Binary Multiplication Calculator
Enter two binary numbers (digits 0 and 1 only) and this calculator multiplies them using the same long-multiplication method taught in computer science and digital electronics courses. You get the product in binary, decimal, and hexadecimal, plus a step-by-step partial-products breakdown that mirrors how a digital circuit or assembly routine performs the operation.
Formula
Worked example
1011 x 0101: The multiplier bits (right to left) are 1, 0, 1, 0. Bit 0 is 1: partial product is 1011 (shifted 0). Bit 1 is 0: partial product is 0. Bit 2 is 1: partial product is 101100 (shifted 2). Bit 3 is 0: partial product is 0. Sum: 1011 + 101100 = 110111, which equals 55 in decimal. Check: 11 x 5 = 55.
How binary multiplication works
Binary multiplication uses the same long-multiplication algorithm you learned for decimal numbers, but because each digit is either 0 or 1 the partial products are trivially simple: if the current multiplier bit is 0 the partial product is 0, and if it is 1 the partial product is just the multiplicand shifted left by the bit position. After computing all partial products you add them together using binary addition to get the final answer. For example, multiplying 1011 (decimal 11) by 0101 (decimal 5) produces four partial products, two of which are non-zero, and their binary sum is 110111, which is decimal 55.
Step-by-step long multiplication method
Write the multiplicand on top and the multiplier below it. Starting from the rightmost (least significant) bit of the multiplier, examine each bit in turn. If the bit is 1, copy the multiplicand and shift it one position to the left for each bit position you have moved from the right. If the bit is 0, write 0. Once you have all partial products, align them by their right edges, including the shift positions, and add the columns using binary addition rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1). The final sum is the product. Carries propagate to the left exactly as in decimal addition.
Decimal and hexadecimal equivalents
This calculator shows the product in three bases: binary (base 2), decimal (base 10), and hexadecimal (base 16). Converting between them is a common step in programming and hardware design. To go from binary to decimal, multiply each bit by the power of 2 it represents (rightmost bit is 2^0, next is 2^1, and so on) and sum the results. To convert to hexadecimal, group binary digits into sets of four from the right, then map each group to the corresponding hex digit (0-9 for 0-9, A-F for 10-15). For instance, 110111 groups as 0011 0111, which is 0x37 in hex and 55 in decimal.
Binary multiplication in computing
Processors perform integer multiplication using variations of binary long multiplication, often optimised with algorithms such as Booth encoding or Wallace trees to reduce the number of addition steps. Shift-and-add is the conceptual foundation: the multiplicand is shifted left once for each bit position in the multiplier and conditionally added to an accumulator. Understanding this method is essential for anyone studying computer architecture, writing low-level code, or designing digital circuits. The number of bits required for the product is at most the sum of the bit-widths of the two operands, which is why multiplying two 8-bit numbers can require a 16-bit result register.
Binary multiplication rules
| Bit A | Bit B | A x B (binary) | A x B (decimal) |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
The four fundamental single-bit multiplication rules used at each position.
Frequently asked questions
How do you multiply two binary numbers?
Use the same long-multiplication method as decimal: examine each bit of the multiplier from right to left. When the bit is 1, write down the multiplicand shifted left by the bit position. When the bit is 0, write 0. Then add all the partial products using binary addition (where 1+1 = 10, carrying 1 to the next column). The result is the binary product.
What are the basic binary multiplication rules?
0 x 0 = 0, 0 x 1 = 0, 1 x 0 = 0, and 1 x 1 = 1. Because each bit is either 0 or 1, each partial product is either 0 or a copy of the multiplicand, which makes binary multiplication simpler to carry out by hand than decimal multiplication.
Why does a binary product need more bits than the inputs?
Multiplying an m-bit number by an n-bit number can produce a result up to m+n bits wide. For example, two 4-bit numbers (maximum value 15 each) can produce a product up to 225, which needs 8 bits. This is why hardware multipliers produce a double-width result and why overflow must be handled carefully in low-level programming.
How do I check my binary multiplication result?
Convert both inputs to decimal, multiply them, then convert the decimal product to binary and compare. For instance, 1011 is 11 in decimal, 0101 is 5, and 11 x 5 = 55. Converting 55 to binary gives 110111. If your binary product matches, the result is correct.
What is the relationship between binary multiplication and bit shifting?
Multiplying a binary number by 2 is identical to shifting all its bits one position to the left (appending a 0 on the right). Multiplying by 4 is a left shift by 2, multiplying by 8 is a left shift by 3, and so on for any power of 2. General multiplication extends this: each partial product is the multiplicand left-shifted by the position of a 1-bit in the multiplier, and the partial products are summed. Modern compilers exploit this by replacing multiplication by powers of 2 with bit-shift instructions, which are faster.
Does binary multiplication work the same for negative numbers?
For unsigned (non-negative) integers, the long-multiplication method above gives the correct result directly. For signed integers in two's complement representation (the standard in modern processors), the same algorithm still works for the lower-order bits of the result, but sign extension and the handling of the most significant bit differ. Hardware uses algorithms such as Booth encoding to handle signed binary multiplication correctly without needing special cases. This calculator uses unsigned binary inputs.