The Quick Answer
| System | Base | Digits Used | Example |
|---|---|---|---|
| Binary | 2 | 0, 1 | 1010 |
| Decimal | 10 | 0-9 | 10 |
| Hexadecimal | 16 | 0-9, A-F | A |
All three represent the same value: ten
Why Different Number Systems?
Decimal (base 10): Humans use it because we have 10 fingers.
Binary (base 2): Computers use it because transistors have two states: on/off.
Hexadecimal (base 16): Programmers use it as a compact way to represent binary (4 binary digits = 1 hex digit).
How Number Systems Work
Decimal (What You Know)
Each position is a power of 10:
2 5 3
│ │ └── 3 × 10⁰ = 3 × 1 = 3
│ └────── 5 × 10¹ = 5 × 10 = 50
└────────── 2 × 10² = 2 × 100 = 200
───
253
Binary (Base 2)
Each position is a power of 2:
1 0 1 1
│ │ │ └── 1 × 2⁰ = 1 × 1 = 1
│ │ └────── 1 × 2¹ = 1 × 2 = 2
│ └────────── 0 × 2² = 0 × 4 = 0
└────────────── 1 × 2³ = 1 × 8 = 8
──
11 (decimal)
Hexadecimal (Base 16)
Each position is a power of 16. Uses A-F for 10-15:
| Hex | Decimal |
|---|---|
| A | 10 |
| B | 11 |
| C | 12 |
| D | 13 |
| E | 14 |
| F | 15 |
2 F
│ └── F × 16⁰ = 15 × 1 = 15
└────── 2 × 16¹ = 2 × 16 = 32
──
47 (decimal)
Converting Between Systems
Binary to Decimal
Add up the powers of 2 where there's a 1:
Binary: 11010
Position: 4 3 2 1 0
Value: 16 8 4 2 1
Binary: 1 1 0 1 0
= 16 + 8 + 0 + 2 + 0 = 26
Decimal to Binary
Repeatedly divide by 2, track remainders:
26 ÷ 2 = 13 remainder 0
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read bottom to top: 11010
Hex to Decimal
Multiply each digit by its power of 16:
Hex: 1A3
= 1 × 16² + A × 16¹ + 3 × 16⁰
= 1 × 256 + 10 × 16 + 3 × 1
= 256 + 160 + 3
= 419
Decimal to Hex
Repeatedly divide by 16:
419 ÷ 16 = 26 remainder 3
26 ÷ 16 = 1 remainder 10 (A)
1 ÷ 16 = 0 remainder 1
Read bottom to top: 1A3
Binary to Hex (Shortcut)
Group binary digits into sets of 4 (from right), convert each group:
Binary: 11010110
Group: 1101 0110
Hex: D 6
Result: D6
Hex to Binary (Shortcut)
Convert each hex digit to 4 binary digits:
Hex: A7
A = 1010
7 = 0111
Result: 10100111
Common Values Reference
| Decimal | Binary | Hex |
|---|---|---|
| 0 | 0000 | 0 |
| 1 | 0001 | 1 |
| 5 | 0101 | 5 |
| 10 | 1010 | A |
| 15 | 1111 | F |
| 16 | 10000 | 10 |
| 100 | 1100100 | 64 |
| 255 | 11111111 | FF |
| 256 | 100000000 | 100 |
Where You'll See These
Binary
- Low-level programming
- Bit flags and permissions
- Network subnet masks
- File permissions (chmod)
Hexadecimal
- Colors in CSS:
#FF5733 - Memory addresses:
0x7fff5fbff8ac - MAC addresses:
00:1A:2B:3C:4D:5E - Unicode:
U+1F600 - Cryptographic hashes
Prefixes in Code
let decimal = 42; // No prefix
let binary = 0b101010; // 0b prefix
let hex = 0x2A; // 0x prefix
let octal = 0o52; // 0o prefix (base 8)
Colors in Hex
CSS colors use hex because each color channel (R, G, B) fits in one byte (0-255 = 00-FF):
#FF5733
││││││
││││└┴── Blue: 33 = 51
││└┴──── Green: 57 = 87
└┴────── Red: FF = 255
Related Tools
- Binary to Decimal - Convert binary to decimal
- Decimal to Hex - Convert decimal to hexadecimal
- ASCII Table - See character codes in all formats