Thursday, August 8, 2019

Number system_class 11



A number system is the set of symbols used to express quantities as the basis for counting, determining order, comparing amounts, performing calculation and representing value
There are two kinds of number system: -
1)     Non positional number system: - In this number system, we have symbols such as I for 1, V for 5, X for 10,L for 50, C=100, D for 500 and  M=1000. Each value represents the same value regardless of its position of a number.
Example: III = 1+1+1=3
                CX=100+10=101
2)     Positional number system: - In a position a number system, there are only few symbols a called digit, the symbols represents different values depending on the position they occupy in a number.
Example, In decimal number system, 245
5 is in unit position             5
4 is in tenth position         40
2 is in hundred position 200
                                    = 245
The position number systems are of the following types
a)      Decimal number system      b) Binary Number system         
b)     Octal number system           d) Hexadecimal number system

Decimal number system: - The number system that uses base or radix as 10 is called decimal number system. The digits used are 0,1,2,3,4,5,6,7,8,9. Each position in decimal number system represents a power of base 10.

Binary number system: - The number system that uses base or radix as 2 is called Binary number system. The digits used are 0 and 1. Each position in binary number system represents a power of base 2.

Octal number system- The number system that uses base or radix as 8 is called Octal number system. The digits used are 0,1,2,3,4,5,6,7. Each position in octal number system represents a power of base 8.

Hexadecimal number system: - The number system that uses base or radix as 16 is called hexadecimal number system. There are 16 digits or symbols. They are 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E and F. Each position in hexadecimal number system represents a power of base 16.

Binary
Decimal
Octal
Hexadecimal
0000
0
0
0
0001
1
1
1
0010
2
2
2
0011
3
3
3
0100
4
4
4
0101
5
5
5
0110
6
6
6
0111
7
7
7
1000
8

8
1001
9

9
1010
10

A
1011
11

B
1100
12

C
1101
13

D
1110
14

E
1111
15

F
                                                                                                                                               
Conversion
1)Decimal to binary
                 I.          Convert (19)10 into binary number system   answer (10011)2
               II.          Convert (45)10 into (?)2   answer (101101)2
             III.          Convert (87)10 into (?)2  answer (1010111)
2)Binary to decimal
                           I.          Convert (1101)2 to (?)10
=1*23+ 1*22+0*21+1*20
=1*8+1*4+0+1
=13
(1101)2 = (13)10
                         II.          Convert  ()2 to (?)10  
3)     Decimal to octal
                 I.          Convert (45)10 to (?)8        answer (55)
               II.          Convert (123)10 to (?)8   answer (173)
4)     Octal to decimal
                           I.          Convert (456)8 to (?)10
=4*82+5*81+6*80
=4*64 + 5*8+6*1
=256 +40+6
=302
(456)8 = (302)10

                         II.          Convert (3176)8 to (?)10
=3*83+1*82+7*81+6*80
=3*512+64+7*8+6*1
=1536+64+56+6
=1662
5)     Convert hexadecimal to decimal number system
(3F7A)16 to (?)10
=3*163+F*162+7*161+A*160
=3*163+15*162+7*161+10*160
=3*4086+ 15*256+7*16+10*1
=12288+3840+112+10
=16250
(3F7A)16 = (16250)10

(7DE)16 to (?)10
=7*162+D*161+E*160
=7*162+13*161+14*160
=7*256+13*16+14*1
=1792+208+14
=2014
(7DE)16= (2014)10

6)     Convert decimal to hexadecimal
i)                 (333)10 into hexadecimal number.  Answer (14D)
ii)               (123)10 into (?)16         ans= (7B)
7)     Convert binary to octal
i)                 Convert (1101010)2 into octal 
Here
Binary number                 001     101      010
Hexadecimal equivalent     1         5        2
 (1101010)2= (152)8

8)     Covert octal to binary
i)                 Convert (562)8 into binary
Here
Octal number                            5             6       2
Binary number equivalent      101     110        010
(562)8 = (101110010)2

ii)               Convert (375)8 into binary
 Here
Octal number                            3         7       5
Binary number equivalent      011      111    101
(375)8 = (011111101)2

9)     Convert binary to hexadecimal number system
i)                 Convert (11010011)2 into hexadecimal
Here
Binary number         1101  0011
Octal equivalent          D       3  
 (1101001)2= (D3)16

ii)               Convert (111011)2 into into hexadecimal
Here
Binary number       0011  1011
Hexadecimal equivalent          3       B  
 (1101001)2= (3B)16

10)  Convert hexadecimal to binary number system
i)                 Convert(2AB)16 into binary
Hexadecimal number  2           A         B    
Binary equivalent       0010   1010     1011
(2AB)16= (001010101011)2

ii)               (COFFEE)16 into binary 
Here
Hexadecimal number  C     0      F         F        E         E
Binary equivalent     1100  0000  1111 1111  1110   1110
(COFFEE)16= (110000001111111111101110)2





Arithmetic operation on binary number
Different arithmetic operations such as additions, subtraction, multiplication and division can be performed on binary numbers.

Adding binary numbers
Rules
0 + 0 = 0
0 +1= 1
1 +0=1
1 +1=10 ( carry 1)
 Note : 1+1+carry=1(carry1)

Adding 11101 and 01101

 11101                         1001                            1011                                        111
 01101                         1010                            1110                                        110


101010                                 10011                          11001                                      1101      


Subtracting binary number
Rules: 
0 -0 = 0
0 -1= 1 ( borrow 1 from the left column.)
1 -0=1
1 -1=0
                                   

 01011             111101           1011      1101001       11001010        1001                   
-00010             -100101           -0110       -  11110      -10011011       -   10


01001              011000            0101        1001011      00101111        0111

Multiplying binary number
0 X 0 = 0
0 X 1= 0
1 X 0=0
1 X 1= 1
                                                                                   
10101                                                              1001
               x101                                             x  1010
            10101                                                              0000   
        00000x                                                            1001X                            
      10101xx                                                          0000XX   
      1101001                                                        1001XXX
                                                                              1011010   
Dividing binary numbers

Complement of binary number system: complement of the binary number system is the reverse or just opposite of given umber system. In binary number system, there are two complement.
a)    1's complement
b)    2's complement
1's complement: - 1's complement of binary number is just reverse or opposite of given which is obtained by replacing 1 into 0 and 0 into 1.
Example 101001=010110

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