Solve binary multiplication and inspect the partial products
Use binary multiplication to check a product and understand how its shifted rows combine. Enter two binary integers, then Calculate to see the full result alongside decimal and hexadecimal checks. The default 1011 × 110 produces 1000010, and each trace row identifies one contribution from the multiplier.
The negative sample demonstrates how binary multiplication handles a sign when you multiply binary numbers as mathematical integers. Spaces and underscores can group long operands, and a leading plus or minus sign is permitted. Editing either operand clears the previous result until you calculate again, so the displayed answer always belongs to the submitted pair.
To check binary multiplication with a fresh pair, use Clear before you multiply binary numbers again, or Copy to retain the current product. The calculation runs locally in your browser using exact integer arithmetic.
Understand each multiplier bit as a place-value contribution
Binary multiplication reduces the elementary digit products to two choices: a zero contributes zero, and a one contributes the multiplicand. Position determines the weight of that contribution. A one in position k adds the multiplicand shifted left by k places, so the correct alignment matters as much as the bit value.
When you multiply binary numbers, binary multiplication assigns weights one, two, four, and eight to successive multiplier positions from the right. A one in multiplier position k contributes the multiplicand shifted left by k places, equivalent to multiplying it by two to the power k. Add all those contributions to get the product.
If you multiply binary numbers by the shift-and-add method, a zero still keeps its place in binary multiplication even though its partial product is zero. This is the same positional principle used in ordinary decimal long multiplication.
Binary multiplication example: 1011 × 110
Follow binary multiplication with 1011 as the multiplicand and 110 as the multiplier. Their decimal values are eleven and six, giving an independent expected product of sixty-six. Reading the multiplier from the right produces a zero row, a one-place shift, and a two-place shift.
The lowest bit is zero when you multiply binary numbers 1011 and 110, so binary multiplication begins with a zero partial product. The next bit is one in the twos position, contributing 10110, which is 1011 shifted left once and equals decimal twenty-two. The final bit is one in the fours position, contributing 101100, which equals decimal forty-four.
To finish this binary multiplication example, add 0010110 and 0101100; you multiply binary numbers by summing the weighted rows, yielding 1000010. The decimal check is sixty-six, matching eleven times six; hexadecimal gives 42. The trace presents these contributions as individual equations so you can align them on paper without losing the place associated with each multiplier bit.
Align the shifted rows and check their carries
The last stage of binary multiplication is a binary sum, not a concatenation of the partial rows. Align the least significant positions and handle carries wherever contributions overlap. If a handwritten answer differs, checking the shifts separately from the addition makes the error easier to locate.
When you multiply binary numbers, the addition inside binary multiplication still writes zero and carries one whenever a column totals two. Three ones in a column write one and also carry one. For example, 11 multiplied by 11 produces partial products 11 and 110; their sum is 1001, decimal nine.
If you multiply binary numbers by hand, check copied digits, row shifts, and carries separately to find a binary multiplication error. The separate binary addition page can help inspect that last stage. The multiplication trace identifies contributions but does not expand every carry of their sum into a separate animation.
Binary multiplication with signs and fixed-width registers
For signed binary multiplication on this page, a minus sign has its ordinary mathematical meaning. The trace multiplies magnitudes and the final answer applies the sign rule; it does not infer a hardware word length. That distinction matters when you compare a complete integer product with a fixed-width register.
For signed binary multiplication, you can multiply binary numbers -101 and 11 to obtain -1111, or decimal minus fifteen. The partial-product rows explain the magnitudes, so the leading minus sign is applied after their sum. This is not a two’s-complement bit-vector multiplier.
Before you multiply binary numbers for a fixed-width circuit, distinguish its sign extension and retained bits from the complete binary multiplication result. Those choices require a declared width and overflow policy. Do not interpret a negative mathematical result as a ready-to-store register bit pattern without making those decisions.
Check long results, zero, and the supported input range
Long binary multiplication should preserve the accepted bits rather than round them through a floating-point value. This calculator returns the complete integer product and uses decimal and hexadecimal fields as independent representations of it. The per-operand input limit controls calculation size, not the number of significant bits retained in the answer.
The 256-character operand limit keeps binary multiplication manageable when you multiply binary numbers with long bit patterns. Leading zeros do not change the numeric value and are removed from the normalized result. Multiplication by zero returns zero; multiplication by one returns the other operand with its mathematical sign.
To multiply binary numbers here, enter integers containing only binary digits and a permitted leading sign; binary multiplication rejects fractions, exponent notation, and prefixes such as 0b. Use the decimal field as an independent magnitude check and the hexadecimal field for a compact representation. Neither field changes the meaning of the binary product or imposes a hardware word length.