Binary Division

Introduction

The binary division is a method to divide one binary number from another binary number. Binary numbers can undergo the four major arithmetic operations addition, division, subtraction, and multiplication-like numbers.

The binary division is similar to the division of decimal numbers, while in the case of decimal numbers, there were 0-9, 10 digits. Here, in the case of binary numbers, there are only two digits 0 and 1.The long division method is one of the significant methods used in binary division.

Definition

Binary Numbers:

• The word ‘bi’ means two so, it is a number system that is used to represent numbers with a base number of 2.

• The numbers in the binary number system are represented using two digits or called bits in computer applications 0 and 1. They are represented using subscript 2.

• Each bit is two times more significant than the bit from the right.

• The bit in ten’s place is two times more than the bit in unit’s place. Similarly, the bit in the hundred’s place is two times more than the bit in the ten’s place.

• Example − $\mathrm{(110)_{2}\:,\:(1010)_{2}}$, etc… are a few numbers in the binary number system.

• The one at the utmost right is the least significant bit and the one at the utmost left is the most significant bit.

How to write a number in a binary number system?

• To write a number in a binary system divide the number by 2 from which a quotient and remainder are obtained.

• Now, divide the quotient again until it’s zero and note down all the remainders in each division at last write all the remainders from the last obtained remainder to the first remainder.

Example

Let’s see converting a number say 5 into a binary number.

• First, divide the number by 2 the remainder is 1 and the quotient is 2.

• Now, divide the quotient again by 2 the remainder is 0 and the quotient is 1.

• Now, divide the quotient again by 2 the remainder is 1 and the quotient is 0.

• The quotient is zero so no need to divide again now write all the remainders from the last obtained remainder to the first remainder which is 101.

• Therefore, $\mathrm{5\:=\:(101)_{2}}$

Binary Division

• Binary division is a method to divide one binary number from another binary number. The one divided is called a dividend and the one that divides is called the divisor.

• Binary division can be done either by the long division method or by converting the binary numbers into decimal numbers and dividing them and converting the resulted decimal number into a binary number.

The Long Division Method

• The first step is to compare the divisor and dividend if the divisor is larger than the dividend writes 0 in the quotient and bring the second bit of the dividend down.

• If the divisor is smaller than the dividend writes 1 in the quotient and multiply the divisor with 1 it becomes the subtrahend, then subtract it from the minuend of the dividend to get the remainder.

• Bring the next bit down and continue the process until the dividend is divided.

Rules

• There were four rules followed in binary division similar to the division of decimal numbers when the divisor is zero it’s meaningless.

• In the case of binary numbers, there are only two digits 0 and 1.

• Hence there were only four possibilities in binary subtractions which are

Dividend	Divisor	Result
0	1	0
1	0	Meaningless
0	0	Meaningless
1	1	1

Comparison with Decimal Values

Converting a binary number into a decimal number:

• A given binary number with 𝑛 bits can be converted into a decimal number by multiplying the most significant bit or leftmost (MSB) with $\mathrm{2^{n\:-\:1}}$ and reducing the power of 2 by 1 with each bit moving right so that the least significant bit or rightmost (LSB) is multiplied with the power of 20, and adding all these products gives the decimal number.

• If the binary number has 𝑘 bits then its decimal number is

$$\mathrm{=\:MSB\times\:2^{k\:-\:1}\:+\:(k\:-\:1)^{th}bit\:\times\:2^{k\:-\:2}\:+\:.....\:+\:LSB\times\:2^{0}}$$

Two binary numbers can be divided by converting them into decimal numbers and dividing the decimal numbers and converting the result into a binary number.

Solved Examples

1) $\mathrm{110010\:\div\:101}$?

$$\mathrm{\:\:\:\:\:\:\:1010\\\:101)\overline{110010}\:\:\:\\\:\:\:\:\:\:\:\:\:\underline{101}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:10\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:\underline{00}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:101\:\:\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:\underline{101}\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:00\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:\underline{00}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\\\\\:\:\:\:\:\:\:\:\:\:0}$$

• The divisor 101 is smaller than the dividend so, 1 in the quotient and multiply the divisor with 1 it becomes 101 the subtrahend then subtracts it from the minuend 110 of the dividend to get the remainder 1.

• Bring the next 0 bit down now the divisor 101 is larger than the dividend 10 so, write 0 in the quotient and bring the next bit 1 on the dividend down.

• Now, both are equal. So, multiply by 1 and the remainder is zero.

• Now, bring the last 0 bit down and multiply the divisor with zero the remainder is zero.

• All bits in the dividend are done and the remainder is zero the quotient is 1010.

Conclusion

In this tutorial, we learned about binary systems, binary division, converting the decimal numbers into binary numbers, the long division method, rules of binary division, comparison with decimal values, and converting the binary numbers into decimal numbers, and a few solved examples.

FAQs

1. What is the process of binary division other than the long division method?

Other than the long division method one can convert the dividend and the divisor into decimal numbers and divide them and convert the result into a binary number.

2. What are the four rules of binary division?

The four rules of binary division are −

$$\mathrm{1\div\:0\:=\:meaningless\:,\:1\div\:1\:=\:1\:,\:0\div\:0\:=\:Meaningless\:,\:0\div\:1\:=\:0}$$

3. How to convert a decimal number into a binary number?

To convert a decimal number into a binary number divide the number by 2 from which a quotient and remainder are obtained. Now, divide the quotient again until it’s zero and note down all the remainders in each division at last write all the remainders from the previously obtained remainder to the first remainder. It is the binary representation of the number.

4. What is the least significant bit and most significant bits in a binary number?

The one at the utmost right is the least significant bit, and the one at the utmost left is the most significant bit.

5. What is the formula for converting a binary number into a decimal number?

If the binary number has 𝑘 bits, then its decimal number is

$$\mathrm{=\:MSB\times\:2^{k\:-\:1}\:+\:(k\:-\:1)^{th}bit\:\times\:2^{k\:-\:2}\:+\:.....\:+\:LSB\times\:2^{0}}$$