How-To
Beginner
2 min read
331 words
Number Base Conversion: Binary, Octal, Decimal, and Hexadecimal
Understanding number bases is essential for programming, networking, and digital electronics. This guide explains conversion methods between binary, octal, decimal, and hexadecimal systems.
Key Takeaways
- Repeatedly divide by the target base and collect remainders in reverse order.
- Multiply each digit by the base raised to its position power (counting from 0 at the right).
- Each hex digit maps to exactly 4 binary digits:
- Binary**: CPU operations, bitwise flags
Featured Tool
Prime Number Tools
Check, factorize, generate, and find prime numbers
The Four Common Bases
| Base | Name | Digits | Prefix |
|---|---|---|---|
| 2 | Binary | 0, 1 | 0b |
| 8 | Octal | 0-7 | 0o |
| 10 | Decimal | 0-9 | (none) |
| 16 | Hexadecimal | 0-9, A-F | 0x |
Decimal to Any Base
Repeatedly divide by the target base and collect remainders in reverse order.
Example: Convert 156 to binary:
- 156 / 2 = 78 remainder 0
- 78 / 2 = 39 remainder 0
- 39 / 2 = 19 remainder 1
- 19 / 2 = 9 remainder 1
- 9 / 2 = 4 remainder 1
- 4 / 2 = 2 remainder 0
- 2 / 2 = 1 remainder 0
- 1 / 2 = 0 remainder 1
Read remainders bottom-up: 10011100
Any Base to Decimal
Multiply each digit by the base raised to its position power (counting from 0 at the right).
Example: 0xFF to decimal:
15 * 16^1 + 15 * 16^0 = 240 + 15 = 255
Binary-Hex Shortcut
Each hex digit maps to exactly 4 binary digits:
| Hex | Binary | Hex | Binary |
|---|---|---|---|
| 0 | 0000 | 8 | 1000 |
| 1 | 0001 | 9 | 1001 |
| 2 | 0010 | A | 1010 |
| 3 | 0011 | B | 1011 |
| 4 | 0100 | C | 1100 |
| 5 | 0101 | D | 1101 |
| 6 | 0110 | E | 1110 |
| 7 | 0111 | F | 1111 |
To convert binary to hex: group bits in fours from the right, then replace each group. 10011100 → 1001 1100 → 9C.
Where Each Base Is Used
- Binary: CPU operations, bitwise flags
- Octal: Unix file permissions (chmod 755)
- Hex: Colors (#FF6B35), memory addresses, MAC addresses, cryptographic hashes
- Decimal: Human-facing values