This document discusses different number systems including decimal, binary, octal, and hexadecimal. It provides details on: - The base and possible digits of each number system. Binary has a base of 2 and digits of 0-1. Octal has a base of 8 and digits of 0-7. Hexadecimal has a base of 16 and uses 0-9 plus A-F. - Algorithms for converting between decimal, binary, octal, and hexadecimal. This includes division for converting to binary and multiplication for converting from binary. - Additional details on binary like each digit being called a bit and common groupings of bits into bytes, halfwords, words, and doublewords. -

1. NUMBER SYSTEM
Dr. (Mrs.) Gargi Khanna
Associate Professor
Electronics & Communication Engg. Deptt..
National Institute of Technology Hamirpur (HP)
Chapter-I

2. INTRODUCTION
 Decimal number system (Base 10).
 Some other number systems :
Number System Base/Radix No of possible
Digits
Decimal 10 10
Binary 2 2
Octal 8 8
Hexadecimal 16 16
 The number system with weights on position is called
weighted number system. e.g. Binary, Octal, Decimal, etc.
 Non-weighted number system e.g. gray code excess-3 code
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3. Characteristics of Numbering
Systems
 The number of digits is equal to the size of the
base.
 Zero is always the first digit and digits are
consecutive.
 The base number is never a digit.
 When 1 is added to the largest digit, a sum of
zero and a carry of one results.
 Numeric values determined by sum of the each
digit multiplied by positional values of the digits.
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4. Decimal Number System
Possible digits 0,1,2,3,4,5,6,7,8,9
Number d3d2 d1 d0. d-1d-2
(Integer) (fractional)
D = d3×103+d2×102 + d1×101 +d0×100 + d -1×10-1 +d -2×10-2
The value of the number is the sum of each digit multiplied by
the corresponding power of the radix
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5. Significant Digits
Binary: 1101101
Most significant digit Least significant digit
Decimal :4566
Hexadecimal: 196CA7A
Most significant digit Least significant digit
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6. Binary Number System
“Base 2 system”
 The binary number system is used to model
the series of electrical signals computers use
to represent information
 0 represents the no voltage or an off state
 1 represents the presence of voltage or an
on state
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7.  Computer perform all of their operations using the binary (base
2).
– Program code and data are stored and manipulated in binary.
– Each digit in a binary number is known as a bit (value 0 or 1).
– Bits are commonly stored and manipulated in groups of:
• 8 bit: Byte.
• 16 bit : Halfword.
• 32 bit: Word.
• 64 bit: Doubleword
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8. Binary Numbering Scale
Base 2 Number
Base 10
Equivalent
Power
Positional
Value
000 0 20 1
001 1 21 2
010 2 22 4
011 3 23 8
100 4 24 16
101 5 25 32
110 6 26 64
111 7 27 128
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9. Decimal to Binary Conversion
 Division Algorithm
 This method repeatedly divides a decimal
number by 2 and records the quotient and
remainder
– The remainder digits (a sequence of zeros and
ones) form the binary equivalent in least
significant to most significant digit sequence
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10. Division Algorithm
Convert 67 to its binary equivalent:
6710 = x2
Step 1: 67 / 2 = 33 R 1 Divide 64 by 2. Record quotient in next row
Step 2: 33 / 2 = 16 R 1 Again divide by 2; record quotient in next row
Step 3: 16 / 2 = 8 R 0 Repeat again
Step 4: 8 / 2 = 4 R 0 Repeat again
Step 5: 4 / 2 = 2 R 0 Repeat again
Step 6: 2 / 2 = 1 R 0 Repeat again
Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0
1 0 0 0 0 1 12
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11. Binary to Decimal Conversion
 The easiest method for converting a
binary number to its decimal equivalent
is to use the Multiplication Algorithm
 Multiply the binary digits by increasing
powers of two, starting from the right
 Then, to find the decimal number
equivalent, sum those products
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12. Multiplication Algorithm
Convert (10101111)2 to its decimal equivalent:
Binary 1 0 1 0 1 1 1 1
Positional Values
xxxxxxxx
2021222324252627
128 + 32 + 8 + 4 +2+1Products
17510
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13. Octal Number System
 Base 8 System
 Uses symbols 0 - 7
 Ease of convertion to binary
 Groups of three binary bits can be used
to represent each octal symbol
 Multiplication and division algorithms for
conversion to and from base 10
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14. Decimal to Octal Conversion
Convert 42910 to its octal equivalent:
429 / 8 = 53 R 5 Divide by 8; R is LSD
53 / 8 = 6 R 5 Divide Q by 8; R is next digit
6 / 8 = 0 R 6 Repeat until Q = 0
6558
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15. Octal to Decimal Conversion
Convert 6538 to its decimal equivalent:
6 5 3
xxx
82 81 80
384 + 40 + 3
42710
Positional Values
Products
Octal Digits
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16. Octal to Binary Conversion
Each octal number converts to 3 binary digits
475.038 =(100111101.000011) 2
To convert 6538 to binary, just
substitute code:
6 5 3
110 101 011
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17. Hexadecimal Number System
 Base 16 system
 Uses digits 0-9 &
letters A,B,C,D,E,F
 Groups of four bits
represent each
base 16 digit
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18. Decimal to Hexadecimal
Conversion
Convert 83110 to its hexadecimal equivalent:
831 / 16 = 51 R 15
51 / 16 = 3 R 3
3 / 16 = 0 R 3
33F16
= F in Hex
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19. Hexadecimal to Decimal
Conversion
Convert (3B4A)16 to its decimal equivalent:
Hex Digits 3 B 4 F
xxx
163 162 161 160
12288 +2816 + 64 +10
15,17810
Positional Values
Products
x
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20. Binary to Hexadecimal
Conversion
 The easiest method for converting binary to
hexadecimal is to use a substitution code
 Each hex number converts to 4 binary digits
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21. Convert 0111001010101111011010112 to hex
using the 4-bit substitution code :
0111 0010 1010 1111 0110 1011
Substitution Code
7 2 A F 6 B
76AF6B16
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22. Substitution code can also be used to convert
binary to octal by using 3-bit groupings:
010 101 101 010 111 001 101 010
Substitution Code
2 5 5 2 7 1 5 2
255271528
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23. G.Khanna, NITH
Number Decimal Binary Octal Hexadecimal
------ ------- ------- ----- -----------
Zero 0 0 0 0
One 1 1 1 1
Two 2 10 2 2
Three 3 11 3 3
Four 4 100 4 4
Five 5 101 5 5
Six 6 110 6 6
Seven 7 111 7 7
Eight 8 1000 10 8
Nine 9 1001 11 9
Ten 10 1010 12 A
Eleven 11 1011 13 B
Twelve 12 1100 14 C
Thirteen 13 1101 15 D
Fourteen 14 1110 16 E
Fifteen 15 1111 17 F
Sixteen 16 10000 20 10
Seventeen 17 10001 21 11
Eighteen 18 10010 22 12
Nineteen 19 10011 23 13
Twenty 20 10100 24 14

24. G.Khanna, NITH