The study of binary number system is very much important from the viewpoint of understanding how data are represented before they are processed by any digital system including a digital computer. Binary number system got many applications in digital world.

In the decimal number system, ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are required to express a number. The base or radix of a number system is the total number of digits used in the system. Clearly, the decimal number has the base ten. Well, the binary number system is like another system, it uses the exact methodology but its based on the number 2. In this section we will be learning more about binary numbers and its operations.

In the binary system, a number is expressed by two digits, 0 and 1. The base of the binary system is thus two. In this system, the individual digits, 0 and 1 represent the coefficient of power of 2.
For example, the decimal number 6 is written as

$6 = 4+2 = 1 \times 2^{2}+1 \times 2^{1}+ 0 \times 2^{0}$
The binary representation of the decimal number 6 is thus 110(read as one one zero not one hundred ten).
Decimal number is equivalent to binary number which is represented as below. The number 0 and 1 are common in both binary and decimal number system.

Binary number 001 010011100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000
Decimal number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
When one add two binary numbers together, one can use the same methodology that is used in the decimal number system, but using 2 as a base. In the decimal system $1+1 = 2$, but in binary system $1+1 =10$ (remember that the binary number 10 represents the same quantity as the decimal number 2).

The binary addition is performed with the help of following four rules:
  1. $0+0 = 0$
  2. $1+0 = 1$
  3. $0+1 = 1$
  4. $1+1 = 10$, implying that one added to one gives 2, the decimal equivalent of $10$ being $2$.
Some binary addition examples are shown below:
Example 1:
10 +
11
101

Example 2:

10001 +
111
11000


Addition of negative numbers are shown below

-2 +
-19
-21

Example 3:
-10 +
-10011
-10101
Binary subtraction is based on the following four rules
  1. $0-0 = 0$
  2. $1-0 = 1$
  3. $1-1 = 0$
  4. $10-1 = 1$
Example for binary substation is given below:

Example 1:
1011
1001
0010

Example 2:

110
11
011

Addition and subtraction of binary numbers is done column wise.
Multiplication of binary numbers obeys the following four rules:
  1. $0 \times 0 = 0$
  2. $1 \times 0 = 0$
  3. $0 \times 1 = 0$
  4. $1 \times 1 = 1$
Binary multiplication of two large numbers consisting of several digits is performed in a manner similar to decimal multiplication. As an example let us multiply a binary number 10111 by the binary number 110 as follows.

10111
110
00000
10111
10111
10001010
Division of binary number is also carried out along the same line as the division of decimal number.

In below example, the binary number 10100 (twenty) is divided by 100 (four)

100)10100 (101
100
100
100
000

The quotient is 101 (five)
A binary number can be converted into its decimal equivalent by noting that the successive digits from the extreme right of a binary number are the coefficients of ascending power of 2, beginning with the zeroth power of 2 at the extreme right. For example the binary number 10111 is written as

$10111 = 1 \times 2^{4}+0 \times 2^{3}+1 \times 2^{2}+1 \times 2^{1}+1 \times 2^{0}$
= $16 +0+4+2+1$ = $23$

Thus the decimal equivalent of the binary number 10111 is 23.similarly,

$111$ = $1 \times 2^{2}+1 \times 2^{1}+1 \times 2^{0}$ = $4+2+1$ = $7$

$1111$ = $1 \times 2^{3}+1 \times 2^{2}+1 \times 2^{1}+1 \times 2^{0}$ = $8+4+2+1$ = $15$

The decimal equivalent of a binary fraction is found by multiplying each digit in the fraction successively by $2^{-1},2^{-2},2^{-3}...$ etc. beginning with first digit after the binary point. The decimal equivalent of binary digit $0.110$ is

$1 \times 2^{-1}+1 \times 2^{-2}+0 \times 2^{-3}$ = $\frac{1}{2}$ + $\frac{1}{4}$ + $0$ = $0.75$

The decimal equivalent of binary number 1111.110 is 15.75, here 1111 has decimal equivalent of 15 and the fraction 0.110 has the decimal equivalent 0.75