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Maths Chapter 1 Number Systems Class 9 Notes FREE PDF Download
Number Systems Class 9 Notes: CBSE Maths Chapter 1
ShareShareReal numbers
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 a 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 fractions 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
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
Remainder never becomes 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 the 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 the decimal point but the remainder never becomes 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
The sum or difference between a rational number and an irrational number is irrational.
The product or quotient of a non-zero rational number with an irrational number is irrational.
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 to rational is called rationalizing the denominator.
It is obtained by multiplying the numerator and denominator by the irrational term present in the denominator but with the 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
${{x}^{m}}.{{x}^{n}}={{x}^{m+n}}$
$\dfrac{{{x}^{m}}}{{{x}^{n}}}={{x}^{m-n}}$
${{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}$
${{x}^{m}}{{y}^{m}}={{\left( xy \right)}^{m}}$
Important Formulas of Class 9 Chapter 1 Maths Number Systems You Shouldn’t Miss!
Important Formulas of Class 9 Chapter 1 Maths: Number Systems and properties are fundamental for understanding number systems and solving problems in Class 9 Maths. Make sure to familiarise yourself with them to build a strong foundation.
1. Euclid’s Division Lemma:
For any two positive integers $a$ and $b$, where $a > b$, there exist unique integers $q$ and $r$ such that:
$ a = bq + r $, where $0 \leq r < b$.
2. Fundamental Theorem of Arithmetic:
Every composite number can be expressed as a product of prime numbers in a unique way, except for the order of factors. For example:
$ 30 = 2 \times 3 \times 5 $
3. Properties of Real Numbers:
Closure Property: For any two real numbers $a$ and $b$, their sum $a + b$ and product $a \times b$ are also real numbers.
Property: $a + b = b + a$ and $a \times b = b \times a$
Associative Property: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$
4. Rational Numbers:
A rational number can be written as:
$ \frac{p}{q} $
where $p$ and $q$ are integers, and $q \neq 0$.
5. Irrational Numbers:
Numbers that cannot be expressed as a simple fraction, e.g., $\sqrt{2}$ and $\pi$.
6. Decimal Representation:
Terminating Decimals: Decimal numbers that end after a finite number of digits, e.g., $0.75$.
Non-Terminating Repeating Decimals: Decimal numbers that repeat a pattern of digits infinitely, e.g., $0.\overline{3}$.
Importance of Chapter 1 Number Systems Class 9 Notes
Number System Class 9 Notes PDF provides a comprehensive understanding of number systems, which is essential for academic success and practical problem-solving.
Foundation for Higher Mathematics: Understanding number systems is crucial as it forms the base for more advanced topics in mathematics, such as algebra, calculus, and number theory.
Conceptual Clarity: The chapter covers essential concepts like types of numbers (natural, whole, integers, rational, and irrational), which are fundamental for solving various mathematical problems.
Application of Theorems: Learning about Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic helps in understanding how numbers are structured and how they can be decomposed or analysed.
Problem-Solving Skills: Mastering number systems improves your ability to tackle problems involving different types of numbers and their properties, which is vital for both exams and practical applications.
Real-World Relevance: Knowledge of number systems is useful in real-life scenarios such as computer science (binary system), finance (decimal system), and engineering.
Tips for Learning the Class 9 Maths Chapter 1 Number Systems
Here are some tips for learning Class 9 Maths Chapter 1: Number Systems:
Understand the Basics: Start by getting the fundamental concepts of different types of numbers, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Learn Key Formulas: Know about the important formulas and theorems, such as Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic. Understanding these will help you solve problems more effectively.
Use Visual Aids: Draw number lines or use diagrams to visualise different types of numbers and their relationships. This can make abstract concepts more concrete.
Practice Problems: Regularly solve practice problems to reinforce your understanding. Focus on problems related to comparing, ordering, and performing operations with different types of numbers.
