Transcript Chapter 2

Digital Fundamentals
CHAPTER 2
Number Systems, Operations, and Codes
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
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Slide 1
Number Systems
Floyd
Digital Fundamentals, 9/e
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Slide 2
Decimal Numbers
• The decimal number system has ten
digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
• The decimal numbering system has a
base of 10 with each position weighted by
a factor of 10:
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Slide 3
Binary Numbers
• The binary number system has two digits:
0 and 1
• The binary numbering system has a base of 2
with each position weighted by a factor of 2:
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Slide 4
Decimal-to-Binary Conversion
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Slide 5
Decimal-to-Binary Conversion
• Sum-of-weights method
• Repeated division-by-2 method
• Conversion of decimal fractions to
binary
Floyd
Digital Fundamentals, 9/e
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Slide 6
Binary Arithmetic
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Slide 7
Binary Arithmetic
•
•
•
•
Binary addition
Binary subtraction
Binary multiplication
Binary division
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Slide 8
Complements of Binary Numbers
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Slide 9
Complements of Binary Numbers
• 1’s complements
• 2’s complements
Floyd
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Slide 10
Complements of Binary Numbers
• 1’s complement
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Digital Fundamentals, 9/e
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Slide 11
Complements of Binary Numbers
• 2’s complement
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Slide 12
Signed Numbers
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Slide 13
Signed Numbers
•
•
•
•
•
Signed-magnitude form
1’s and 2’s complement form
Decimal value of signed numbers
Range of values
Floating-point numbers
Floyd
Digital Fundamentals, 9/e
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Slide 14
Signed Numbers
• Signed-magnitude form
– The sign bit is the left-most bit in a signed
binary number
– A 0 sign bit indicates a positive magnitude
– A 1 sign bit indicates a negative magnitude
Floyd
Digital Fundamentals, 9/e
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Slide 15
Signed Numbers
• 1’s complement form
– A negative value is the 1’s complement of
the corresponding positive value
• 2’s complement form
– A negative value is the 2’s complement of
the corresponding positive value
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 16
Signed Numbers
• Decimal value of signed numbers
– Sign-magnitude
– 1’s complement
– 2’s complement
Floyd
Digital Fundamentals, 9/e
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Slide 17
Signed Numbers
• Range of Values
2’s complement form:
– (2n – 1) to + (2n – 1 – 1)
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Slide 18
Signed Numbers
• Floating-point numbers
– Single-precision (32 bits)
– Double-precision (64 bits)
– Extended-precision (80 bits)
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Digital Fundamentals, 9/e
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Slide 19
Arithmetic Operations with Signed Numbers
•
•
•
•
Addition
Subtraction
Multiplication
Division
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Digital Fundamentals, 9/e
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Slide 20
Arithmetic Operations with Signed Numbers
Addition of Signed Numbers
• The parts of an addition function are:
– Addend
– Augend
– Sum
Numbers are always added two at a time.
Floyd
Digital Fundamentals, 9/e
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Slide 21
Arithmetic Operations with Signed Numbers
Four conditions for adding numbers:
• Both numbers are positive.
• A positive number that is larger than a
negative number.
• A negative number that is larger than a
positive number.
• Both numbers are negative.
Floyd
Digital Fundamentals, 9/e
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Slide 22
Arithmetic Operations with Signed Numbers
Signs for Addition
• When both numbers are positive, the
sum is positive.
• When the larger number is positive and
the smaller is negative, the sum is
positive. The carry is discarded.
Floyd
Digital Fundamentals, 9/e
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Slide 23
Arithmetic Operations with Signed Numbers
Signs for Addition
• When the larger number is negative and
the smaller is positive, the sum is
negative (2’s complement form).
• When both numbers are negative, the
sum is negative (2’s complement form).
The carry bit is discarded.
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Digital Fundamentals, 9/e
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Slide 24
Arithmetic Operations with Signed Numbers
Subtraction of Signed Numbers
• The parts of a subtraction function are:
– Subtrahend
– Minuend
– Difference
Subtraction is addition with the sign of the
subtrahend changed.
