Number Systems for Grade 9
Introduction to Number Systems
Number systems are ways to represent numbers. In Grade 9, we focus on four main types of number systems: Decimal, Binary, Octal, and Hexadecimal.
1. Decimal System
The decimal system is the most common number system, used in everyday life. It is a base-10 system, meaning it uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Example: The number 1234 in decimal is represented as:
1234 = 1×10³ + 2×10² + 3×10¹ + 4×10⁰
2. Binary System
The binary system is a base-2 system, used extensively in computer science. It uses only two digits: 0 and 1.
Example: The number 101 in binary is represented as:
101₂ = 1×2² + 0×2¹ + 1×2⁰ = 4 + 0 + 1 = 5 in decimal
3. Octal System
The octal system is a base-8 system, using eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.
Example: The number 75 in octal is represented as:
75₈ = 7×8¹ + 5×8⁰ = 56 + 5 = 61 in decimal
4. Hexadecimal System
The hexadecimal system is a base-16 system, using sixteen digits: 0-9 and A-F (where A=10, B=11, ..., F=15).
Example: The number 2A in hexadecimal is represented as:
2A₁₆ = 2×16¹ + 10×16⁰ = 32 + 10 = 42 in decimal
Converting Between Systems
Understanding how to convert numbers between different systems is crucial. For instance, converting a decimal number to binary involves repeatedly dividing by 2 and noting the remainders.
Example: Convert 13 from decimal to binary:
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading remainders from bottom to top, 13 in decimal is 1101 in binary.
Applications of Number Systems
Number systems are fundamental in various fields such as computer science, digital electronics, and cryptography. Understanding these systems helps in coding, data representation, and problem-solving.