Binary Division

A binary number system represents the number with the base 2, it uses the digits 1 and 0. As it uses only two digits 0 and 1 and has a base of 2, it is called binary. The way we perform arithmetic operations on the decimal number system, similarly we can perform all arithmetic operations on binary numbers. Operations such as addition, subtraction, multiplication, and division.

What is a Binary Division?

The binary division is an important part of binary arithmetic. The binary division is similar to that of a decimal division operation. In this article, we will study step-by-step methods to make binary division understand as much as possible. Long division is one of the easiest and most efficient ways to solve binary division.

Rules of Binary Division

Simplifying binary division is almost as easy as multiplying binary numbers, and involves our knowledge of binary multiplication. Just we have to take note of some rules while dividing two binary numbers. There are  four rules associated with binary division. The binary division rules are as follows.

• 1÷1 = 1

• 1÷0 = 0

• 0÷1 = Meaningless

• 0÷0 = Meaningless

As binary numbers include only two digits i.e. 0 and 1, these four rules are all the possible conditions for the division of binary numbers.

Here is the stepwise procedure of how to divide two binary numbers.

Four Steps to Binary Division

Likewise, decimal division binary division also carries out four steps for the division of numbers.

• Division: First take the leftmost digit of the dividend, we attempt to divide it by the divisor which must be smaller than the dividend digits. This results in a quotient.

• Multiplication: Once we have found the quotient we use it to multiply the divisor to obtain a product.

• Subtraction: Having calculated the product in the previous step, we subtract that from the working dividend to calculate a remainder.

• Bring Down: The final step is then to bring down the next digit in our original dividend, combine it with the remainder in the previous step and form a new working dividend. At this point, the process is repeated.

Let’s look some examples of applying this process for binary division,

Binary Division Examples

Example: Divide 01111100 ÷ 0010

Solution:

Here the dividend is 01111100 and the divisor is 0010

The zero’s in the Most Significant Bit  in both the dividend and divisor doesn’t change the value of the number. So remove the zero’s.

So the dividend becomes 1111100 and the divisor becomes 10.

Now, use the long division method.

10) 1 1 1 1 1 0 0 ( 1 1 1 1 1 0

    - 1 0

____________

        1 1

        1 0

     _________

          1 1

         -1 0

        ___________

             1 1

              -1 0

  ___________

                   1 1

                  -1 0

                ____________

          0 0

          0 0

                 _______________

          0 0 

• Step 1: First, compare the first two numbers in the dividend with the divisor. Add the number 1 in the quotient place. Multiply, write it under the dividend and then subtract the value, you get 1 as remainder.

• Step 2: Then bring down the next number from the dividend portion, now you have remainder and the value from dividend now do the step 1 process again

• Step 3: Repeat the process until the remainder becomes zero.

• Step 4: After you get the remainder value as 0, you have zero left in the dividend portion, so bring that zero to the quotient portion.

Therefore, the resultant value is quotient value which is equal to 111110

So, 01111100 ÷ 0010 = 111110

Solved Example

Example 1: 101 Divide By 10

10) 1 0 1 ( 1

  -   1 0

________

      0 0 1

Example 2: Divide 11010 By 101

101 ) 1 1 0 1 0 ( 101

              1 0 1

    ________

          0 0 1 1 0

          -     1 0 1

          _________

                0  0 1

Quiz Time

• Divide 111111 by 11

• Divide 10001 by 10

Key Notes Regarding Binary Division

Some of the crucial points to be kept in mind about the process of binary division are:

• First of all, binary division involves the application of two other arithmetic operations - multiplication and subtraction.

• Secondly, in order to perform a bithe nary division, a student is expected to follow the same procedure with which we divide regular numbers. The only difference is , in case of binary division, we need to decide if it's going to be a 1 or a 0 as placeholder in the quotient and this makes calculation quite easy.

• Lastly, mathematical problems in binary division can be simplified with the help of the long division method, which is easy and one of the most efficient ways to solve in binary division.

Today, this method finds its application in the field of computer technology.

Important Point: Binary division is a process also known as the long division method, which is used to find the resultant in an easy way. Hence it is used in place of other division methods (e.g. double division method)

How to Interpret Decimal Results in Binary Division

During the division process, when the quotient obtained is a non-integer and the division process is extends beyond the decimal point, one of two scenario is likely:

• In the first case, the division process terminates , which means that a remainder of 0 is obtained ultimately; or

• In the second case, a remainder is reached that is identical to a previous remainder or non-identical to the previous remainder (digit which has occurred after the decimal points were written). In the latter case, further going on with the process will not be productive, because from that point onward the same sequence of digits would reappear in the quotient over and over again. To deal with such cases a mathematical notation called the bar is used. In this, a bar is drawn over the repeating sequence to indicate that it repeats forever (i.e., every rational number is either a terminating or repeating decimal).