Number System:

In every day life we encounter with objects that are to be counted and written to be understood by all. The represented number also may require addition, subtraction etc. So, number system is a way of representing the count of some objects or countable things using some well define symbols and on which some kind of operation can be performed.

Number may be called as real numbers. The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.  The real numbers are “all the numbers” on the number line.

All numbers can be categorized as shown in figure-1.

Classification of Number System

These are the codes that are used in representing the information in a computer system. The classification of these codes are shown in figure-2

Analytical Codes

These are the code whose decimal value can be calculated using the analytical relation

N=  wⁿx bn ………………..+……..w²x b2    +        w¹ x b1      +       w⁰xbo

Decimal System:

Decimal system has a base of 10; it use 10 symbols (0-to-9) to form any number

Binary

Binary system uses only two symbols to represent any number. The symbols used are ‘0’ and  a ‘1’.

 

 

Octal

Octal system has a base of 8, total of 8 symbols are used to represent any number. These are 0-to-7. Any digit greater than 7 i.e 8,9 etc are invalid in octal.

In Octal system : 0+1=1                         1+1=2                      2+1=3                      3+1=4

4+1=5                       5+1=6                       6+1 =7

7+1= 10; note since there no symbol to represent 8, so the resets to zero with a carry to make decimal 8 = 10 in octal.

Hexadecimal:

Base is 16. It uses 16 symbols to represent any number. The valid symbols are 0-to-9, and ‘A’, ‘B’,’C’, ‘D’, ‘E’, ‘F’

some of the valid numbers in hexadecimal are:

B68A, 567, 1FB2, ABCD, CDEF, etc.

 

Decimal	Binary	octal	Hexadecimal
0	0000	0 –> 000	0
1	0001	1  –> 001	1
2	0010	2  –> 010	2
3	0011	3  –> 011	3
4	0100	4  –> 100	4
5	0101	5  –> 101	5
6	0110	6  –> 110	6
7	0111	7  –> 111	7
8	1000	10  –> 001 000	8
9	1001	11  –> 001 001	9
10	1010	12  –> 001 010	A
11	1011	13  –> 001 011	B
12	1100	14  –> 001 100	C
13	1101	15  –> 001 101	D
14	1110	16  –> 001 110	E
15	1111	17  –> 001 111	F

Conversion of Number System:

Conversion from Decimal to any other system:

Follow the procedure of division by the base to which conversion is required, Every time a division is done keep the remainder on the right. When done, read the number from bottom-to-top

Example:

Convert  :   58₁₀ =    (                    )₂

 

Divisor	Dividend	Remainder
2	58
2	29        —————>	0
2	14       —————>	1
2	7        —————>	0
2	3        —————>	1
2	1        —————>	1
	0        —————>	1

 

Fraction part:

Multiplying the fraction by the base in which conversion is required. After required precession read integer values from top to bottom.

0.55₁₀ =   0.55 x 2    =   1.10——>  1

.10 x 2     =    0.20 —>    0

58.55₁₀ = 111010.10

From any number system to Decimal:

For converting from any other system to decimal system require use the analytical relation

Octal to Decimal

(567)₈  =  (         )₁₀

= 8²x 5     +        8¹ x 6      +       8⁰x7

= 64 x 5   +     8×6           +         1×7

=  320      +      48           +          7

=  375

Hexadecimal to to Decimal

(AC8)₁₆   =  (                )₁₀

= 16²x A     +        16¹x C      +       16⁰ x 8

= 256 x 10  +       16 x 12      +        1×8

= 2560        +          192         +        8

=  2760

Binary to Decimal

Convert (11011)₂ to decimal

2⁴x1    +     2³x1    +      2²x0     +        2¹x 1      +       2⁰x1

= 16    +     8         +      0            +       2             +      1

=27₁₀

Convert 101.101 to Decimal

=2²x1  + 2¹x0  + 2⁰x1       .  2^-1x1  +  2^-2x0    +  2^-3x1

=     4        +    0    +   1      .    0.5   + 0           + 0.125

= 5.625

Convert (A29C)₁₆ to Decimal

=    16³xA   +  16²x2   +  16¹x9    +  16⁰xC

=    4096×10  + 256×2  + 16×9   +  1×12

=    40960    + 512        + 144     + 12

=    (41628)₁₀

Alphanumeric Code

(i)Extended Binary Coded Decimal Interchange Code (EBCIC)

(ii)American Standard Code for Infromation Interchange (ASCII)

The EBCDIC is used by IBM computer systems

The ASCII is used by almost all other computer systems.

In either case, the information that we feed to a computer system or extract some information out of the computer is alphanumeric i.e. in the form of letters, digits or some other special symbols.

ASCII format is a standard format commonly used as alphanumeric code. Its usage allows the manufacturers to standardize the i/o hardware such as KB, printer, display units etc. The ASCII code is of 7-bit or 8-bit format. With 7-bit format we can have 128 different codes. The table below shows the ASCII 7-bit code.

High bits 6,5,4,

Lower 4 bits of ASCII Code (3,2,1,0)

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

000

nul

soh

stx

etx

eot

enq

ack

bel

bs

ht

lf

vt

ff

cr

so

si

001

dle

dc1

dc2

dc3

dc4

nak

syn

etb

can

em

sub

esc

fs

gs

rs

us

010

SP

!

“

#

$

%

&

‘

(

)

*

+

‘

–

.

/

011

0

1

2

3

4

5

6

7

8

9

:

;

<

=.

>

?

100

@

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

101

P

Q

R

S

T

U

V

W

X

Y

Z

[

\

]

^

_

110

`

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

111

p

q

r

s

t

u

v

w

x

y

z

{

|

}

~

del

We can easily trace the ASCII code for any digit, letter and printable characters from the table. The rows of the table gives 3-most significant bits of the ASCII code and column gives the lower 4-bits, thus we get an equivalent 7-bit code of any printable ASCII code.
Numbers are transmitted to and from a computer as a sequence of ASCII-code digits. The number for example 234 would be transmitted as 32, 33, 34. Code of ‘A’ is 41h while that of ‘a’ is 61h

The code may be stored without modification, in which case the number will be in unpacked BCD format; or two digits can be packed together by stripping the high order; or it may convert the number to binary format. The choice of the format depends on the requirement of the program that will use this data.

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