NUMBER SYSTEM – DETAILED NOTES (CLASS 9)
1. Introduction
The number system is a way to represent and classify numbers. In Class 9, we mainly study natural
numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
2. Natural Numbers (■)
Natural numbers are counting numbers starting from 1. Example: 1, 2, 3, 4, 5, ...
3. Whole Numbers (■)
Whole numbers include all natural numbers along with zero. Example: 0, 1, 2, 3, 4, ...
4. Integers (■)
Integers include positive numbers, negative numbers, and zero. Example: -3, -2, -1, 0, 1, 2, 3
5. Rational Numbers (■)
A rational number is a number that can be written in the form p/q, where p and q are integers and q
≠ 0. Example: 1/2, -3/5, 0.75
6. Irrational Numbers
Irrational numbers cannot be expressed in the form p/q. Their decimal expansion is non-terminating
and non-repeating. Examples: √2, √3, π
7. Real Numbers (■)
All rational and irrational numbers together form the set of real numbers.
8. Decimal Expansion of Rational Numbers
The decimal expansion of a rational number is either terminating or non-terminating recurring.
• If denominator has only factors 2 and/or 5 → terminating decimal.
• If denominator has factors other than 2 or 5 → non-terminating recurring decimal.
9. Laws of Exponents
• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)
• (a^m)^n = a^(mn)
• (ab)^m = a^m b^m
• a^0 = 1 (a ≠ 0)
10. Important Exam Points
1. Introduction
The number system is a way to represent and classify numbers. In Class 9, we mainly study natural
numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
2. Natural Numbers (■)
Natural numbers are counting numbers starting from 1. Example: 1, 2, 3, 4, 5, ...
3. Whole Numbers (■)
Whole numbers include all natural numbers along with zero. Example: 0, 1, 2, 3, 4, ...
4. Integers (■)
Integers include positive numbers, negative numbers, and zero. Example: -3, -2, -1, 0, 1, 2, 3
5. Rational Numbers (■)
A rational number is a number that can be written in the form p/q, where p and q are integers and q
≠ 0. Example: 1/2, -3/5, 0.75
6. Irrational Numbers
Irrational numbers cannot be expressed in the form p/q. Their decimal expansion is non-terminating
and non-repeating. Examples: √2, √3, π
7. Real Numbers (■)
All rational and irrational numbers together form the set of real numbers.
8. Decimal Expansion of Rational Numbers
The decimal expansion of a rational number is either terminating or non-terminating recurring.
• If denominator has only factors 2 and/or 5 → terminating decimal.
• If denominator has factors other than 2 or 5 → non-terminating recurring decimal.
9. Laws of Exponents
• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)
• (a^m)^n = a^(mn)
• (ab)^m = a^m b^m
• a^0 = 1 (a ≠ 0)
10. Important Exam Points
