Number Systems Overview

Definition of Number Systems

• A number system is a consistent way of representing numbers using digits or symbols.
• The value of each digit in a number is determined by its position and the base of the number system.
• Number systems are used for converting between different representations of numbers.
• Number systems help represent numbers using a limited set of symbols.

Types of Number Systems

• There are four main types of number systems:
  • Binary (Base 2)
  • Octal (Base 8)
  • Decimal (Base 10)
  • Hexadecimal (Base 16)

Octal Number System

• Uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7.
• Has a base of 8.
• Has fewer digits compared to other systems, resulting in fewer computational errors.
• Examples: 358, 9238, 14183588

Decimal Number System

• Uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Has a base of 10.
• Is the system used in everyday life.
• Numbers without a specified base are assumed to be decimal (base 10).
• Examples: 72310, 524710, 3210, 983710

Converting Binary to Decimal

• Step 1: Identify the base of the binary number.
• Step 2: Multiply each digit in the binary number by the corresponding power of 2, starting from the rightmost digit with an exponent of 0 and increasing by 1 for each position to the left.
• Step 3: Add the results from Step 2 to obtain the decimal equivalent.

Conversion Examples

• Example 1: Converting (100111)2 to decimal:
  • (1 × 2^5) + (0 × 2^4) + (0 × 2^3) + (1 × 2^2) + (1 × 2^1) + (1 × 2^0) = 32 + 0 + 0 + 4 + 2 + 1 = 39
• Example 2: Converting (111111)2 to decimal:
  • (1 × 2^5) + (1 × 2^4) + (1 × 2^3) + (1 × 2^2) + (1 × 2^1) + (1 × 2^0) = 32 + 16 + 8 + 4 + 2 + 1 = 63