Number system

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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

    83_h                                                   57_h                                          21_h                                            AB_h                           

_{1000  0011b                               0101  0111b                              0010  0001b                           1010  1011b}

    3C_h                                                   4F_h                                          DB_h                                            EE_h                           

_{0011  1100b                               0100  1111b                                1101  1011b                           1110  1110b}

0011  1111_b                                     0001  1000_b                               0101  1010_b                         1101   0111

   _{3            15                                                1              8                                            5              10                                 13             7}                                 

   3        F                               1         8                             5         A                       D        7

      3F_h                                                        18_h                                                         5A_h                                               D7_h

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.