About

Binary multiplication follows the same shift-and-add algorithm taught for decimal long multiplication, but each partial product is trivially 0 or a shifted copy of the multiplicand. This simplicity is exactly why hardware ALUs implement multiplication as iterated addition with bit-shifting. A single misaligned partial product, however, propagates carry errors through every subsequent bit position. For inputs exceeding 53 bits the IEEE 754 double-precision mantissa overflows, silently truncating your result in naive implementations. This tool uses arbitrary-precision integer arithmetic internally, so it handles operands of any practical length without loss.

The calculator renders each partial product row, its positional shift, and the running summation so you can trace exactly where each output bit originates. Decimal equivalents of both operands and the product are displayed for cross-verification. Note: the tool assumes unsigned (non-negative) binary integers. Two's complement signed multiplication requires additional sign-extension logic not covered here.

Formulas

Binary long multiplication mirrors decimal long multiplication. For two unsigned binary numbers A (multiplicand) and B (multiplier), the product P is computed as:

P = n − 1∑i = 0 B_i ⋅ (A « i)

where n is the number of bits in B, B_i is the i-th bit of the multiplier (from LSB), and « denotes a left bit-shift by i positions (equivalent to multiplication by 2ⁱ). Each partial product is either 0 (when B_i = 0) or A shifted left by i places (when B_i = 1).

The maximum bit-width of the product is len(A) + len(B). Carry propagation during summation of partial products follows standard binary addition: 1 + 1 = 10₂, producing a carry of 1 into the next bit position.

Reference Data

Decimal	Binary	Bit Width	Max Unsigned Value	Common Use
0	0	1	1	Boolean flag
15	1111	4 (nibble)	15	Hex digit, BCD
255	11111111	8 (byte)	255	Pixel channel, ASCII
1023	1111111111	10	1023	ADC resolution
4095	111111111111	12	4095	12-bit sensor data
65535	1111111111111111	16	65,535	TCP port, UInt16
16777215	111111111111111111111111	24	16,777,215	RGB color
2³² − 1	32 × 1	32	4,294,967,295	IPv4 address, UInt32
2⁵³ − 1	53 × 1	53	9,007,199,254,740,991	JS Number.MAX_SAFE_INTEGER
2⁶⁴ − 1	64 × 1	64	18,446,744,073,709,551,615	UInt64, memory addressing
2¹²⁸ − 1	128 × 1	128	≈ 3.4 × 10³⁸	IPv6 address, UUID
2²⁵⁶ − 1	256 × 1	256	≈ 1.16 × 10⁷⁷	SHA-256 hash space

Frequently Asked Questions

The largest value representable in n bits is 2ⁿ − 1. Multiplying two such values yields (2ⁿ − 1)² = 2²ⁿ − 2ⁿ⁺¹ + 1, which requires exactly 2n bits. Hardware multipliers always allocate double the operand width for the result register to avoid truncation.

JavaScript's Number type uses IEEE 754 double-precision, limiting integer precision to 2⁵³ − 1. This tool converts binary strings to native BigInt values, which support arbitrary-precision integer arithmetic. The binary result string is then derived from the BigInt product via toString(2), ensuring zero precision loss regardless of input length.

No. The tool implements unsigned binary multiplication. Two's complement multiplication requires sign-extension of partial products to the full result width and a final subtraction step for the MSB partial product. If you need signed multiplication, first determine the result sign (XOR of both sign bits), multiply the absolute magnitudes here, then apply the sign.

Each partial product in binary long multiplication is simply the multiplicand shifted left by the bit-position of a 1 in the multiplier. Hardware exploits this: a barrel shifter produces each partial product in one clock cycle, and an adder tree sums them. Booth's algorithm further optimizes by encoding runs of 1s as a subtraction and a shift, reducing the number of additions for multipliers with long runs of consecutive 1-bits.

A partial product row is all zeros when the corresponding multiplier bit B_i is 0. The formula multiplies the entire multiplicand by that bit: 0 × A = 0. Hardware multipliers typically skip these rows entirely (zero-skip optimization) to save adder cycles.

Binary addition at each column follows: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10₂ (write 0, carry 1). When summing multiple partial products, a single column may accumulate carries greater than 1. The tool resolves this by summing all column values as integers and propagating the full carry chain from LSB to MSB.