1. Copyright (c) 2004 Professor Keith W. Noe
Number Systems & Codes
Part I
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2. 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
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3. 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.
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4. 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.
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5. 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) Professor Keith W. Noe

6. 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) Professor Keith W. Noe

7. 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) Professor Keith W. Noe

8. 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
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9. 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.
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10. 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:
10,000 1,
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11. 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 = ,000
4 x 102 = 4 x =
2 x 101 = 2 x =
8 x 100 = 8 x =
Copyright (c) Professor Keith W. Noe

12. The Decimal Number System
Express this base 10 number in positional notation:
56,782.45
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13. The Decimal Number System
Solution
5 x 104 = 5 x 10,000 = 50,000
6 x 103 = 6 x 1,000 = ,000
7 x 102 = 7 x =
8 x 101 = 8 x =
2 x 100 = 2 x =
4 x 10-1 = 4 x =
+ 5 x = 5 x =
56,782.45
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14. Other Number Systems Used in Digital Electronics & Computers
Binary (Base 2)
Octal (Base 8)
Hexadecimal (Base 16)
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15. 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.
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16. The Binary Number System
Base 2
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17. 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) Professor Keith W. Noe

18. 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.
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19. The Binary Number System
Counting in Binary 1 10 11
100
101
110
111
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20. The Binary Number System
Counting in Binary
Write the next 16 counts beginning with
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21. The Binary Number System
You should have written -
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22. 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, , 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) Professor Keith W. Noe

23. 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, , 2-2, etc.
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24. The Binary Number System
Positional Notation
For example:
1 x 24 = 1 x 16 = 16
1 x 23 = 1 x 8 =
0 x 22 = 0 x 4 =
1 x 21 = 1 x 2 =
+ 1 x 20 = 1 x 1 = 27
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25. The Binary Number System
Express this binary number in positional notation:
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26. The Binary Number System
1 x 25 = 1 x 32 =
0 x 24 = 1 x 16 =
1 x 23 = 1 x 8 =
1 x 22 = 1 x 4 =
0 x 21 = 0 x 2 =
1 x 20 = 1 x 1 =
0 x 2-1 = 0 x 0.5 =
+ 1 x 2-2 = 0 x .25 =
45.25 S O L U T I N
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27. The Octal Number System
Base 8
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28. 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) Professor Keith W. Noe

29. 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.
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30. The Octal Number System
Counting in Base 8
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31. The Octal Number System
Counting in Base 8
Write the next 23 counts beginning with:
608
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32. The Octal Number System
Counting in Base 8
You should have written:
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33. The Octal Number System
Positional Notation
The positional multipliers for the octal number system are:
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34. The Octal Number System
Positional Notation
For Example:
7 x 83 = 7 x = 3,584
4 x 82 = 4 x =
6 x 81 = 6 x =
x 80 = 2 x =
3,890
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35. The Octal Number System
Express this octal number using positional notation:
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36. The Octal Number System
4 x 83 = 4 x = 2,
7 x 82 = 7 x =
1 x 81 = 1 x =
2 x 80 = 2 x =
+ 5 x 8-1 = 5 x = 2,
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37. The Hexadecimal Number System
Base 16
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38. 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:
A B C D E F
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39. 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) Professor Keith W. Noe

40. 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) Professor Keith W. Noe

41. 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) Professor Keith W. Noe

42. The Hexadecimal Number System
Counting in Hexadecimal
A B C D E F
A 1B 1C 1D 1E 1F
A 2B 2C 2D 2E 2F
A 3B 3C 3D 3E 3F
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43. The Hexadecimal Number System
Counting in Hexadecimal
Write the next 32 counts beginning with 4016
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44. The Hexadecimal Number System
You should have written:
A 4B 4C 4D 4E 4F
A 5B 5C 5D 5E 5F
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45. 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) Professor Keith W. Noe

46. The Hexadecimal Number System
The powers of 16 used for the positional notation system for base 16 are: 4,
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47. 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) Professor Keith W. Noe

48. The Hexadecimal Number System
Positional Notation
Look at this example: B95F16
B x 163 = 11 x 4,096 = 45,056
9 x = 9 x = 2,304
5 x = 5 x =
+ F x = 15 x =
47,455
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49. The Hexadecimal Number System
Express this base 16 number in positional notation: 3C9F.B16
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50. The Hexadecimal Number System
3 x 163 = 3 x 4, = 12,
C x 162 = 12 x = 3,
9 x 161 = 9 x =
F x 160 = 15 x =
B x 16-1 = 11 x =
15,
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51. Copyright (c) 2004 Professor Keith W. Noe
The End
Copyright (c) Professor Keith W. Noe