Number Systems

 Table: - Various number systems and their bases

S. No
Name of Number System
Base/Radix
1.
Binary
2
2.
Octal
8
3.
Decimal
10
4.
Hexadecimal
16

Table: - number systems
S. No
Name of Number System
Base/Radix
First Digit
Last Digit
All Digits/ Characters
1.
Binary
2
0
1
0,1
2.
Octal
8
0
7
0,1,2,3,4,5,6,7
3.
Decimal
10
0
9
0,1,2,3,4,5,6,7,8,9
4.
Hexadecimal
16
0
F
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Table: - Number Formats
S. No
Name
Size (bits)
Example
1.
Bit
1
1
2.
Nibble
4
1010
3.
Byte
8
1000 1100
4.
Word
16
1010 1010 1010 1010
5.
Double Word
32
1100 1100 1100 1100 1100 1100 1100 1100

Table: - Decimal, Binary, Octal and Hexadecimal Number
S. No
Decimal
Binary
Octal
Hexadecimal
1.
0
0000
0
0
2.
1
0001
1
1
3.
2
0010
2
2
4.
3
0011
3
3
5.
4
0100
4
4
6.
5
0101
5
5
7.
6
0110
6
6
8.
7
0111
7
7
9.
8
1000
10
8
10.
9
1001
11
9
11.
10
1010
12
A
12.
11
1011
13
B
13.
12
1100
14
C
14.
13
1101
15
D
15.
14
1110
16
E
16.
15
1111
17
F
17.
16
0001 0000
20
10
18.
17
0001 0001
21
11
19.
18
0001 0010
22
12
20.
19
0001 0011
23
13
21
20
0001 0100
24
14


Base Conversions
·         Binary to Decimal
(1110.01)2= (?)10
= 1×23 + 1x22 + 1×21+ 0x20 + 0x2-1 + 1×2-2 
= 1× 8+1×4+1×2+0×1+0×0.5+1×0.25
= 8+4+2+0+0.5+0.25 = 14.25
(1110.01)2 = (14.25)10
  •   Octal to Decimal
(4057.06)8= (?)10
=4 x 83 + 0 x 82 +5 x 81 + 7 x 80 +0×8-1+6 x 8-2 
= 4×512+0×64+5×8+7×1+0×0.125+6×0.015625
= 2048+0+40+7+0+0.0937
= (2095.0937)10

(4057.06)8 = (2095.0937)10
  • Hexadecimal to Decimal
(A0F9.0EB) 16= (?)10
= A x 163 + 0 x 162 +F x 161 + 9 x 160 +0×16-1+E x 16-2 +B x 16-3
= 10 x 163 + 0 x 162 +15 x 161 + 9 x 160 +0×16-1+14 x 16-2 +11 x 16-3
= 10 x 4096 + 0x 256 +15x 16 + 9x 1 +0×0.0625+14 x0.00390625 +11x0.000244140625
= 40960+0+240+9+0+0.0546+0.0026
= (41209.0572)10
(A0F9.0EB) 16= (41209.0572)10

Note MSB = Most significant bit & LSB = Least significant bit 
·         Decimal to Binary
(52)10= (?)2

2
52
0 LSB
2
26
0
2
13
1
2
6
0
2
3
1
1
1 MSB

(52)10= (110100)2

·         Decimal to Binary

(105.75)10= (?)2
2
105
1 L SB
2
52
0
2
26
0
2
13
1
2
6
0
2
3
1
1
1 MSB

Given Fraction                                    0.75     
Multiply 0.75 by 2                              1.50     MSB
Multiply 0.50 by 2                              1          LSB

(105.75)10= (1101001.11)2

·         Decimal to Octal

(5497.93)10= (?)8

8
5497
1 L.SB
8
687
7
8
85
5
8
10
2
1
1 M.S.B

Given Fraction                                    0.93     
Multiply 0.93 by 8                              7.44     MSB
Multiply 0.44 by 8                              3.52
Multiply 0.52 by 8                              4.16
Multiply 0.16 by 8                              1.28
Multiply 0.28 by 8                              2.24
Multiply 0.24 by 8                              1.92
Multiply 0.92 by 8                              7.36
Multiply 0.36 by 8                              2.88     L.SB

