NUMBER SYSTEMS IN DIGITAL SYSTEM

INTRODUCTION TO DIGITAL SYSTEM

A Digital system is an interconnection of digital modules and it is a system that manipulates discrete elements of information that is represented internally in the binary form.

Now a day’s digital systems are used in a wide variety of industrial and consumer products such as automated industrial machinery, pocket calculators, microprocessors, digital computers, digital watches, TV games and signal processing and so on.

Characteristics of Digital systems

• Digital systems manipulate discrete elements of information.

• Discrete elements are nothing but the digits such as 10 decimal digits or 26 letters of the alphabet and so on.

• Digital systems use physical quantities called signals to represent discrete elements.

• In digital systems, the signals have two discrete values and are therefore said to be binary.

• A signal in a digital system represents one binary digit called a bit. The bit has a value of either 0 or 1.

NUMBER SYSTEM

The number system is a basis for counting various items. Modern computers communicate and operate with binary numbers which use only the digits 0 &1. Basic number system used by humans is the Decimal number system.

For Ex: Let us consider the decimal number 18. This number is represented in binary as 10010.

We observe that the binary number systems take more digits to represent the decimal number. For large numbers, we have to deal with very large binary strings. So this fact gave rise to three new number systems.

i) Octal number systems

ii) Hexa Decimal number system

iii) Binary Coded Decimal number(BCD) system To define any number system we have to specify

• Base of the number system such as 2,8,10 or 16.

• The base decides the total number of digits available in that number system.

• First digit in the number system is always zero and the last digit in the number system is always base-

Binary number system:

The binary number has a radix of 2. As r = 2, only two digits are needed, and these are 0 and 1. In a binary system, weight is expressed as a

power of 2.

The left-most bit, which has the greatest weight is called the Most Significant Bit (MSB). And the right most bit which has the least weight is called the Least Significant Bit (LSB).

For Ex: 1001.012 = [ ( 1 ) × 23 ] + [ ( 0 ) × 22 ] + [ ( 0 ) × 21 ] + [ ( 1 ) × 20 ] + [ ( 0 ) × 2-1 ] + [ ( 1 ) × 22 ]

1001.012 = [ 1 × 8 ] + [ 0 × 4 ] + [ 0 × 2 ] + [ 1 × 1 ] + [ 0 × 0.5 ] + [ 1 × 0.25 ]

1001.012 = 9.2510

Decimal Number system:

The decimal system has ten symbols: 0,1,2,3,4,5,6,7,8,9. In other words, it has a base of 10.

Octal Number System:

Digital systems operate only on binary numbers. Since binary numbers are often very long, two shorthand notations, octal and hexadecimal, are used for representing large binary numbers. Octal systems use a base or radix of 8. It uses first eight digits of decimal number system. Thus it has digits from 0 to 7.

Hexa Decimal Number System:

The hexadecimal numbering system has a base of 16. There are 16 symbols. The decimal digits 0 to 9 are used as the first ten digits as in the decimal system, followed by the letters A, B, C, D, E and F, which represent the values 10, 11,12,13,14 and 15 respectively. 

                                                      

                                                    Decimal   Binary     Octal      Hexadecimal

0          0000          0                 0

1          0001          1                 1

2          0010          2                 2

3          0011          3                 3

4          0100          4                 4

5          0101          5                 5

6          0110          6                 6

7          0111          7                 7

8          1000         10                8

9          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