Number System and Conversion

1. Introduction to Number Systems

A number system is a writing system for expressing numbers. Each number system has a base (or radix) that determines the number of unique digits used.

The four main number systems used in computing are: Decimal (10), Binary (2), Octal (8), and Hexadecimal (16).

2. Decimal Number System (Base 10)

Uses digits 0-9. Each position represents a power of 10.

Example: 3456 = 3x10³ + 4x10² + 5x10¹ + 6x10⁰

3. Binary Number System (Base 2)

Uses digits 0 and 1. Each position represents a power of 2. Binary is the fundamental language of computers.

Example: 1101₂ = 1x2³ + 1x2² + 0x2¹ + 1x2⁰ = 8+4+0+1 = 13₁₀

4. Octal Number System (Base 8)

Uses digits 0-7. Each position represents a power of 8. Often used in Unix file permissions.

Example: 125₈ = 1x8² + 2x8¹ + 5x8⁰ = 64+16+5 = 85₁₀

5. Hexadecimal Number System (Base 16)

Uses digits 0-9 and letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15).

Example: 2F₁₆ = 2x16¹ + 15x16⁰ = 32+15 = 47₁₀

6. Number System Conversions

Decimal to Binary (Repeated Division by 2)

Convert 25₁₀ to binary:
25 ÷ 2 = 12 remainder 1 (LSB)
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1 (MSB)
Result: 11001₂

Binary to Decimal

Convert 10101₂ to decimal:
= 1x2⁴ + 0x2³ + 1x2² + 0x2¹ + 1x2⁰
= 16 + 0 + 4 + 0 + 1 = 21₁₀

Binary to Octal (Group of 3 bits)

Convert 110101₂ to octal:
Group (from right): 110 101
110₂ = 6₈, 101₂ = 5₈
Result: 65₈

Binary to Hexadecimal (Group of 4 bits)

Convert 11010110₂ to hexadecimal:
Group (from right): 1101 0110
1101₂ = D₁₆, 0110₂ = 6₁₆
Result: D6₁₆

7. Binary Arithmetic

Binary Addition

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (carry 1)

Binary Subtraction

0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1)