Binary Multiplication Calculator

What the binary multiplication calculator does

This multiplies two binary numbers and shows the product in both binary and decimal. Enter the two numbers, press the button, and read off the result.

Binary multiplication uses the same long multiplication layout you already know, but there is a pleasant surprise waiting in it: the times tables vanish completely, which makes the whole thing simpler than its decimal cousin.

How to use it

1. Choose a bit size. This sets how wide your numbers are, with 8 as the default. A product can be wide, so allow some room.
2. Enter the two binary numbers. Type each one using only 0s and 1s.
3. Press Calculate. The product appears as both a binary result and a decimal result.

Press Reset to clear the inputs.

Why binary multiplication is easier than decimal

In decimal long multiplication, you multiply by each digit of the bottom number, and that means knowing your times tables up to nine. In binary, every digit of the bottom number is either 0 or 1, so each row of the working can only be one of two things. Multiplying by 0 gives a row of zeros. Multiplying by 1 gives back the top number unchanged. There is nothing in between, so you never multiply at all, you just copy or skip, shift each row left by its position, and add the rows together. Binary multiplication is really shift and add wearing a multiplication hat.

A worked example with real numbers

Let us multiply 101 by 11. We take each digit of 11, the bottom number, in turn.

• The rightmost digit of 11 is 1. Multiplying 101 by 1 gives 101.
• The next digit of 11 is also 1. Multiplying 101 by 1 gives 101 again, but because this digit sits one place to the left, the row shifts one place over to become 1010.
• Now add the two rows: 101 + 1010 = 1111.

So the product is 1111. To check it, 101 is 5 and 11 is 3, and 5 times 3 is 15, which is exactly what 1111 comes to. Every row was just a copy of 101, shifted and added, with no multiplying needed.

How wide the answer gets

Products grow quickly, and it helps to know roughly how wide to expect. Multiplying a number of one width by a number of another gives a result whose width is about the two added together, so a 4-bit number times a 4-bit number can reach 8 bits. This is the same reason a three-digit number times a three-digit number can run to six digits in decimal. When you are working with the calculator, choose a bit size with enough room for the product to land. The decimal result always shows the true value alongside.

Questions people ask

How does binary multiplication work?

You use long multiplication, taking each digit of the bottom number in turn. Since each digit is 0 or 1, every row is either a row of zeros or a copy of the top number, shifted to its place, and you add the rows together.

Do I need times tables for binary?

No, and that is the nice part. Multiplying by 0 gives zeros and multiplying by 1 gives the number back, so there is nothing to memorize. The work is just shifting and adding.

Why is the product so much longer than the numbers?

Because multiplying two numbers gives a result whose width is roughly their widths added together. A 4-bit times a 4-bit number can reach 8 bits, just as multiplying two three-digit numbers can give six digits.

Why does each row shift to the left?

Because each digit of the bottom number stands for a higher place value as you move left, so its row is worth more and lines up one column further over, exactly as in decimal long multiplication.

References

1. Weisstein, Eric W. "Binary." From MathWorld, A Wolfram Resource. https://mathworld.wolfram.com/Binary.html