Binary Division

We can perform all arithmetic operations such as addition, subtraction, multiplication, and division on binary numbers, in the same way as we perform arithmetic operations on the decimal number system. Binary subtraction, binary multiplication, binary addition, and binary division are the four types of arithmetic operations that are performed here. We just need to follow some rules while dividing two binary numbers. There are four rules to be followed while performing binary division. Similar to the decimal system (or in any other number system), division by 0 is meaningless in Binary division. The binary division rules are as follows:

The four rules given above are all the possible conditions for the division of binary numbers as binary numbers include only two digits 0 and 1.

Binary division problems can be solved by using the long division method, which is one of the most efficient and easiest ways to divide binary numbers. These are the steps to be followed in a binary division operation:

• Step 1: Compare the divisor with the dividend. If the divisor is larger, place 0 as the quotient, then bring the second bit of the dividend down. If the divisor is smaller, multiply it with 1 and the result becomes the subtrahend. Then, subtract the subtrahend from the minuend to get the remainder.
• Step 2: Then bring down the next number bit from the dividend portion and perform step 1 again.
• Step 3: Repeat the same process until the remainder becomes zero or the whole dividend is divided.

Let us understand binary division operation better using the following example:

Example: Consider two binary numbers, B = \(011010_{2}\) and C = \(0101_{2}\) where we want to divide B by C.

Given: Dividend, \(011010_{2}\) and the divisor,C = \(0101_{2}\).

Step1: Since the zero in the most significant bit position doesn't change the value of the number, let's remove it from both the dividend and divisor. So the dividend becomes \(11010_{2}\), and the divisor becomes \(101_{2}\).

Step 2: Let us use the long-division method. In this step, compare the divisor \(101_{2}\) with the first digit in the dividend \(11010_{2}\), since the divisor is smaller, it will be multiplied with 1 and the result will be the subtrahend.

As per the binary multiplication rules:

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

So, 101 × 1 = \(101_{2}\), and this result is written below.

Step 3: Subtract the subtrahend \(101_{2}\) from the minuend \(110_{2}\).

As per the binary subtraction rules,

• 0 - 1 = 1, we need to borrow 1 from the next more significant bit.
• 0 - 0 = 0
• 1 - 1 = 0
• 1 - 0 = 0

When we apply the above rules, this is how the calculation is done:

• For the first digit on the right, we have to subtract (0 - 1). So, we borrow a 1 from the digit on the left or the next higher order digit. Therefore, the result is 1.
• Then, (0 - 0 = 0) since the number in the next higher order digit becomes 0 after borrowing.
• 1 - 1 = 0 in the second next higher order digit.
• So, \(110_{2}\) - \(101_{2}\) = \(001_{2}\), and this result is written below.

Step 4: As per the rules of division, the next least significant bit comes down, and the divisor is multiplied by 1. Since the result, \(101_{2}\) is bigger than the minuend \(0011_{2}\), this step cannot be completed. Then, we have to go to the next step

Step 5: We write 0 as the next bit of the quotient and then, the least significant bit 0 comes down.

Step 6: Again the divisor is multiplied by 1 and the result is written as 101 × 1 = \(101_{2}\).

Step 7: Now we are at the final step. As per the binary subtraction, we subtract \(101_{2}\) from \(110_{2}\). We get, \(110_{2}\) - \(101_{2}\) = \(001_{2}\). The remainder is similar to Step 3, as all the numbers are the same.

The binary division operation is completed now and we get the following result.

• Quotient = \(101_{2}\)
• Remainder = 001 = 1

Related Topics

Check out these interesting articles to learn more about binary division and its related topics.

• Binary Calculator
• Binary Subtraction
• 12 in Binary
• 15 In Binary
• 100 In Binary
• 32 in Binary
• Decimals and Fractions

Important Notes

Here are some of the important points to be kept in mind regarding binary division.

• Binary division involves two other binary arithmetic operations - multiplication and subtraction.
• In order to perform a binary division, we follow the same process which we follow for dividing regular numbers. But, in this case, we only need to decide if it's going to be a 1 or a 0 and this makes calculation far easier.
• Binary division problems can be solved by using the long division method, which is one of the most efficient and easiest ways to solve binary division.
• The most common application of the binary number system can be found in computer technology.