Binary Long Multiplication Calculator
Multiply two unsigned binary numbers using the traditional long multiplication method. See each multiplier-bit decision, shifted partial product, running sum, decimal equivalent and the final binary product.
Enter values and calculate.
| Step | Multiplier Bit | Bit Position | Base Partial Product | Shift | Shifted Partial Product | Partial Decimal | Running Sum |
|---|---|---|---|---|---|---|---|
| Enter binary values and calculate to view multiplication steps. | |||||||
What Is Binary Long Multiplication?
Binary long multiplication is the base-2 version of the long multiplication method commonly used with decimal numbers. Because binary contains only the digits 0 and 1, each multiplier digit produces a particularly simple partial product.
When a multiplier bit is 1, the corresponding partial product is a shifted copy of the multiplicand. When the multiplier bit is 0, the partial product is zero.
All shifted partial products are then added together to obtain the final binary product.
How Binary Long Multiplication Works
Read the Multiplier Bits
Start with the least significant multiplier bit and continue toward the most significant bit.
Multiply by Each Bit
A multiplier bit of 1 copies the multiplicand, while a bit of 0 produces a zero partial product.
Shift the Partial Product
Each successive multiplier position shifts its partial product one place farther to the left.
Add Partial Products
The shifted partial products are added to form the complete binary product.
Binary Multiplication Rules
| A | B | A × B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Binary Long Multiplication Algorithm
Running Sum = 0
Read multiplier bits from right to left.
For each multiplier bit:
If bit = 0:
Partial Product = 0
If bit = 1:
Partial Product = Multiplicand
Shift the partial product left by its bit position.
Add it to the running sum.
After all multiplier bits are processed:
Running Sum = Final Product
Worked Example: 1011 × 0110
Binary 1011 represents decimal 11 and binary 0110 represents decimal 6.
1011 = 11
Multiplier:
0110 = 6
11 × 6 = 66
Binary product:
1000010
Verification:
1000010 binary = 66 decimal
Paper-Style Example: 1011 × 0110
× 0110
————
0000
10110
101100
+ 0000000
————
1000010
The multiplier is read from right to left. Its bits are 0, 1, 1 and 0, so the second and third shifted copies of the multiplicand contribute to the final sum.
Worked Example: 101 × 11
11 binary = 3 decimal
5 × 3 = 15
Binary product:
1111
Partial products:
101
1010
101 + 1010 = 1111
Worked Example: 1111 × 0011
0011 = 3
15 × 3 = 45
Binary product:
101101
What Happens When a Multiplier Bit Is Zero?
A zero multiplier bit contributes no numeric value to the product. Its partial-product row therefore contains zeros even though its positional shift still corresponds to that multiplier position.
Multiplier bit = 0
Base partial product:
0000
Shifted partial product:
0
Why Partial Products Shift Left
Each binary position represents a power of two. Moving one position left therefore multiplies a binary value by 2.
Shift left 1 position:
10110 = 22
Shift left 2 positions:
101100 = 44
11 × 2 = 22
11 × 4 = 44
Binary Long Multiplication Examples
| Multiplicand | Multiplier | Decimal | Product | Product Decimal |
|---|---|---|---|---|
| 0010 | 0011 | 2 × 3 | 0110 | 6 |
| 0011 | 0100 | 3 × 4 | 1100 | 12 |
| 0101 | 0011 | 5 × 3 | 1111 | 15 |
| 0110 | 0110 | 6 × 6 | 100100 | 36 |
| 1011 | 0110 | 11 × 6 | 1000010 | 66 |
| 1111 | 0011 | 15 × 3 | 101101 | 45 |
Binary Long Multiplication vs Decimal Long Multiplication
| Feature | Binary | Decimal |
|---|---|---|
| Number base | Base 2 | Base 10 |
| Available digits | 0 and 1 | 0 through 9 |
| Single-digit multiplication | 0 or original value | Can produce many values |
| Partial products | Zeros or shifted multiplicand | Digit multiples |
| Final step | Add partial products | Add partial products |
Long Multiplication vs Direct Binary Multiplication
A direct multiplication calculator may provide only the final product. Binary long multiplication exposes how that product is constructed from individual multiplier bits and shifted copies of the multiplicand.
Direct Result
Useful when the final binary product is the main goal.
Long Multiplication
Useful when you need to understand partial products, shifts and the step-by-step arithmetic process.
How Many Bits Can the Product Need?
If one unsigned operand uses m bits and another uses n bits, their product can require up to m + n bits.
1111 = 15
15 × 15 = 225
225 decimal =
11100001 binary
The product requires 8 bits.
Important Binary Long Multiplication Notes
Only digits 0 and 1 are accepted.
Leading zeros are allowed and do not change numeric value.
A multiplier bit of 1 creates a shifted copy of the multiplicand.
A multiplier bit of 0 creates a zero partial product.
Each step farther left in the multiplier corresponds to an additional left shift of the partial product.
The final product is the sum of all shifted partial products.