Number system

                            tosinrogue!! A blogger

 Number System
Number system: It is an organized and systematic way of representing number.
Bit, Byte, Word and Character
•      A bit is the smallest quantity of information that can be stored or manipulated in a computer system. Digital computer store information in form of “1” or “0” at a defined memory location or address.
A single binary digit is known as a bit. It is the smallest unit of information.
•      A byte is a group of eight bits.
–        Byte = 8 bits
•         A word is a contiguous group of bytes. A word means the basic unit of information operated on by a computer.
Usually a word represents a number or instruction
Typical word sizes: 4, 8, 16, 32, 64, 128 bits
1K = 1024 bytes
  • Characters are alphanumeric (A-Z, a-z,0-9) and symbols (*,-,+,! Etc.) which are assigned binary patterns so that they can be stored and manipulated within the computer system.
Types of Number System
1. Decimal number system: It is number system with base 10 and use digits 0 to 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9).It can be written as subscripted 10.For example: 16710.
2. Binary system: The prefix “bi-” stands for 2. The binary number system is a Base 2 number system: There are 2 symbols that represent quantities: 0, 1.
3. Octal number system: It is a number system with base 8 and uses digits 0 to 7.It can be written as subscripted 8.For example: 17898.
4. Hexadecimal Number system: It is a number system with base 16 and use digits 0 to 9 and alphabet A to F. It can be written as subscripted 16. For example: (ABC)16

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

8
9
1001

9
10
1010

A
11
1011

B
12
1100

C
13
1101

D
14
1110

E
15
1111

F

CONVERT BINARY TO DECIMAL
11002 = (1 x 23 ) + ( 1 x 22 )+ ( 0 x 21 )+ ( 0 x 20 )      
           =       8     +      4       +      0      +      0
           =                             1210
100012 = (1 x 24) + ( 0 x 23 )+ ( 0 x 22 )+ ( 0 x 21 )+ ( 1 x 20 )
            =       16     +      0       +      0      +      0    +      1
            =                                 1710   
1010112= (1 x 25) + (0 x 24) + ( 1 x 23 )+ ( 0 x 22 )+ ( 1 x 21 )+ ( 1 x 20 )
              =    32     +      0       +      8      +      0     +      2       +       1       
              =                                    4310   
1111012= (1 x 25) + (1 x 24) + ( 1 x 23 )+ ( 1 x 22 )+ ( 0 x 21 )+ ( 1 x 20 )
              =    32     +   16       +      8      +      4     +      0       +       1       
              =                                    6110 

1102       = 610
10102     = 1010
1100002 = 4810
111112   = 3110
CONVERT DECIMAL TO BINARY
211
210
29
28
27
26
25
24
23
22
21
20


2048
1024
512
256
128
64
32
16
8
4
2
1

710  =









1
1
1
= 1112
3410 =






1
0
0
0
1
0
= 1000102
8910 =





1
0
1
1
0
0
1
= 10110012
20310 =




1
1
0
0
1
0
1
1
= 110010112
1910 =







1
0
0
1
1
= 100112
2210 =







1
0
1
1
0
= 101102
1510 =








1
1
1
1
= 11112
3610 =






1
0
0
1
0
0
= 1001002
4810 =






1
1
0
0
0
0
= 1100002
3910 =






1
0
0
1
1
1
= 1001112
6410 =





1
0
0
0
0
0
0
= 10000002
10310 =





1
1
0
0
1
1
1
= 11001112
EXAMPLE:      710  =  1112 because    7- 4 = 3        3 – 2 = 1        1 - 1= 0
Or
12510 =?2                                             
2   125
2   62     1
2   31     0
2   15    1
2    7    1
2    3    1
2    1    1
      0    1
12510 = 11111012
710 =?2
2          7
2          3          1
2          1          1
            0          1
710 =1112

8910 =
2          89
2          44        1
2          22        0
2          11        0
2          5          1
2          2          1
2          1          0
            0          1
8910 = 10110012

Converting From Base-8 to Base-10 (Octal - Decimal)

1118 = 1*82+1*81+1*80 = 64+8+1 = 7310
438 = 4*81+3*80 = 32+3 = 3510
61238 = 6*83+1*82+2*81+3*80 = 3072+64+16+3 = 315510
Binary --> octal
    1. Group into 3's starting at least significant symbol  (if the number of bits is not evenly divisible by 3, then add 0's at the most significant end)
    2. Write 1 octal digit for each group
    Example:
100 010 111  (binary)
                4   2   7   (octal)
                10 101 110  (binary)
                2   5   6   (octal)

 
Binary --> hex
    1. Group into 4's starting at least significant symbol
       (If the number of bits is not evenly divisible by 4, then add 0's at the most significant end)
    2. Write 1 hex digit for each group
Examples:

 
 1001 1110  0111  0000
    9         e         7                   0
  1 1111 1010 0011
               1    f    a    3


hex --> binary
     just write down the 4 bit binary code for each hexadecimal digit
     example:
      3    9    c    8
    0011 1001 1100 1000
octal --> binary
   just write down the 8 bit binary code for each octal digit
example:
      5   0   1
     101 000 001
hex --> octal
     do it in 2 steps,       1. hex --> binary 
                             2. binary --> octal
HEXADECIMAL    NUMBER    SYSTEM
Hexadecimal number system is Base Sixteen
Number of symbols = 16. Symbols     0,1,2,3,4,5,6,7,8,9, A,  B,   C,   D,   E,   F
                                                                                                             
1611
1610
169
168
167
166
165
164
163
162
161
160





16,777,216
1,048,576
65,536
4,096
256
16
1

We use hexadecimal numbers as “shorthand” for binary numbers
• Each group of four binary digits can be represented by a single hexadecimal digit.

