📘 Class 9 Maths Chapter 1: Number Systems Notes (Complete & SEO Optimized)

🔹 Introduction to Number Systems

Number system is a way to represent numbers. In this chapter, we study different types of numbers like natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

🔹 Natural Numbers

Natural numbers start from 1 and go to infinity.

Example:
1, 2, 3, 4, 5, …

🔹 Whole Numbers

Whole numbers start from 0 and go to infinity.

Example:
0, 1, 2, 3, 4, …

🔹 Integers

Integers include negative numbers, zero, and positive numbers.

Representation:
… -3, -2, -1, 0, 1, 2, 3 …

🔹 Even and Odd Numbers

✔ Even Numbers

Numbers divisible by 2.

Example: 2, 4, 6, 8 …

✔ Odd Numbers

Numbers not divisible by 2.

Example: 1, 3, 5, 7 …

🔹 Rational Numbers

A number is called a rational number if it can be written in the form:$$\frac{p}{q}$$qp​

Where p and q are integers and q ≠ 0.

Examples:
2/5, 6/3, 0/1, 5/1

🔹 Standard Form of Rational Numbers

A rational number is in standard form if HCF of p and q = 1.

Example:
2/4 = 1/2

🔹 Rational Number Between Two Numbers

A rational number between a and b is:$$\frac{a + b}{2}$$2a+b​

👉 There are infinite rational numbers between any two numbers.

🔹 Decimal Representation of Rational Numbers

Rational numbers can be written in decimal form:

✔ Terminating Decimal

Decimal that ends.

Example:
1/2 = 0.5

✔ Non-Terminating Recurring Decimal

Decimal that repeats.

Example:
1/3 = 0.333…

🔹 Irrational Numbers

Numbers that cannot be written in p/q form are called irrational numbers.

Examples:
√2, √3, √5, π

👉 π is irrational but 22/7 is rational.

🔹 Real Numbers

All rational and irrational numbers together form real numbers.

👉 Real numbers include:

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers

🔹 Representation of Real Numbers on Number Line

• Every real number can be represented on a number line
• Irrational numbers are located using geometrical methods

🔹 Laws of Exponents

For any non-zero numbers:

1. aᵐ × aⁿ = aᵐ⁺ⁿ
2. aᵐ ÷ aⁿ = aᵐ⁻ⁿ
3. (aᵐ)ⁿ = aᵐⁿ
4. a⁰ = 1
5. a⁻ⁿ = 1 / aⁿ

🔹 Prime Numbers

A number having only two factors (1 and itself).

Examples:
2, 3, 5, 7 …

Important Facts:

• Prime numbers (0–50) = 15
• Prime numbers (50–100) = 10
• Prime numbers (0–100) = 25
• 2 is the smallest even prime number

🔹 Composite Numbers

Numbers having more than two factors.

Examples:
4, 6, 8, 9 …

🔹 Important Concepts

• Square root of any prime number is irrational
• Rational numbers between two numbers are infinite