Number Systems & Codes
Part I
Copyright (c) 2004
Professor Keith W. Noe
Reading Assignment
Digital Design with CPLD Applications
and VHDL, by Robert K. Dueck
Chapter 1, Pages 6 through 17
Copyright (c) 2004
Professor Keith W. Noe
Objectives
Upon the successful completion of this lesson, you should
be able to:
• Explain positional notation and write the
positional multipliers for any number base.
• Count in binary, octal, decimal, &
hexadecimal.
• Write a given number in any base using
positional notation.
Copyright (c) 2004
Professor Keith W. Noe
Objectives
Upon the successful completion of this lesson, you should
be able to:
• Convert a binary, octal & hexadecimal
number to decimal.
Copyright (c) 2004
Professor Keith W. Noe
Number System Basics
All number systems have some commonalities:
• The base of the number system identifies
how many unique symbols are used for that
particular number system.
• The base of the number system identifies
the value of the highest symbol.
• All number systems begin counting at Zero.
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
• Has ten unique symbols.
• The ten symbols are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• The value for the highest symbol is
determined using the following formula:
Highest Symbol Value = Base – 1
• (Base) 10 – 1 = 9
• The value for the highest symbol in the
decimal number system is 9.
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
Counting in decimal or Base 10 number system
• When you begin counting in a number
system, always begin with Zero.
• When you have used up all of the symbols,
increment the column to the left by 1 and
begin counting again starting with Zero.
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
Counting in the decimal or Base 10 number system.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
10, 11, 12, 13, 14, 15, 16, 17, 18, 19
20, 21, 22, 23, 24, 25, 26, 27, 28, 29
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
Positional Notation
• All number systems use positional notation.
• The base of the number identifies the base
value to be used when determining the
value for each position.
• All number systems use a POINT to
separate the integer from the factional part.
• For Base 10, this is called the decimal point.
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
• The values of the positional multipliers are
the number system’s base raised to a power.
• For the decimal number system, the
multipliers are the powers of ten:
104 103 102 101 100 . 10-1 10-2
10,000 1,000 100 10 1 . 0.1 0.01
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
Positional Notation
For example: 37,42810
3 x 104 = 3 x 10,000 =
30,000
7 x 103 = 7 x 1, 000 =
7,000
4 x 102 = 4 x
100 =
400
2 x 101 = 2 x
10 =
20
8 x 100 = 8 x
1=
8
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
Express this base 10 number in
positional notation:
56,782.45
Copyright (c) 2004
Professor Keith W. Noe
The Decimal Number System
Solution
5 x 104 = 5 x 10,000 =
50,000
6 x 103 = 6 x 1,000 =
6,000
7 x 102 = 7 x
100 =
700
8 x 101 = 8 x
10 =
80
2 x 100 = 2 x
1 =
2
4 x 10-1 =
4x
0.1 =
0.4
+ 5 x 10-2 = 5 x
0.01 =
0.05
Copyright (c) 2004
Professor Keith W. Noe
56,782.45
Other Number Systems Used in
Digital Electronics & Computers
• Binary
(Base 2)
• Octal
(Base 8)
• Hexadecimal (Base 16)
Copyright (c) 2004
Professor Keith W. Noe
Summary About the Basics
All of the basics discussed as they relate
to the decimal number system applies
directly to the Binary, Octal &
Hexadecimal number systems.
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
Base 2
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Professor Keith W. Noe
The Binary Number System
• Has two unique symbols.
• Remember, the value of the highest symbol
equals the Base of the Number System
minus 1.
• Base 2 – 1 = 1
• Therefore, the highest symbol in the binary
number system is 1.
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
Counting in Binary
• When counting in binary, begin with Zero,
just as you do with any other number
system.
• When you have used all of the unique
symbols, increment the column to the left
by one and start with Zero again.
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
Counting in Binary
0
10
100
1
11
101
110
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Professor Keith W. Noe
111
The Binary Number System
Counting in Binary
Write the next 16 counts beginning with
100002
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
You should have written -
10001
10101
11001
11101
10010
10110
11010
11110
10011
10111
11011
11111
10100
11000
11100
100000
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
Positional Notation
• Each position will be 2 raised to a power.
• The binary number system is based on the powers
of 2.
25, 24, 23, 22, 21, 20 . 2-1, 2-2, 2-3, etc.
• The point that separates the integer part from the
fractional part of the number is called the binary
point.
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
Positional Notation
• Positional notation in the binary number
system is based on powers of two.
• For example:
25, 24, 23, 22, 21, 20 . 2-1, 2-2, etc.