Conclusion
Chapter 1 of Class 9 Maths, Number Systems, provides essential knowledge about different types of numbers and their properties. Understanding this chapter is crucial for building a strong foundation in mathematics, as it supports understanding advanced concepts and problem-solving skills. By these key concepts, practising regularly, and relating them to real-life scenarios, you'll enhance your ability to work with various number systems effectively. Use the notes and practice problems to reinforce your learning and ensure you are well-prepared for both exams and practical applications.
Related Study Materials for Class 9 Maths Chapter 1 Number Systems
Students can also download additional study materials provided by Vedantu for Class 9 Maths Chapter 1 Number Systems.
Chapter-wise Links for Class 9 Maths Notes
Related Important Links for Maths Class 9
Along with this, students can also download additional study materials provided by Vedantu for Maths Class 9–
FAQs on Number Systems Class 9 Notes: CBSE Maths Chapter 1
1. What are the most important concepts to revise in Number Systems for Class 9 Maths Chapter 1?
The core concepts to focus on during revision are:
- Types of numbers: Natural, Whole, Integers, Rational, and Irrational numbers, including their definitions and symbols.
- Decimal representation: Terminating and non-terminating decimals, including recurring and non-recurring categories.
- Properties of real numbers: Closure, commutative, and associative properties.
- Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic
2. How should I structure my quick revision for Number Systems before exams?
To revise efficiently, begin with a concept map of all number types and their relationships. Next, summarize key properties, formulas, and theorems. Practice a few representative problems of each concept, especially on decimal expansions and operations with real numbers. End with a recap of important formulas and common application-based questions.
3. What are the key differences between rational and irrational numbers?
Rational numbers can be expressed as p/q, where p and q are integers and q ≠ 0. Their decimal expansions either terminate or repeat. Irrational numbers cannot be so expressed and have decimal expansions that are non-terminating and non-recurring. Examples: Rational – 3/4, -2/5; Irrational – √2, π.
4. How do the properties of real numbers simplify calculations in Class 9 Number Systems?
The closure, commutative, and associative properties enable flexible calculation strategies. For example, using commutative property (a + b = b + a), you can change the order of addition; associativity allows grouped operations. These properties apply to both addition and multiplication of real numbers, making calculations faster and reducing errors in multi-step problems.
5. Why is understanding the decimal expansion of numbers important in Class 9 Maths?
Recognizing if a decimal is terminating, non-terminating recurring, or non-terminating non-recurring helps determine if a number is rational or irrational. This is fundamental for both Class 9 Board exams and higher-level Mathematics, as it connects concepts of fractions, number types, and practical computations.
6. What is the Fundamental Theorem of Arithmetic and how should I remember it for exams?
The Fundamental Theorem of Arithmetic states that every composite number can be expressed uniquely (apart from the order) as a product of prime numbers. For exams, remember this as the ‘unique prime factorization’ property, which is essential for problems involving divisibility, HCF, and LCM.
7. How can a concept map help during last-minute revision of Number Systems Class 9?
A concept map visually links all types of numbers and their properties, enabling a quick summary of definitions, symbols, and relationships. It is a valuable tool to reinforce memory and identify connections across subtopics in Chapter 1, which is highly useful during last-minute review.
8. What misconceptions should I avoid while studying Number Systems for Class 9 revision?
Avoid confusing irrational numbers with non-terminating decimals that repeat (these are still rational). Remember, zero is a whole number and an integer, but not a natural number. Also, the sum or product of two irrational numbers is not always irrational, so check with examples. Clarity on these points prevents common errors in exams.
9. How often should I revisit key formulas and properties during my Class 9 Number Systems revision?
It is recommended to review key formulas and properties regularly, especially before tests and exams. Frequent short reviews help reinforce memory and identify weak areas needing extra practice.
10. What are some effective tips for rapid revision of Class 9 Maths Chapter 1?
Summarize each subtopic in your own words, focus on example problems, create flashcards for formulas, and attempt a few challenging problems from previous years or sample papers. Prioritize understanding concept flows instead of rote memorization for more efficient and confident recall in the exam.
