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Data Representation and Arithmetic

Objectives


Understand why computers use binary (Base-2) numbering.
Understand how to convert Base-2 numbers to Base-10 or Base-8.
Understand how to convert Base-8 numbers to Base-10 or Base 2.
Understand how to convert Base-16 numbers to Base-10, Base 2 or Base-8.

Why Binary System?


Computers are made of a series of switches
Each switch has two states:  ON or OFF
Each state can be represented by a number – 1 for “ON” and 0 for “OFF”

Number System


The position  of every different digit represents a different multiple of base.
For every general number, we have to multiply each of digit by some power of base(B).
Largest value of digit is always one less than base.


Decimal Number System :

Base 10, digit used : 0 to 9.

Octal Number System : 

Base 8, digit used : 0 to 7.

Hexa Decimal Number System :

Base 16, digit used : 0 to 9 , Letters used : A to F.

Binary Number System :

Base 2, digit used : 0,1.

Concept of MSB & LSB


MSD -(MOST SIGNIFICANT DIGIT) Leftmost digit having the highest weight is called as MSD.
LSD - -(LEAST SIGNIFICANT DIGIT) Rightmost digit having the lowest weight is called as LSD.
In binary we called as MSB (MOST SIGNIFICANT BIT) and LSB -(LEAST SIGNIFICANT BIT).


Decimal Number System


The no. system we have learning from school is called decimal number system.
General counting and mathematics use this system.
It uses base of 10
Largest value 9
How we represent decimal no. e.g. (50)10


Binary Number System


The modern computer systems uses the binary logic for their operation.
The number system which uses only two digits  viz. 0 and 1.
It uses base of 2 hence largest value is 1.
How we represent binary no. e.g. (1010)2


Octal Number System


Base for octal number system is 8
Hence the largest value of digit is 7 
The numbers in octal system are like this
     1,2,3,4,5,6,7,10,11,12,13….
     How we represent octal no. e.g. (141)8


Hexadecimal Number System


 A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. 
  Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. 
   The base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B is 11, C is 12, D is 14, E is 15 and F is 16. 
   How we represent octal no. e.g. (A6)16


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