Number Systems Operation and Codes(Digital Logic Notes)

Number System

A number system is a set of symbols used to represent values derived from a common base or radix(r). As far as computer are concerned, number systems can be classified into four major categories.

• Decimal Number System
• Binary Number System 
• Octal Number System 
• Hexadecimal Number System

A number expressed in base-r-system has coefficient multiplied by power of r.

A_n .r ⁿ + a_n-1.r^n-1 +…. a₂.r² + a₁.r +a₀ + a_-1.r^-1 +a_-2.r² + ….. +a_-m .r^-m

EXAMPLE :-

The first Sixteen Number in the decimal, binary, octal and hexadecimal systems are given in following Table :

Decimal	Binary	Octal	Hexadecimal
Base 10	Base 2	Base 8	Base 16
00	0000	00	0
01	0001	01	1
02	0010	02	2
03	0011	03	3
04	0100	04	4
05	0101	05	5
06	0110	06	6
07	0111	07	7
08	1000	10	8
09	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

 

                                                      Table : 1.1 Number with Different Bases

Decimal Number System

Definition : The term decimal is derived from a Latin prefix deci, which means ten. Decimal number system has ten digits ranging from 0-9. Because this system has ten digits; it is also called a base ten number system or denary number system.

 A decimal number should always be written with a subscript 10 e.g. X10.

Binary Number System

Definition : It uses two digits namely, 1 and 0 to represent numbers, unlike in decimal numbers where the place value goes up in factors of ten, in binary system, the place values increase by the factor of 2. Binary numbers are written as (X)2, consider a binary number such as (1011). 

The right most digit has a place value of 1x2⁰ while the left most has a place value of 1x2³        

Octal Number System

Definition : Consists of eight ranging from 0-7 the place value of octal numbers goes up in factors eight from right to left. A octal number system can be denoted using 8 as a subscript.

For example: - 

25 can be written as (25)8.

Hexadecimal Number System

Definition : This is base 16 number system that consists of sixteen digit ranging from 0-9 and after it count as A-F where A is equivalent to 10, B to 11 up to F which is equivalent to 15 in Base ten system. The place value of hexadecimal numbers goes up in factors of sixteen.

It denoted as 16 as a subscript or capital letter H to the right of the number. 

For example  : 

94B written as (94B)16.

Next Topic -> Binary Arithmetic Operation