Binary Multiplication Calculator
This calculator multiplies two binary numbers and shows you every step of the working, not just the final answer, so you can see exactly how binary long multiplication produces the result. Enter your multiplicand and multiplier as binary digits (0s and 1s only) in the two input fields; each field also displays its decimal equivalent as you type, so you can check you have entered the number you intended. As soon as you enter valid binary values, the calculator returns the product in four formats: binary, decimal, hexadecimal and octal, with the binary result highlighted as the primary answer. Below the results, a full step-by-step working panel lays out each partial product bit by bit, shifted left according to its position in the multiplier, exactly as you would work through it by hand, followed by the binary addition that sums them into the final total. Input and result summary panels list both numbers in binary and decimal, the operation performed, and the bit length of the answer. Use this tool to check homework, verify calculations for computer science study, or understand how processors handle binary arithmetic at the bit level. It accepts positive binary integers of up to 32 bits each; it does not support negative numbers in two's complement form or binary fractions, so keep your inputs to whole, unsigned binary values.
1. Multiplicand (top number)
2. Multiplier (bottom number)
Step-by-Step Working
Input Summary
Result Summary
How Binary Multiplication Works
Binary multiplication uses the same long-multiplication method you would use with decimal numbers, but because binary only has two digits (0 and 1), the rules are much simpler. When you multiply a binary number by 0 the result is 0. When you multiply by 1 the result is a copy of the number unchanged.
For each bit in the multiplier, working from right to left, you write either a row of zeros (if the bit is 0) or a copy of the multiplicand (if the bit is 1). Each successive row is shifted one place to the left compared to the row below it, exactly as in decimal long multiplication. Once all partial products are written, you add them together using binary addition rules: 0+0=0, 0+1=1, 1+1=10 (carry 1), 1+1+1=11 (carry 1).
Worked Example: 1011 × 1101
Using the default values from this calculator:
- Multiplicand:
1011(decimal 11) - Multiplier:
1101(decimal 13) - Expected result: 11 × 13 = 143, which is
10001111in binary
| Step | Multiplier bit | Partial product (shifted) |
|---|---|---|
| Bit 0 (rightmost, value 1) | 1 | 0001011 |
| Bit 1 (value 0) | 0 | 0000000 |
| Bit 2 (value 1) | 1 | 0101100 |
| Bit 3 (value 1) | 1 | 1011000 |
| Sum of all partial products | 10001111 |
The result 10001111 equals 128 + 8 + 4 + 2 + 1 = 143 in decimal, confirming the answer.
Binary Multiplication Rules
| Bit A | Bit B | Product |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Binary multiplication of individual bits follows AND logic: the product of two bits is 1 only when both bits are 1. The complexity arises when carrying during the addition of partial products.
Converting Binary to Decimal
To check your answer, convert the binary result to decimal by multiplying each bit by its positional value (a power of 2) and summing. The rightmost bit is 2^0 = 1, the next is 2^1 = 2, then 2^2 = 4, and so on. For example, 10001111 = 128 + 0 + 0 + 0 + 8 + 4 + 2 + 1 = 143.
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Method: Standard binary long multiplication: for each bit of the multiplier (right to left), form a partial product by ANDing with the multiplicand and shifting left by the bit position, then sum all partial products using binary addition. Inputs are treated as unsigned integers. Conversion to decimal uses the positional value formula: sum of (bit × 2^position) for all bit positions.
This calculator works with unsigned binary integers. Negative numbers (two's complement) and binary fractions are not supported. The maximum supported input is 32 bits per number. For very large numbers, verify the result using the decimal equivalents shown.