In this article, the binary multiplication of unsigned binary numbers is explained using the examples. And you will also learn how to multiply two fractional binary numbers.
Understanding Binary Multiplication:
The binary multiplication is similar to the conventional decimal multiplication. In Binary Multiplication, the each digital of the second number is multiplied with each digit of the first number. The first number in the multiplication is known as the “Multiplicand” and the second number is called as “Multiplier”.
In binary multiplication, each digit of the multiplier is multiplied with the multiplicand. Each multiplication gives the partial products. The partial products are adjusted according to their weights. And the summation of the partial products gives us the final result of the multiplication.
Since the binary numbers involves only two digits (1s and 0s), the binary multiplication process in much simpler than decimal multiplication.
In binary multiplication,
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1
Using these simple rules of binary multiplication, it is possible to multiply two numbers.
Let’s take one example and through that let’s understand the process of binary multiplication.
Example 1
Let’s say, we want to multiply 6 (110) and 5 (101) in binary.
To perform the binary multiplication, starting from the LSB of the multiplier, we will multiply each digit to the multiplicand.
So, if we start from 1 then first, we will multiply it with 0, then 1 and at the last with this MSB (1).
Now, when we are multiplying the binary digits, then actually we are also multiplying their weights. And according to their weights, we will place the result in a particular column.
So, here first let’s multiply this 1 with 0. So, here the weight of both 0 and 1 is 20. So, let’s multiply their weights separately.
So, as you know 1 x 0 is 0. And this 20 x 20 is also 20. That means we will place the result in 20 column.
Similarly, when we multiply 1 with 1 then actually it is (1 x 1) ( 20 x 21 ) = 1 x 21 .
Therefore, the result will be placed in 21 column.
Likewise, when we multiply 1 (the LSB of multiplier) with 1 (MSB of multiplicand), then actually it is (1 x 1) (20 x 22) = 1 x 22 . Therefore, the result will be placed in 22 column.
The result of the multiplication is known as the partial product. The first partial product is shown below.
Similarly, the second digit of the multiplier will be multiplied with the each digit of the multiplicand. And the result will be placed in appropriate columns according to their weights. The same is shown below.
And finally, the MSB of the multiplier will be multiplied with the each digit of the multiplicand. And once again the result will be placed in an appropriate columns according to their weights. The same is shown blow.
In this way, in the multiplication process, we got three partial products. As you can observe, the value of partial product is either 0 or, its value is same as the multiplicand. If the digit of the multiplier is 0 then partial product is 0 and if the digit of the multiplier is 1 then partial product is same as the multiplicand.
Also, as you can observe, as we move from LSB to MSB of the multiplier, each partial product is left shifted by 1 bit position. So, once we get all our partial products then final step is to add all the partial products to get the final result of the multiplication.
Once we add all the partial products, then result is 11110. Its decimal equivalent is 30. And in this way, we can perform the binary multiplication of two unsigned binary numbers. Let’s take another example to understand the multiplication procedure.
Example 2
Let’s say, we want to multiply 12 (1100) and 13 (1101). So, here 12 is multiplicand and 13 is a multiplier. Starting of the LSB of the multiplier, let’s multiply each digit with the multiplicand.
Since the first digit is of the multiplier is 1, the partial product is same as the multiplicand. If we move to the next digit of the multiplier, then it is 0. Since it is 0, the partial product will be 0. But it will be left shifted by 1 bit position.
Similarly, the next two bits are 1. So, partial product will be same as the multiplicand. But each partial product will be left shifted by 1 bit position. The same is shown below.
The final step is to add all the partial products to get the result of the multiplication. If we add all the partial products then result is 10011100.
In this way, it is possible to multiply two unsigned binary numbers. Similarly, now let’s see the multiplication procedure for fractional binary numbers.
Multiplication of Fractional Binary Numbers
The multiplication of fractional binary number is similar to the binary integers. Let’s take one example and through that, let’s understand the multiplication procedure. Let’s say, we want to multiply 6.25 and 3.5.
In the binary, two numbers are 110.01 and 11.10. In the first number (110.01), the number of digits after the binary point are 2. Similarly, in the second number (11.10), the number of digits after the binary point are 1.
First of all, let’s write the two numbers without their binary points. And let’s multiply the two numbers like a binary integers. For the multiplication, align the two numbers from the LSB positions.
After multiplying the two numbers, the result will be 10101111.
Now, we just need to put the binary point at the appropriate location. In the first number 6.25 (110.01) the number of digits after the binary point are 2. And similarly, in the second number 3.5 (11.1), the number of digits after the binary point are 1. The total number of digits are 3 (2 +1). Therefore, in the obtained result, put the binary point after the 3 digits from the LSB.
And therefore, the result after the multiplication is 10101. 111. If we see its binary equivalent then it is equal to 21.875. In this way, we can multiply two fractional binary numbers.