\sqrt{6}$

Divide $10\sqrt{15}$ by $\sqrt{5}$ 

$\dfrac{10\sqrt{15}}{\sqrt{5}}=\dfrac{10\sqrt{3}\times \sqrt{5}}{\sqrt{5}}=10\sqrt{3}$

• Some common facts of operation on real numbers are

1. The sum or difference of a rational number and an irrational number is irrational.

2. The product or quotient of a non-zero rational number with an irrational number is irrational.

3. If we add, subtract, multiply or divide two irrationals, then the result may be rational or irrational.

Rationalizing Denominator

• When the denominator is irrational then the process of converting the denominator rational is called rationalizing the denominator.

• It is obtained by multiplying numerator and denominator by the irrational term present in the denominator but with opposite sign.

• For example: Rationalizing $\dfrac{1}{\sqrt{2}+3}$ 

$\dfrac{1}{\sqrt{2}+3}\times \dfrac{\sqrt{2}-3}{\sqrt{2}-3}$

$=\dfrac{\sqrt{2}-3}{{{\left( \sqrt{2} \right)}^{2}}-{{3}^{2}}}$ 

$=\dfrac{\sqrt{2}-3}{2-9}$ 

$=\dfrac{\sqrt{2}-3}{-7}$ 

Laws of Exponents for Real Numbers

There are some laws of exponent for real numbers such as

1. ${{x}^{m}}.{{x}^{n}}={{x}^{m+n}}$ 

2. $\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}$

3. ${{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}$ 

4. ${{x}^{m}}{{y}^{m}}={{\left( xy \right)}^{m}}$ 

Number System Class 9 PDF – Free PDF Download

9th Class Maths Notes Chapter 1 Number Systems Free PDF

The CBSE Class 9 Maths Notes Chapter 1 Number Systems are available on the official website of Vedantu in a PDF format for free. Students can avail benefits from the downloading of these PDFs in both soft copy and hard copy. The students can understand and practice it during their feasible time and need not bother about the interrupted internet connection. 

Class 9 Maths Chapter 1 Number Systems 

Introduction:- 

The Number Systems Class 9 Notes has recalled all the knowledge of numbers learned in the earlier classes. It has been explained by taking a funny scenario of having a bag and collecting different numbers. Let's say that the first natural numbers started from 1 and ask to collect the 0 also into the bag, which makes the set into whole numbers. It is denoted by W. Later on, moving back to the negative integers on the number line, which recalls the memory of integers Z. And the next important classification of numbers is rational numbers. It is in p/q form and denoted by Q.

Irrational Numbers:- 

After understanding certain solved examples, the PDF has directed to solve an exercise related to all the classifications of numbers studied so far. Class 9 Maths Notes Chapter 1 has taken the initiation to introduce irrational numbers to the students, which takes a form that doesn't equal the rational number. The point is that all the numbers can be represented on the number line at any point. Similarly, every point of the number line denotes a number irrespective of its classification. This principle is called a real number line because those numbers are called real numbers.

Real Numbers and their Decimal Expansion

To explore more about the real numbers, these Notes of Class 9 Maths Chapter 1 have explained both rational and irrational numbers in a new way that is nothing but the decimal point method.

So every fraction can be represented using decimals by dividing exactly where two cases will arise. 

They are:

• The quotient will be zero

• The quotient will not be zero

if the numerator is exactly divided by the denominator, then the quotient will be 0 and if the numerator is not divided by the denominator exactly then the quotient may not be 0.

Representing Real Numbers on The Number Line

In this section, Notes for Class 9 Maths Chapter 1 has experimented while explaining the representation of real numbers on the number line. As the real numbers are very nearby, including all the points of the number line, they can be seen using a magnifying glass. This visualization of representing real numbers using a magnifying glass is known as the process of successive magnification.

Operations on Real Numbers

Class 9th Maths Chapter 1 Notes has explained all the properties like commutative, closure, identity, distributive, etc., concerning addition, subtraction, multiplication, and division of rational and irrational numbers. Few observations had been made from this. They are as follows:

• The sum or difference between a rational number and the irrational number is irrational.

• All four mathematical operations done by two irrational numbers might be either rational or irrational.

• The product or quotient of a non-zero rational number with an irrational number is irrational.

Laws of Exponents For Real Numbers

The Notes of Mathematics Class 9 Chapter 1 have focused more on this area and covered the idea of exponents in detail. The exponent is the number or variable that appears at the top of a digit. It is not also referred to as power. The original digit is known as the base. It can be written as mn. They have supplied some solid and unsolved examples to help you learn these exponentials in depth. The last of these exponents were wider apart than the next classes. The square roots and cube roots produced from these exponents are also described in the Chapter 1 Mathematics Class 9 notes.

Benefits of Class 9 Maths Chapter 1 Revision Notes

• A great reference tool to strengthen the important concepts of the chapter.

• Students become more confident to solve even the complex question of the chapter during the exam.

• Exam stress and anxiety are reduced.

• Help students minimise the chance of making silly mistakes.

• Saves precious time during exams as these notes will help you to revise all the important topics of the chapter quickly.

• The accuracy of answers attempted during the exams is higher.

Conclusion

CBSE Class 9 Mathematics Chapter 6 Revision Notes are one of the most useful resources you may utilize since each chapter's topic is given in an easy-to-read style. Viewing these notes helps students remember all of the ideas covered in the chapter. As a result, it will save time during tests. They will also receive all of the chapter's major formulae in one spot.