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Quick Answer
Binary 1011 + 1101 = 11000 (which is 24 in decimal). Binary 11111111 equals 255 in decimal and FF in hexadecimal.
Binary Arithmetic
Digits 0 and 1 only
Base Conversion
Type in any field to update all others
Common Examples
| Input | Result |
|---|---|
| 1011 + 1101 | 11000 (decimal 24) |
| 10000 - 0111 | 1001 (decimal 9) |
| 110 * 101 | 11110 (decimal 30) |
| Binary 11111111 | Decimal 255, Hex FF, Octal 377 |
| Binary 1010 | Decimal 10, Hex A, Octal 12 |
How It Works
Binary Number System
Binary is a positional number system with base 2. Each digit (called a bit) is either 0 or 1. The value of a binary number is calculated by multiplying each bit by 2 raised to its positional power (starting from 0 on the right):
value = b(n) x 2^n + b(n-1) x 2^(n-1) + … + b(1) x 2^1 + b(0) x 2^0
For example, binary 1011 = 1 x 2^3 + 0 x 2^2 + 1 x 2^1 + 1 x 2^0 = 8 + 0 + 2 + 1 = 11 in decimal.
Binary Arithmetic
Addition: Follows the same column-by-column approach as decimal addition, but carries at 2 instead of 10. The key rules are: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1).
Subtraction: Borrows at 2 instead of 10. The key rules are: 0-0=0, 1-0=1, 1-1=0, 10-1=1 (borrow from the next column).
Multiplication: Works like long multiplication in decimal. Multiply by each bit of the second number and shift left for each position, then add the partial products.
Division: Integer division, returning the whole-number quotient. Works like long division in decimal but with base 2.
Worked Example
To add binary 1011 and 1101: start from the rightmost column. 1+1=10, write 0, carry 1. 1+0+1(carry)=10, write 0, carry 1. 0+1+1(carry)=10, write 0, carry 1. 1+1+1(carry)=11, write 11. Result: 11000. Verification: 11 + 13 = 24, and binary 11000 = 16 + 8 = 24.
To convert binary 1010 to decimal: 1 x 8 + 0 x 4 + 1 x 2 + 0 x 1 = 10. To hex: 10 in decimal = A in hex. To octal: 10 / 8 = 1 remainder 2, so octal 12.