(5497.93)10= (12571.73412172)8

·         Decimal to Hexadecimal
(3481.55)10= (?)16

16
3481
9 LSB
16
217
9
13
13 MSB

Given Fraction                                    0.55    
Multiply 0.55 by 16                            8.8       MSB
Multiply 0.8 by 16                              12.8     LSB
Where 13= D & 12=C

(3481.55)10= (D99.8C)16
·         Binary to Octal
(1110.01)2= (?)8
= 1×23 + 1x22 + 1×21+ 0x20 + 0x2-1 + 1×2-2 
= 1× 8+1×4+1×2+0×1+0×0.5+1×0.25
= 8+4+2+0+0.5+0.25 = 14.25
(1110.01)2 = (14.25)10

8
14
6 L.SB
1
1 M.S.B

Given Fraction                                    0.25     
Multiply 0.25 by 8                              0.08     MSB
Multiply 0.08 by 8                              0.64
Multiply 0.64 by 8                              5.12
Multiply 0.12 by 8                              0.96
Multiply 0.96 by 8                              7.68
Multiply 0.68 by 8                              5.44
Multiply 0.44 by 8                              3.52
Multiply 0.52 by 8                              4.16     L.SB

            (1110.01)2= (61.00507534)8

·         Binary to Hexadecimal
(11110.01)2= (?)16
=1×24 + 1×23 + 1x22 + 1×21+ 0x20 + 0x2-1 + 1×2-2 
= 1× 16+1× 8+1×4+1×2+0×1+0×0.5+1×0.25
= 16+8+4+2+0+0.5+0.25 = 30.25
(11110.01)2 = (30.25)10

16
30
14  LSB
1
1     MSB

Given Fraction                                    0.25    
Multiply 0.25 by 16                            4.00       MSB
Where 13= D, 14= E & 12=C

(11110.01)2= (1E.4)16

·         Octal to Binary
(4057.06)8= (?)2
=4 x 83 + 0 x 82 +5 x 81 + 7 x 80 +0×8-1+6 x 8-2 
= 4×512+0×64+5×8+7×1+0×0.125+6×0.015625
= 2048+0+40+7+0+0.0937
= (2095.0937)10

(4057.06)8 = (2095.0937)10

2
2095
1 L SB
2
52
0
2
26
0
2
13
1
2
6
0
2
3
1
1
1 MSB

Given Fraction                                    0.75     
Multiply 0.75 by 2                              1.50     MSB
Multiply 0.50 by 2                              1          LSB

(105.75)10= (1101001.11)2

·         Octal to Hexadeximal
(4057.06)8= (?)16
=4 x 83 + 0 x 82 +5 x 81 + 7 x 80 +0×8-1+6 x 8-2 
= 4×512+0×64+5×8+7×1+0×0.125+6×0.015625
= 2048+0+40+7+0+0.0937
= (2095.0937)10

(4057.06)8 = (2095.0937)10
16
2095
15 LSB
16
130
2
8
8    MSB

Given Fraction                                    0.0937    
Multiply 0.0937 by 16                        1.499   MSB
Multiply 0.499 by 16                          7.98
Multiply 0.98 by 16                            15.68
Multiply 0.68 by 16                            10.88
Multiply 0.88 by 16                            14.08
Multiply 0.08 by 16                            1.28     LSB

Where 10=A, 15=F, 14=E, 13= D & 12=C

(4057.06)8 = (F28.17FAE1)16

·         Hexadecimal to Binary
(A0F9.0EB) 16 = (?)2
= A x 163 + 0 x 162 +F x 161 + 9 x 160 +0×16-1+E x 16-2 +B x 16-3
= 10 x 163 + 0 x 162 +15 x 161 + 9 x 160 +0×16-1+14 x 16-2 +11 x 16-3
= 10 x 4096 + 0x 256 +15x 16 + 9x 1 +0×0.0625+14 x0.00390625 +11x0.000244140625
= 40960+0+240+9+0+0.0546+0.0026
= (41209.0572)10
(A0F9.0EB) 16= (41209.0572)10

2
41209
1 LSB
2
20604
0
2
10302
0
2
5151
1
2
2575
1
2
1287
1
2
643
1
2
321
1
2
160
0
2
80
0
2
40
0
2
20
0
2
10
0
2
5
1
2
2
0
1
1 MSB

Given Fraction                                    0.0572     
Multiply 0.0575 by 2                          0.1144     MSB
Multiply 0.1144 by 2                          0.2288
Multiply 0.2288 by 2                          0.4576
Multiply 0.4576 by 2                          0.9152
Multiply 0.9152 by 2                          1.83
Multiply 0.83 by 2                              1.66
Multiply 0.66 by 2                              1.32          LSB

