Number System Class 9 Notes Maths - Chapter 1
Understanding the Number System Class 9 Notes Maths is the first step to mastering mathematics in CBSE Class 9. These notes cover all key topics from Chapter 1, including natural numbers, whole numbers, integers, rational and irrational numbers, and their placement on the number line.
With our number system class 9 notes, students can easily grasp the decimal system, real numbers, and various operations. These notes are designed to simplify learning and strengthen your basics.
Whether you are revising for exams or building your concepts, our number system class 9 notes pdf download offers a handy and clear reference for quick learning.
Start your maths journey with confidence and clarity through these well-structured notes.
CBSE Class 9 Maths Chapter 1 Number System Notes
Number System: A number system is a set of sybols or digits used to represent numbers and epxress mathematical concepts. The most commonly used system is the decimal system is the decimal system (base 10), which includes digits from 0 to 9. Other systems are binary, octal and hexadecimal.
Decimal Number System: Decimal number system is the most widely used number system in every day life. It consists of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each place value represents a power of 10, with the right most digit representing 100 (ones), the next representing 101 (tens) and so on. Its used for counting measurements and general computation
Understanding Real Numbers: Real numbers are the numbers we use in everyday life for counting, measuring and doing calculations. They include rational and irrational numbers. Rational numbers are numbers that can be written as fractions. These include whole numbers (like 5), negative numbers (like – 3) and decimals that either stop (like 0.75) or repeat (like 2.666…).
Examples are 4, – 7, 1/2 and 3.5
Irrational numbers can not be writen as fractions. Their decimals parts go on forever without repeating. Some well-known examples are √2, √3 and π These numbers cannot be written exactly but are used often in math and science
Together, rational and irrational numbers from the set of real numbers, represented by the symbol R. Real numbers can be shown on a number line, with each point representing a real number. They are important in all areas of mathematics such as algebra, geometry and calculus.
Precisely Marking Real Numbers on a Number line: Representing real numbers on a number line invalues a process called magnification, which helps us pinpoint the exact location of a real number with increasing accuracy.
This process allows us to display any real number clearly, as shown in the images above, where magnification helps zoom into smaller and smaller intervals to locate the real number precisely
Important Concepts to Remember:
1. Natural numbers: A natural number is a positive integer used for counting and ordering. It starts from 1 and goes on indefinitely. They are represented by N.
N = {1, 2, 3, 4, 5, …}
2. Whole numbers: 0 and natural numbers together make the set of whole numbers. Whole numbers are represented by W.
W = {0, 1, 2, 3, 4, 5, …}
3. Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The smallest prime number is 2. P denotes the set of prime numbers.
P = {2, 3, 5, 7, 11, 13, …}
4. Coposite numbers: A composite number is a natural number greater than 1 that has more than two positive divisors (i.e., it can be divide by numbers other than 1 and itself). They are represented by C.
C = {4, 6, 8, 9, 12, …}
[Note: 1 is neither prime nor composite number]
5. Co-prime numbers: Co-prime numbers (also known as relatively prime numbers) are two numbers that have no common divisors other than 1.
Example: 4 and 5, 8 and 9, 8 and 15 etc
6. Integers: The collection of negative numbers, positive numbers and zero (excluding fractions) are called integers. Integers are represented by Z.
$$Z = \left\{ \ldots, -3, -2, -1, \underbrace{0}_{\text{Neither } \oplus \text{ nor } \ominus}, 1, 2, 3, \ldots \right\} \\ \quad \underbrace{\hspace{3.2cm}}_{\text{-ve integers}} \quad \underbrace{\hspace{2.4cm}}_{\text{+ve integers}}$$
Real Numbers and their Decimal Expansions
Real numbers can have different types of decimal expansions:
1. Terminating Decimals: When the remainder becomes zero, the decimal stops.
Example: 0.72, 1.25, 0.875 etc.
2. Non-terminating Decimals: These numbers never end. They can be
(a) Recurring: The digits are repeat.
Example: 1/11 = 0.090909…
(b) Non-Recurring: The digits never repeat, like p = 3.141592653589793…, where the digits continue without a pattern.
Operations on Real Numbers
The operations on real numbers are of three types:
(i) Operation on two rational numbers: When we perform arithmetic operations on two rational numbers like addition, subtraction, division and multiplication then the result will be rational numbers.