CONVERT HEXADECIMAL TO DECIMAL
7216    = (7 x 161 ) + ( 2 x 160 )      
           =       112   +      2     
           =              11410
235916 = ( 2 x 163 )+ ( 3 x 162 )+ ( 5 x 161 )+ ( 9 x 160 )      
           =     8192    +    768      +      80      +      9
           =                             904910
C2916 = ( C x 162 )+ ( 2 x 161 )+ ( 9 x 160 )      
           =     3072   +      32      +      9
           =                   311310
12AB16 = ( 1 x 163 )+ ( 2 x 162 )+ ( 10 x 161 )+ ( 11 x 160 )      
           =     4096    +    512       +      160      +       11  
 =         477910
3AC216 = ( 3 x 163 )+ ( 10 x 162 )+ ( 12 x 161 )+ ( 2 x 160 )      
           =     12288   +    2560      +      192      +      2
           =                             1504210
CONVERT DECIMAL TO HEXADECIMAL
Allowable symbols  0, 1, 2, 3, 4,5, 6, 7, 8, 9,  A,   B,   C,  D,   E,   F
                                                                             
166
165
164
163
162
161
160


16,777,216
1,048,576
65,536
4096
256
16
1

67210  =




2
A
0
= 2A016
1,76310 =




6
E
3
= 6E316
1,32410 =




5
2
C
= 52C16
43,98110 =



A
B
C
D
= ABCD16
993,85110 =


F
2
A
3
B
= F2A3B16
12,76010 =



3
1
D
8
= 31D816

For example:
                         672                                                    1763
     2 @ 256      -512                                6 @  256    -1536
                          160                                                      227      
   10 @ 16        -160                                 14 @ 16      -224
                               0                                                         3 
           2A016                                                                                  6E316

                          1324                                                     43,981
     5 @ 256      -1280                            10 @  4096    - 40,960
                              44                                                        3021 
     2 @ 16        -    32                            11 @ 256         -  2816
                              12                                                         205 
                 52C16                                    12  @ 16             -192
                                                                                              13
                                                                                                                                                                     ABCD16
Binary to Hexadecimal
1010   11002                                 1110   1001 2                  0110   1011 2               0100    1100
     10            12                                            14            9                                            6              11                                       4             12
   A         C                              E       9                             6         B                          4         C 
        AC16                                                     E916                                         6B16                                                  4C16
Hexadecimal to Binary
    83h                                                   57h                                          21h                                            ABh                           
1000  0011b                               0101  0111b                              0010  0001b                           1010  1011b


    3Ch                                                   4Fh                                          DBh                                            EEh                           
0011  1100b                               0100  1111b                                1101  1011b                           1110  1110b

0011  1111b                                     0001  1000b                               0101  1010b                         1101   0111
   3            15                                                1              8                                            5              10                                 13             7                                 
   3        F                               1         8                             5         A                       D        7
      3Fh                                                        18h                                                         5Ah                                               D7h
Convert A3D7 to binary.
• A 3 D 7
• 1010 0011 1101 0111
Character Representation: ASCII
n  Alphanumeric data such as names and addresses are represented as strings of characters containing letters, numbers and symbols.
n  Each character has a unique code or sequence of bits to represent it.  As each character is entered from a keyboard it must be converted into its binary code.
n  Character code sets contain two types of characters:
•         Printable (normal characters)
•         Non-printable, i.e. characters used as control codes. For example:
–        CTRL G  (beep)
–        CTRL Z   (end of file)
n  ASCII :American Standard Code for Information Interexchange
•         strictly speaking a 7-bit code (128 characters)
•         has an extended 8-bit version
•         used on PC’s and non-IBM mainframes
•         widely used to transfer data from one computer to another
•         codes 0 to 31 are control codes

An ASCII subset
Symbol    Code                                   
A             41
 B             42
 C             43
 D             44
 E             45
 F             46
 0             30
 1             31
 2             32
 3             33
 4             34
 5             35
 6             36
 7             37
“BAD” = 42414416
             = 0100 0010 0100 0001 0100 01002
“F1” = 463116
         =  0100 0110 0011 00012
“3415” = 3334313516
                   = 0011 0011 0011 0100 0011 0001
                0011 01012

Note that this is a text string and no arithmetic may be done on it.  

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