Binary Multiplication Calculator
Multiply two binary numbers and see every partial product.
Updated
Binary Multiplication Calculator
Our Binary Multiplication Calculator multiplies two binary numbers and shows the product in binary, decimal and hexadecimal. It lists every partial product with the running total, so you can follow the long multiplication step by step, and it tells you whether the product fits in the bit width you choose or overflows.
Quick answer
Multiply binary numbers like decimal long multiplication: for each 1 in the second number, write the first number shifted left to that position, then add the rows. Each 0 adds nothing. For example, 110 × 101 = 110 + 11000 = 11110, which is 6 × 5 = 30. Enter any two binary numbers above and the calculator shows the product and every step instantly.
- Rules: 0 × 0 = 0 × 1 = 1 × 0 = 0, and 1 × 1 = 1
- Times 2: add a 0 on the right (a left shift), so 1011 × 10 = 10110
- Size: an n-bit × m-bit product needs at most n + m bits
- Overflow: in 8 bits, 11111111 × 10 = 111111110 needs 9 bits, so 11111110 is kept
Binary Multiplication Rules
| Bits | Product |
|---|---|
| 0 × 0 | 0 |
| 0 × 1 | 0 |
| 1 × 0 | 0 |
| 1 × 1 | 1 |
Single-bit multiplication is the same as the AND operation: the result is 1 only when both bits are 1. Unlike addition, a single column never produces a carry; the carries come later, when the partial products are added.
Binary Long Multiplication
Write the first number (the multiplicand) on top and the second (the multiplier) below. Go through the multiplier’s bits from right to left. For a 1, copy the multiplicand, shifted left by the bit’s position; for a 0, write a row of 0s. Then add all the rows with binary addition.
× 1 0 1 (5)
-------
1 1 0 bit 0 is 1: copy 110
0 0 0 bit 1 is 0: zeros, shifted 1 place
1 1 0 bit 2 is 1: copy 110, shifted 2 places
---------
1 1 1 1 0 (30)
Example Calculations
| Binary | Decimal |
|---|---|
| 110 × 101 = 11110 | 6 × 5 = 30 |
| 101 × 11 = 1111 | 5 × 3 = 15 |
| 111 × 111 = 110001 | 7 × 7 = 49 |
| 1101 × 1011 = 10001111 | 13 × 11 = 143 |
| 1111 × 1111 = 11100001 | 15 × 15 = 225 |
| 1100 × 100 = 110000 | 12 × 4 = 48 |
Multiplying by Powers of Two
Multiplying by 10 (2) in binary works like multiplying by 10 in decimal: add a 0 on the right. So 1011 × 10 = 10110 (11 × 2 = 22), 1011 × 100 = 101100 (× 4) and 1011 × 1000 = 1011000 (× 8). Computers do this with a left shift, which is much faster than a full multiplication.
Overflow and Product Size
A product can need up to as many bits as both factors together: 4 bits × 4 bits fits in 8 bits, and 32 bits × 32 bits fits in 64. If the register is smaller than that, the top bits are lost. In 8 bits, 11111111 × 10 = 111111110 (255 × 2 = 510) needs 9 bits, so the register keeps 11111110 (254). The calculator shows the true product, the wrapped result and an overflow flag for the width you pick.
Key Terms
- Multiplicand
- The number being multiplied (Factor A).
- Multiplier
- The number you multiply by (Factor B); its bits decide which rows are added.
- Partial product
- One row of long multiplication: the multiplicand shifted into place, or zeros.
- Product
- The result of the multiplication.
- Overflow
- When the product needs more bits than the register has and the top bits are lost.
How to Use the Binary Multiplication Calculator
- Enter the first factor. Type the first binary number in the “Factor A (binary)” box, for example 110. Spaces, underscores and a 0b prefix are ignored. Pressing Enter moves you to the next box.
- Enter the second factor. Type the second binary number in the “Factor B (binary)” box, for example 101.
- Set the bit width. Use the 8, 16, 32 or 64 chips, or type any width from 1 to 64. The width is the size of the register the product has to fit in, and decides whether the multiplication overflows.
- Read the product. The product appears as soon as both numbers are filled, with the working in decimal underneath (for example 110 × 101 = 11110, 6 × 5 = 30). Below it, a table shows each partial product and the running total, and a second table gives the product in binary, decimal and hexadecimal, the result wrapped to the bit width and whether it overflowed.
- Copy the result. Press Copy to copy the binary product. Press Clear or Esc to start again.
Example: enter 1101 and 1011 and the product is 10001111 (13 × 11 = 143), built from three partial products.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you multiply binary numbers?
Use long multiplication. For each bit of the second number, starting from the right, write down the first number if the bit is 1 or a row of 0s if it is 0, shifting each new row one place further left. Then add all the rows in binary. For example, 110 × 101 = 110 + 11000 = 11110, which is 6 × 5 = 30.
What are the rules of binary multiplication?
There are only four: 0 × 0 = 0, 0 × 1 = 0, 1 × 0 = 0 and 1 × 1 = 1. They are the same as the AND operation. The hard part of binary multiplication is the addition of the partial products, not the multiplying.
Why is binary multiplication easier than decimal?
Each partial product is either the first number (shifted) or zero, so there are no times tables to remember. You just copy the first number for every 1 in the second number and add the copies.
How do you multiply a binary number by 2?
Add a 0 on the right, which shifts every bit one place left: 1011 × 10 = 10110 (11 × 2 = 22). Multiplying by 100 (4) adds two 0s, by 1000 (8) adds three, and so on. The Bit Shift Calculator does this as a left shift.
How many bits does a binary product need?
At most the two lengths added together. A 4-bit number times a 4-bit number fits in 8 bits: even the largest, 1111 × 1111 = 11100001 (15 × 15 = 225), is 8 bits long. That is why processors often store the product of two 32-bit numbers in 64 bits.
What is overflow in binary multiplication?
Overflow happens when the product needs more bits than the register holds, so the top bits are lost. In 8 bits, 11111111 × 10 = 111111110 (255 × 2 = 510) needs 9 bits, and the register keeps only 11111110 (254). The calculator shows the true product, the wrapped result and whether it overflowed.
How do computers multiply binary numbers?
The simplest hardware uses shift-and-add, exactly like the table this calculator shows: for each 1 bit of one number, add the other number shifted into place. Faster chips add many partial products at once with array or Wallace-tree multipliers, and Booth’s algorithm is used to handle signed numbers and reduce the number of additions.
How do you check a binary multiplication?
Convert to decimal: 110 = 6, 101 = 5, 6 × 5 = 30, and 30 is 11110 in binary. Or divide the product by one factor with the Binary Division Calculator: 11110 ÷ 101 = 110 with no remainder.
Is the Binary Multiplication Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Binary Multiplication Calculator to multiply any two binary numbers instantly.