(A0F9.0EB) 16= (101000011111001.0000111)2

·         Hexadecimal to Octal
(A0F9.0EB) 16 = (?)8
= A x 163 + 0 x 162 +F x 161 + 9 x 160 +0×16-1+E x 16-2 +B x 16-3
= 10 x 163 + 0 x 162 +15 x 161 + 9 x 160 +0×16-1+14 x 16-2 +11 x 16-3
= 10 x 4096 + 0x 256 +15x 16 + 9x 1 +0×0.0625+14 x0.00390625 +11x0.000244140625
= 40960+0+240+9+0+0.0546+0.0026
= (41209.0572)10

(A0F9.0EB) 16= (41209.0572)10

8
41209
1 L.SB
8
5151
7
8
643
3
8
80
0
8
10
2
1
1 M.S.B

Given Fraction                                    0.0572     
Multiply 0.0572 by 8                          0.4576     MSB
Multiply 0.4576 by 8                          3.66
Multiply 0.66 by 8                              5.28
Multiply 0.28 by 8                              2.24
Multiply 0.24 by 8                              1.92
Multiply 0.92 by 8                              7.36
Multiply 0.36 by 8                              2.88
Multiply 0.88 by 8                              7.04     L.SB

(A0F9.0EB) 16= (120371.03521721)8


Table: - Complement
S. No
Decimal Number
Signed Binary
1’s Complement
2’s Complement
1.
+7
0111
0111
0111
Positive number in their normal form
2.
+6
0110
0110
0110
3.
+5
0101
0101
0101
4.
+4
0100
0100
0100
5.
+3
0011
0011
0011
6.
+2
0010
0010
0010
7.
+1
0001
0001
0001
8.
+0
0000
0000
0000
Unique Zero
9.
-0
1000
1111
1000
Negative numbers in their 1’s and 2’s Complement form
10.
-1
1001
1110
1111
11.
-2
1010
1101
1110
12.
-3
1011
1100
1101
13.
-4
1100
1011
1100
14.
-5
1101
1010
1011
15.
-6
1110
1001
1010
16
-7
1111
1000
1001
17
-8
1000
 1111
1000

Table: - Decimal, Binary, Octal and Hexadecimal Number
S. No
Decimal
Binary Code
BCD Code
Excess-3 Code
(BCD+3)
Gray Code
1.
0
0000
0000
0011
0000
2.
1
0001
0001
0100
0001
3.
2
0010
0010
0101
0011
4.
3
0011
0011
0110
0010
5.
4
0100
0100
0111
0110
6.
5
0101
0101
1000
0111
7.
6
0110
0110
1001
0101
8.
7
0111
0111
1010
0100
9.
8
1000
1000
1011
1100
10.
9
1001
1001
1100
1101
11.
10
1010
0001 0000
0001 0011
1111
12.
11
1011
0001 0001
0001 0100
1110
13.
12
1100
0001 0010
0001 0101
1010
14.
13
1101
0001 0011
0001 0110
1011
15.
14
1110
0001 0100
0001 0111
1001
16.
15
1111
0001 0101
0001 1000
1000
17.
16
0001 0000
0001 0110
0001 1001
0001 1000
18.
17
0001 0001
0001 0111
0001 1010
0001 1001
19.
18
0001 0010
0001 1000
0001 1011
0001 1011
20.
19
0001 0011
0001 1001
0001 1100
0001 1010
21
20
0001 0100
0010 0000
0001 1101
0001 1110


Figure:- Classification of codes.

REFERENCES
1.        Digital Design by M. Morris Mano, Michael D Ciletti, Pearson
2.        Digital Fundamentals by Thomas L. Floyd, R. P. Jain, Pearson
3.        Digital Circuits and Design by S Salivahanan, S Arivazhagan, Vikas Publishing House Pvt Ltd.
4.        Digital Systems by Ronald J. Tocci, Neal S. Widmer, Gregory L. Moss, Pearson
5.        Digital Electronics Principles And Integrated Circuits by Anil K. Maini
6.        Fundamentals of Digital Circuits by Anand Kumar
7.        Digital Electronics by John Morris
8.        Digital Electronics : An Introduction To Theory And Practice By William Gothmann

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