Binary Division

Introduction to the Binary Number System

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).

FAQs on Binary Division

1. What is Binary Addition?

Binary addition is the addition of binary numbers. Binary addition islike addition in the decimal number system the difference is only of the base. Addition of binary numbers is carried out with the binary addition rules.


Binary Addition Rules

Addition of two binary numbers is as easy as the addition of a decimal number system. Just we have to understand some rules while adding two binary numbers. There are  four rules associated with binary addition. The binary addition rules are as follows.

  • 0 + 0 = 0

  • 0 + 1 = 1

  • 1 + 0 = 1

  • 1 + 1 =10 

2. How to Convert Binary Numbers to Decimal Number Systems?

To convert a binary number (e.g. 101101) to decimal number system we can use the following steps:

  • Using the positional method of conversion, each number is given an exponential value of the place value starting from zero (right to left) to the base 2. For the given example, this can be denoted as 25 24 23 22 21 20

  • Now we multiply each digit in the binary number from its corresponding base value of 2. This will result in: 25 x 1= 32

24 x 0= 0

23 x 1= 8

22 x 1= 4

21 x 0= 0

20 x 1= 1

  • In the end, we sum up all the values. The end result is 45 (10) for the binary number 101101.

3. What is a polynomial expression? Can long division be used to simplify a polynomial?

A polynomial can be defined as an algebraic expression that consists of indeterminates and their coefficients. The terms and their variables may be mathematically in relation such as through multiplication, division, addition, or subtraction. Polynomials have a non-negative integer as their variable exponent.


Yes, the long division method is absolutely useful in simplifying the polynomial and one polynomial can be divided from other polynomials in the same manner as numbers in a conventional binary division.

4. What is the difference between the binary division (long division) method and the double division method?

Both binary division and double division are division methods taught to students to simplify the division in mathematics. The main difference is that double division does not depend upon memorizing multiple tables. It also takes the whole dividend into consideration instead of going digit by a digit (as in the case of binary division). However, for the initial few divisions, students experience fatigue as it is slightly time-consuming.

5. What are some of the common errors that are committed during the binary division method?

During the process of the binary division method, students often commit the following errors:

  • Often students forget to use zero as a placeholder, which simply exhibits error in understanding the long division method.

  • Sometimes they place numbers in the wrong place value.

  • In some cases, the pupils are unable to recall tables of a number (which is due to poor revision) and can be corrected by going through the tables every once in a while. This helps important mathematical tools to remain in fresh memory.

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