数制
About Number System Conversion
Number systems are methods of representing quantities using different bases—a concept that dates back thousands of years. The Babylonians used base-60 (giving us 60-minute hours), the Mayans used base-20, and various cultures developed base-12 systems. While we typically use decimal (base-10) in daily life—likely because we have ten fingers—computers rely on binary (base-2), and programmers frequently work with hexadecimal (base-16) and octal (base-8).
Understanding number systems is fundamental to computer science, digital electronics, programming, and cybersecurity. Each system has specific practical advantages: binary directly mirrors the on/off states of transistors in computer hardware, hexadecimal provides a compact human-readable representation of binary data (essential for memory addresses, color codes, and debugging), and octal simplifies Unix file permissions and was historically important in early computing systems.
Our converter handles all standard bases from binary to base-36 and helps visualize how the same quantity appears in different number systems, making it invaluable for students, programmers, and anyone working with digital systems.
Common Number System Conversions
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0000 | 0 | 0 |
| 1 | 0001 | 1 | 1 |
| 8 | 1000 | 10 | 8 |
| 10 | 1010 | 12 | A |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 100 | 1100100 | 144 | 64 |
| 255 | 11111111 | 377 | FF |
| 256 | 100000000 | 400 | 100 |
| 1000 | 1111101000 | 1750 | 3E8 |
Number System Reference
Binary (Base-2) – Uses only 0 and 1, directly representing the two states of digital electronics (on/off, high/low voltage). Foundation of all digital computing from the simplest calculator to supercomputers. Each digit is called a "bit" (binary digit). 8 bits = 1 byte, the standard unit for computer memory. Powers of 2 (2, 4, 8, 16, 32, 64, 128, 256...) appear constantly in computing. Example: 1010₂ = 10₁₀
Octal (Base-8) – Uses digits 0-7. Each octal digit corresponds exactly to 3 binary digits, making conversion straightforward. Historically important in early computers with 12-bit, 24-bit, or 36-bit architectures. Still used today in Unix/Linux file permissions (chmod 755 means rwxr-xr-x) and C/C++ character escape sequences. Example: 17₈ = 15₁₀
Decimal (Base-10) – Uses digits 0-9. The standard human number system, almost certainly because we have ten fingers (digits). Each position represents a power of 10. While intuitive for humans, decimal doesn't align neatly with binary hardware, requiring conversion in computers.
Hexadecimal (Base-16) – Uses 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). The most important base for programmers because each hex digit represents exactly 4 binary bits—a byte is always exactly 2 hex digits. Essential for memory addresses, RGB color codes (#FF5733), MAC addresses, cryptographic hashes, and low-level debugging. Prefixed with 0x in most programming languages. Example: FF₁₆ = 255₁₀
Other Bases – Base-36 uses 0-9 and A-Z, maximizing information density with alphanumeric characters (common in short URLs). Base-64 encoding represents binary data as printable ASCII text for email attachments and data URLs. Mathematically, any positive integer greater than 1 can serve as a valid base.
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