Binary Calculator
Valid characters: 0-9
Base Systems
Related Calculators
Understanding Number Systems
What is Binary?
Binary is a base-2 number system that uses only two digits: 0 and 1. It's the fundamental language of computers because electronic circuits can easily represent two states: on (1) and off (0).
Number Systems Comparison
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 5 | 101 | 5 | 5 |
| 10 | 1010 | 12 | A |
| 15 | 1111 | 17 | F |
| 16 | 10000 | 20 | 10 |
| 100 | 1100100 | 144 | 64 |
| 255 | 11111111 | 377 | FF |
Converting Binary to Decimal
Each position in binary represents a power of 2, starting from 2⁰ on the right:
Example: 1101 to Decimal
Position: 3 2 1 0
Binary: 1 1 0 1
Power of 2: 2³ 2² 2¹ 2⁰
Value: 8 + 4 + 0 + 1 = 13
Converting Decimal to Binary
Divide by 2 repeatedly and read the remainders bottom-to-top:
Example: 13 to Binary
13 ÷ 2 = 6 remainder 1 ↑
6 ÷ 2 = 3 remainder 0 |
3 ÷ 2 = 1 remainder 1 |
1 ÷ 2 = 0 remainder 1 |
Read upward: 1101
Why Hexadecimal?
Hexadecimal (hex) is popular in computing because:
- One hex digit represents exactly 4 binary digits (a nibble)
- Two hex digits represent 8 bits (a byte): 00 to FF
- It's much more compact than binary while still easy to convert
- Common in colors (#FF5733), memory addresses, and programming
Hex Digits
Common Prefixes
0b- Binary (0b1010)0o- Octal (0o12)0x- Hexadecimal (0xA)
Real-World Applications
- • Colors: #RRGGBB in hexadecimal (e.g., #FF0000 = red)
- • IP Addresses: Sometimes shown in hex
- • Memory: RAM addresses in hexadecimal
- • Permissions: Unix permissions in octal (chmod 755)
- • ASCII: Character codes in hex
Frequently Asked Questions
How do you convert binary to decimal?
Each binary digit represents a power of 2, starting from 2 to the 0 on the right. Add up the values where there's a 1. For example, 1101 = (1x8) + (1x4) + (0x2) + (1x1) = 8 + 4 + 0 + 1 = 13 in decimal.
How do you convert decimal to binary?
Repeatedly divide by 2 and record remainders. Read remainders from bottom to top. For 13: 13/2=6 R1, 6/2=3 R0, 3/2=1 R1, 1/2=0 R1. Reading upward: 1101. Alternatively, subtract the largest powers of 2 that fit.
Why do computers use binary?
Binary (base-2) uses only 0 and 1, which matches the two states of electronic circuits: on/off, high/low voltage. This makes binary naturally suited for digital electronics and makes computer hardware simpler and more reliable.
What is octal and when is it used?
Octal is base-8 using digits 0-7. Each octal digit represents exactly 3 binary digits. It's used in Unix/Linux file permissions (chmod 755) and some older computer systems. It's more compact than binary but less common than hexadecimal today.