Complete guide to understanding different number systems, their properties, and applications in computing
A number system is a systematic way to represent numbers using symbols or digits. The value of each digit in a number depends on:
The base of a number system determines how many different digits are available and the value of each position. For example:
In positional notation, the value of a digit is determined by its position in the number. Each position represents a power of the base.
dₙdₙ₋₁...d₂d₁d₀ = dₙ×bⁿ + dₙ₋₁×bⁿ⁻¹ + ... + d₂×b² + d₁×b¹ + d₀×b⁰
Where:
3×10² + 2×10¹ + 5×10⁰ = 300 + 20 + 5 = 325
1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11
A single binary digit (0 or 1)
8 bits (can represent 256 values)
Group of bits processed together (16, 32, or 64 bits)
| Hex | Decimal | Binary | Hex | Decimal | Binary |
|---|---|---|---|---|---|
| 0 | 0 | 0000 | 8 | 8 | 1000 |
| 1 | 1 | 0001 | 9 | 9 | 1001 |
| 2 | 2 | 0010 | A | 10 | 1010 |
| 3 | 3 | 0011 | B | 11 | 1011 |
| 4 | 4 | 0100 | C | 12 | 1100 |
| 5 | 5 | 0101 | D | 13 | 1101 |
| 6 | 6 | 0110 | E | 14 | 1110 |
| 7 | 7 | 0111 | F | 15 | 1111 |
Use positional notation:
dₙ×bⁿ + dₙ₋₁×bⁿ⁻¹ + ... + d₁×b¹ + d₀×b⁰
Use repeated division:
Group binary digits in sets of 4:
1101 0111₂ = D7₁₆
Group binary digits in sets of 3:
101 110 011₂ = 563₈
Binary is fundamental to CPU operations, memory addressing, and digital logic circuits.
Hexadecimal colors: #RRGGBB where each pair represents red, green, and blue intensity.
Unix uses octal notation for file permissions: 755 = rwxr-xr-x
IP addresses, subnet masks, and MAC addresses often use hexadecimal representation.
All data in computers is ultimately stored and processed in binary format.
Hexadecimal is used in memory dumps and low-level debugging for compact representation.
