Binary Multiplication Calculator

Multiply two binary numbers with partial products, step-by-step long multiplication, and decimal verification.

Enter Binary Numbers

Result

Binary Product (A x B)
10001111
binary
A (decimal)13
B (decimal)11
Product (decimal)143
Bits in result8

Step-by-Step Solution

1101 x 1011 = 10001111

Understanding Binary Multiplication

Binary multiplication follows the same principles as decimal long multiplication but is simpler because each digit is either 0 or 1. This means each partial product is either a copy of the multiplicand (shifted appropriately) or zero. Binary multiplication is a core operation in digital processors, forming the basis for more complex computations in signal processing, cryptography, and computer graphics.

Binary Multiplication Rules

0 x 0 = 0

Zero times zero is zero.

Partial product = 0

0 x 1 = 0

Zero times one is zero.

Partial product = 0

1 x 0 = 0

One times zero is zero.

Partial product = 0

1 x 1 = 1

One times one is one.

Partial product = multiplicand

How Binary Long Multiplication Works

The process of binary long multiplication involves these steps:

  1. Write the multiplicand (top) and multiplier (bottom) as in decimal long multiplication.
  2. For each bit of the multiplier (starting from the rightmost), create a partial product.
  3. If the multiplier bit is 1, the partial product is the multiplicand shifted left by the bit position.
  4. If the multiplier bit is 0, the partial product is all zeros.
  5. Add all partial products together to get the final result.

Shift-and-Add Method

In hardware, binary multiplication is often implemented using the "shift-and-add" method. For each bit of the multiplier, if it is 1, the multiplicand is shifted left by the appropriate amount and added to a running sum. This is exactly how long multiplication works on paper, and it's efficiently implemented using shift registers and adders.

Result Size

When multiplying an n-bit number by an m-bit number, the result can have at most n + m bits. For example, multiplying two 4-bit numbers (maximum 1111 x 1111 = 15 x 15 = 225 = 11100001) can produce up to an 8-bit result.

Practical Applications

Binary multiplication is essential in digital signal processing (DSP), graphics rendering, cryptographic algorithms (like RSA), and scientific computing. Modern CPUs have dedicated hardware multiplier units that can perform binary multiplication in a single clock cycle for common operand sizes.

Advanced Multiplication Algorithms

  • Booth's Algorithm: Handles signed binary multiplication efficiently by encoding runs of 1s in the multiplier.
  • Wallace Tree: A hardware technique that reduces partial products in parallel for faster multiplication.
  • Karatsuba Algorithm: Reduces the number of single-digit multiplications for very large numbers.
  • Array Multiplier: A straightforward hardware implementation that computes all partial products simultaneously.