32 16 8
4 2
1 .5 .25
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
Positional Notation
For example: 110112
1 x 24 = 1 x 16 =
16
1 x 23 = 1 x 8 =
8
0 x 22 = 0 x 4 =
0
1 x 21 = 1 x 2 =
2
+ 1 x 20 = 1 x 1 =
1
Copyright (c) 2004
Professor Keith W. Noe
27
The Binary Number System
Express this binary number in positional
notation:
101101.012
Copyright (c) 2004
Professor Keith W. Noe
The Binary Number System
S
1 x 25 = 1 x 32 =
32.00
O
0 x 24 = 1 x 16 =
0.00
1 x 23 = 1 x 8 =
8.00
U
1 x 22 = 1 x 4 =
4.00
T
0 x 21 = 0 x 2 =
0.00
1 x 20 = 1 x 1 =
1.00
O
0 x 2-1 = 0 x 0.5 =
0.00
N
+ 1 x 2-2 = 0 x .25 =
0.25
L
I
Copyright (c) 2004
Professor Keith W. Noe
45.25
The Octal Number System
Base 8
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
• Is based on powers of 8.
• The value of the highest symbol is 7.
• The octal point separates the integer portion
of the number from the fractional portion of
the number.
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
Counting in Base 8
• When counting in the octal number system,
begin with Zero.
• When you have used all of the unique
symbols, increment the column to the left
by one and begin with zero again.
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
Counting in Base 8
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
20 21 22 23 24
25 26 27
30
35
31
32
33
34
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Professor Keith W. Noe
36
37
The Octal Number System
Counting in Base 8
Write the next 23 counts beginning
with:
608
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
Counting in Base 8
You should have written:
60
61
62
63
64
65
66
67
70
71
72
73
74
75
76
77
100 101
102 103 104 105 106 107
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
Positional Notation
• The positional multipliers for the octal number
system are:
84 83 82 81 80 .
8-1
8-2
4096 512 64 8
1 . 0.125 0.015625
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
Positional Notation
For Example:
74628
7 x 83 = 7 x 512 = 3,584
+
4 x 82 = 4 x
64 =
256
6 x 81 = 6 x
8 =
48
2 x 80 = 2 x
1 =
2
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Professor Keith W. Noe
3,890
The Octal Number System
Express this octal number using
positional notation:
4712.58
Copyright (c) 2004
Professor Keith W. Noe
The Octal Number System
4 x 83 = 4 x 512
= 2,048.000
7 x 82 = 7 x 64
=
448.000
1 x 81 = 1 x
8
=
8.000
2 x 80 = 2 x
1
=
2.000
=
0.625
+ 5 x 8-1 = 5 x 0.125
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Professor Keith W. Noe
2,506.625
The Hexadecimal Number System
Base 16
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
• The base of this number system is 16.
• There are 16 unique symbols for this
number system.
• The sixteen symbols are:
0 1 2 3 4 5 6 7 8 9 A B C D E F
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
Some Additional Information
• A numeric symbol must occupy only one
place in a number.
• Numbers such as 12, 15, 24, etc uses two
symbols as two places are occupied.
• Since there are only 10 symbols defined
because of the decimal number system, six
additional symbols must be selected.
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
Some Additional Information
• The six extra symbols needed are borrowed
from the alphabet.
• The six letters borrowed from the alphabet
are: A B C D E F
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
Counting in Hexadecimal
• This number system begins counting at
zero.
• After counting from 0 to 9, the next six
counts are A, B, C, D, E, F.
• After using the 16 possible symbols,
increment the next column to the left by one
and start counting with zero again.
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
Counting in Hexadecimal
0 1 2 3 4 5 6 7 8 9 A B C D E F
10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F
30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
Counting in Hexadecimal
Write the next 32 counts beginning
with 4016
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
You should have written:
40 41 42 43 44 45 46 47 48 49 4A 4B 4C 4D 4E 4F
50 51 52 53 54 55 56 57 58 59 5A 5B 5C 5D 5E 5F
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
• The hexadecimal number system uses
positional notation just like the other
number systems studied so far.
• The hexadecimal number system is based
on the number 16.
• The Hexadecimal Point separates the
integer portion of the number from the
fractional portion.
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
• The powers of 16 used for the positional
notation system for base 16 are:
163 162
161
160 . 16-1
4,096 256
16
1 . 0.0625
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
• Usually technicians and engineers in the
digital electronics field often refer to the
hexadecimal number system simply as Hex.
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
Positional Notation
Look at this example: B95F16
B x 163 = 11 x 4,096 = 45,056
9 x 162 = 9 x 256 =
2,304
5 x 161 = 5 x
16 =
80
+ F x 160 = 15 x
1 =
15
Copyright (c) 2004
Professor Keith W. Noe
47,455
The Hexadecimal Number System
Express this base 16 number in positional
notation: 3C9F.B16
Copyright (c) 2004
Professor Keith W. Noe
The Hexadecimal Number System
3 x 163 = 3 x 4,096
= 12,288.0000
C x 162 = 12 x
256
=
3,072.0000
9 x 161 = 9 x
16
=
144.0000
F x 160 = 15 x
1
=
15.0000
B x 16-1 = 11 x 0.0625 =
0.6875
Copyright (c) 2004
Professor Keith W. Noe
15,519.6875
Copyright (c) 2004
Professor Keith W. Noe