Binary Multiplication Calculator - Multiply Binary Numbers with Step-by-Step Partial Products

The binary multiplication calculator outlines how to multiply binary numbers (which you can generate with the binary converter). In this article, you will also learn the similarities and differences between the binary and decimal numeral systems and see step-by-step instructions for the multiplication of binary numbers.

🔎 Go beyond multiplying. You can add, subtract, and divide these types of numbers using our other binary calculators.

How do I multiply binary numbers?

While the decimal numeral system, which we are all familiar with, is based on the powers of 10, the binary system has the base 2. This means that every digit of a binary number, a so-called bit, can only represent two logical values: 0 or 1. Therefore, binary numbers are commonly used in digital electronics and communications, representing the two states on and off.

We can convert binary numbers to the decimal system. They also allow the application of arithmetic operations, like addition, subtraction, division, and, as we will see in this binary calculator, multiplication. Binary numbers furthermore allow operations unique to the binary system, like bit shifts and the bitwise operations AND, OR, and XOR.

Those operations can also be executed with negative binary numbers, in which the first digit indicates the sign of the number. This calculator uses the signed representation, which means that the first bit of your input numbers can be considered a signed bit. Take care not to receive false results due to this, and increase the bit representation if needed.

What are the rules of binary multiplication?

Binary multiplication has 4 basic rules:

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 × 1 = 1

Based on those rules, binary multiplication is very similar to decimal long multiplication. We can even consider it slightly easier since we only have to deal with the digits 0 and 1.

Multiplication is a commutative operation, which means that the product does not depend on the order of factors. Nevertheless, it is recommended to set the longer number as the multiplier (factor 1) and the shorter number as the multiplicand (factor 2) to reduce the number of steps. Multiply the multiplier by each digit of the multiplicand to achieve intermediate products, whose last digit is in the position of the corresponding multiplicand digit. The final product is the sum of those intermediate products.

As an example, let us look at the multiplication of 1011 and 0101 (11 and 5 in the decimal system):

        1 0 1 1        (multiplier = 11)
      × 0 1 0 1        (multiplicand = 5)
      ───────────
        1 0 1 1        1011 × 1
      0 0 0 0          1011 × 0, shifted 1
    1 0 1 1            1011 × 1, shifted 2
  0 0 0 0              1011 × 0, shifted 3
  ─────────────
    0 1 1 0 1 1 1      (product = 55)

The step-by-step procedure for the multiplication of those binary numbers is:

1. Set the longer number as the multiplier. 1011 has four significant bits and is therefore set as factor 1.
2. Multiply the multiplier with the multiplicand's last digit: following the main rules above, 1011 × 1 = 1011. This is the first intermediate product, starting from the position of the multiplicand's last digit.
3. Multiply the multiplier with the multiplicand's second-to-last digit: 1011 × 0 = 0000. This is the second intermediate product, shifted one position to the left.
4. Multiply the multiplier with the multiplicand's third-to-last digit: 1011 × 1 = 1011, shifted two positions to the left.
5. Multiply the multiplier with the multiplicand's fourth-to-last (first) digit: 1011 × 0 = 0000, shifted three positions to the left.
6. The final product is the sum of the intermediate products: 1011 + 00000 + 101100 + 0000000 = 110111 (which is 55 in decimal, and 11 × 5 = 55 ✓).

How to use the binary multiplication calculator?

You now know how to perform the multiplication of binary numbers, so let's learn to use the binary multiplication calculator. As an example, let's investigate the correctness of the step-by-step procedure above and multiply 1011 and 101:

1. Input your factors. Since the multiplication operation is commutative, the order does not matter — enter 1011 and 101 (only 0s and 1s, up to 32 bits each).
2. Click "Multiply Binary Numbers" to see the result.
3. The binary multiplication calculator presents your product in both the binary and decimal system, along with a step-by-step table of the partial products.

These are the results of the multiplication of those binary numbers:

Binary:  0011 0111
Decimal: 55

In case your binary result has a value of 1 on the most significant bit, it could be understood as a positive result in unsigned notation or a negative result in signed notation. When this happens, our calculator displays both interpretations so you can pick the one that matches your context.

FAQs

What are the rules of binary multiplication?

There are four rules: 0 × 0 = 0, 0 × 1 = 0, 1 × 0 = 0, and 1 × 1 = 1. Because a bit can only be 0 or 1, multiplying by a bit either copies the multiplier (when the bit is 1) or produces a row of zeros (when the bit is 0).

How is binary multiplication similar to decimal multiplication?

Both use long multiplication: you multiply the top number by each digit of the bottom number, shift each partial product one place further left, and add all partial products together. Binary is actually simpler because every digit is 0 or 1, so each partial product is either the multiplier itself or all zeros — there is no multiplication table to memorise.

Why should the longer number be the multiplier?

Multiplication is commutative, so the order of factors does not change the product. However, the number of partial products equals the number of bits in the multiplicand. Choosing the shorter number as the multiplicand keeps the number of rows — and therefore the number of additions — to a minimum.

Can binary multiplication overflow?

Yes. Multiplying an m-bit number by an n-bit number can produce a result of up to m + n bits. In a fixed-width computer register this would overflow, but our calculator always shows the full, untruncated product. If you need signed results, increase the bit representation to leave room for the sign bit.

How do I convert the binary product back to decimal?

Multiply each bit by 2 raised to its position (counting from 0 on the right) and add the results. For example, 110111 = 1×32 + 1×16 + 0×8 + 1×4 + 1×2 + 1×1 = 55. Our calculator shows the decimal equivalent of every number automatically for quick verification.