#### Number system

**Base 2: Binary numbering system**

1)Only have 2 values

2)Work on binary logic(0,1)

**Base 8:Octal numbering system**

1)Ranges from 0-7 total 8

2)Represented by 3 bits by using 2^n while n=3 so 2^3=8.

**Base 10:Decimal numbering system**

1)Ranges from 0-9

2)Most commonly used in daily life.

**Base 16:Hexadecimal numbering system**

1)Ranges from 0-15 Where 10=A,11=B and so on till 15=F

2)Represented by 4 bits.

### 653

The number given above 6 is the most significant number and 3 is the least significant number based on radix of system.

#### Conversion

Decimal Binary

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

#### conversion system

### Arithmetic operations

#### addition

Add (365)8 and(234)8

1 1

(3 6 5)8

(2 3 4)8

————-

(6 2 1)8

————-

Note:Addition process first we add 5 and 4 which is 9 but 9 is not present in octal number so subtract base 8 from 9 and forward a carry to next value and so on.

For binary it is following

1 1 1 1 1

(1 0 1 1 0 1)2

(1 1 0 0 1 1)2

(1 1 1 0 0 0 0 0)2

#### Subtraction

Subtraction is different from addition based on compliment technique.

10’s Compliment: Lets understand this concept by an example

7 2 8 4

0 3 9 2

————

————

so its 10’s compliment is subtracting 0392 by 10000 because first value is always 1 bit more than second one.

1 0 0 0 0

0 3 9 2

————-

9 6 0 8 is 10’s compliment

————

now add 7284 and 9608 to get answer

9’s Compliment same as above for above example

9 9 9 9

-0 3 9 2

————

9 6 0 7 is 9’s compliment

————-

now add 7284 with 9607.

1’s Compliment of 10101 is 01010 means invert numbers.

2’s Compliment add 1 in 1’s compliment.

A=11001,B=10010 Find A-B

2’s compliment of B is 01110

1 1 0 0 1

0 1 1 1 0

————-

ignore<–1 0 0 1 1 1

—————————

#### Binary codes

**1)Alpha-Numeric Code**

They can be alphabets and numerical there are total 128 alpha-numeric characters from which 94 are printable and 34 are unprintable.

**2)Decimal Code**

a)BCD: stands for Binary Coded Decimal where decimal numbers are represented by fixed number of bits.

b)Excess 3 Code it means the binary list started of decimal numbers starts from 3 to 12 so unused states are 1,2,3,13,14,15.

**3)Error Detection Code**

To find error we used even and odd parity they helped in error detection for this we use additional bits.

Example: for even parity we require 0’s in even quantity and for odd parity 0’s in odd quantity.

**4)Grey Code**

Is the ordering of the binary number system such that two successive values differ in only one bit. They are designed to prevent the sudden output from electric switches.

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