Example: (0.25 + 0.25) = 0.50 can be written as 50/100
(0.30 – 0.10) = 0.20 can be written as 20/100
0.4 × 184 = 73.6 and can be written as 736/10
0.252/0.4 = 0.63 and can be written as 63/100 (All results are form of p/q)
(ii) Operation on two irrational numbers: When we perform arithmetic operations on two irrational numbers like addition, subtraction multiplication and division then the result will be an irrational or a rational number.
Example:
$$\sqrt{3}+\sqrt{3}=2\sqrt{3}\quad{\text{(irrational number)}}\\ 5\sqrt{2}-\sqrt{2}=4\sqrt{2}\quad{\text{(irrational number)}}\\ 2\sqrt{2}-2\sqrt{2}=0\quad{\text{(rational number)}}\\\sqrt{2}\times\sqrt{2}=2\quad{\text{(rational number)}}\\\sqrt{2}\times\sqrt{3}=\sqrt{6}\quad{\text{(irrational number)}}\\\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{2}}{\sqrt{3}}\quad{\text{(irrational number)}}\\\frac{\sqrt{3}}{\sqrt{3}}=1\quad{\text{(rational number)}}$$
(iii) Operations on a Rational and an irrational number: When we perform arithmetic operations on a non-zero rational number and an irrational number like addition, subtraction, multiplication and division then the result is always irrational.
Example:
$$2+5\sqrt{3}=2+5\sqrt{3}\quad\text{(irrational)}\\3-5\sqrt{2}=3-5\sqrt{2}\quad\text{(irrational)}\\2\times\sqrt{3}=\sqrt{7}\quad\text{(irrational)}\\\frac{5\sqrt{7}}{5}=\sqrt{7}\quad\text{(irrational)}\\0 \times\sqrt{3}=0\quad\text{(rational)}\\\frac{0}{\sqrt{3}}=0\quad\text{(rational)}$$
Rationalizing the Denominator
Rationalizing the denominator is a process used removing irrational numbers from the denominator.
Example:
$$\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}\quad\text{(Denominator is rational)}\\\frac{1}{\sqrt{2}+3}=\frac{1}{\sqrt{2}+3}\times\frac{\sqrt{2}-3}{\sqrt{2}-3}\frac{\sqrt{2}-3}{(\sqrt{2}+3)(\sqrt{2}-3)}=\frac{\sqrt{2}-3}{2-9}=\frac{\sqrt{2}-3}{-7}\quad\text{(denominator is ationalized)}$$
Identities involving square roots for real numbers:
(i) √mn=√m.√n
(ii) (√m + √n) (√m - √n) = m - n
(iii) (p+ √n) (p- √n) = p2 - n
(iv) (√m + √n)( (√q + √r) = √mq + √mr + √nq + √nr
(v) (√m - √n)( (√q - √r) = √mq + √mr - √nq - √nr
(vi) (√m + √n)2 = m + 2√mn + n
(vii) √(m/n) = √m/√n
Laws for Exponents:
(i) When multiplying two powers with the same base, add the exponents.
Example:
$$a^m\times a^n=a^{m+n}\\2^3\times2^4=2^{3+4}=2^7$$
(ii) When dividing two powers with the same base, subtract the exponents
Example:
$$\frac{a^m}{a^n}=a^{m-n},\quad\text{where }a\ne0\\\frac{5^6}{5^2}=5^{6 - 2}=5^4$$
(iii) When raising a power to another power, multiply the exponents
Example:
$$(a^m)^n=a^{mn}\\(3^2)^4=3^{2\times 4}=3^8$$
(iv) When raising a product to a power, distribute the exponent to each factor.
Example:
$$(ab)^m=a^m\cdot b^m\\(2\cdot5)^3=2^3\cdot5^3$$
(v) When raising a quotient to a power, apply the exponent to both numerator and denominator
Example:
$$\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m},\quad b\ne 0\\\left(\frac{4}{3}\right)^2=\frac{4^2}{3^2}$$
(vi) Any non-zero number raised to the power 0 is 1.
Example:
$$a^0=1,\quad a\ne0\\7^0=1$$
(vii) A negative exponent means the reciprocal of the base raised to the positive exponent
Example:
$$a^{-m}=\frac{1}{a^m},\quad a\ne0\\2^{-3}=\frac{1}{2^3}$$
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