Floyd
Digital Fundamentals, 9/e
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Slide 25
Arithmetic Operations with Signed Numbers
Subtraction
• The sign of a positive or negative binary
number is changed by taking its 2’s
complement
• To subtract two signed numbers, take
the 2’s complement of the subtrahend
and add. Discard any final carry bit.
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
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Slide 26
Arithmetic Operations with Signed Numbers
Multiplication of Signed Numbers
• The parts of a multiplication function are:
– Multiplicand
– Multiplier
– Product
Multiplication is equivalent to adding a
number to itself a number of times equal to
the multiplier.
Floyd
Digital Fundamentals, 9/e
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Slide 27
Arithmetic Operations with Signed Numbers
There are two methods for multiplication:
• Direct addition
• Partial products
The method of partial products is the most
commonly used.
Floyd
Digital Fundamentals, 9/e
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Slide 28
Arithmetic Operations with Signed Numbers
Multiplication of Signed Numbers
• If the signs are the same, the product is
positive.
• If the signs are different, the product is
negative.
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 29
Arithmetic Operations with Signed Numbers
Division of Signed Numbers
• The parts of a division operation are:
– Dividend
– Divisor
– Quotient
Division is equivalent to subtracting the
divisor from the dividend a number of
times equal to the quotient.
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 30
Arithmetic Operations with Signed Numbers
Division of Signed Numbers
• If the signs are the same, the quotient is
positive.
• If the signs are different, the quotient is
negative.
Floyd
Digital Fundamentals, 9/e
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Slide 31
Hexadecimal Numbers
Floyd
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Slide 32
Hexadecimal Numbers
• Decimal, binary, and hexadecimal
numbers
Floyd
Digital Fundamentals, 9/e
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Slide 33
Hexadecimal Numbers
• Binary-to-hexadecimal conversion
• Hexadecimal-to-decimal conversion
• Decimal-to-hexadecimal conversion
Floyd
Digital Fundamentals, 9/e
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Slide 34
Hexadecimal Numbers
•
Binary-to-hexadecimal conversion
1. Break the binary number into 4-bit
groups
2. Replace each group with the
hexadecimal equivalent
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 35
Hexadecimal Numbers
•
Hexadecimal-to-decimal conversion
1. Convert the hexadecimal to groups of 4-bit
binary
2. Convert the binary to decimal
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
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Slide 36
Hexadecimal Numbers
•
Decimal-to-hexadecimal conversion
– Repeated division by 16
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
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Slide 37
Binary Coded Decimal (BCD)
Floyd
Digital Fundamentals, 9/e
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Slide 38
Binary Coded Decimal (BCD)
Decimal and BCD digits
Floyd
Digital Fundamentals, 9/e
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Slide 39
Digital Codes
Floyd
Digital Fundamentals, 9/e
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Slide 40
Digital Codes
• Gray code
• ASCII code
Floyd
Digital Fundamentals, 9/e
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Slide 41
Digital Codes
• Gray code
Floyd
Digital Fundamentals, 9/e
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Slide 42
Digital Codes
• ASCII code (control characters)
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Slide 43
Digital Codes
• ASCII code (graphic symbols 20h – 3Fh)
Floyd
Digital Fundamentals, 9/e
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Slide 44
Digital Codes
• ASCII code (graphic symbols 40h – 5Fh)
Floyd
Digital Fundamentals, 9/e
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Slide 45
Digital Codes
• ASCII code (graphic symbols 60h – 7Fh)
Floyd
Digital Fundamentals, 9/e
Copyright ©2006 by Pearson Education, Inc.
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Slide 46
Digital Codes
Extended ASCII code (80h – FFh)
• Non-English alphabetic characters
• Currency symbols
• Greek letters
• Math symbols
• Drawing characters
• Bar graphing characters
• Shading characters
Floyd
Digital Fundamentals, 9/e
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Slide 47
Error Detection and Correction Codes
Floyd
Digital Fundamentals, 9/e
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Slide 48
Error Detection and Correction Codes
• Parity error codes
• Hamming error codes
Floyd
Digital Fundamentals, 9/e
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Slide 49
Error Detection and Correction Codes
• Parity error codes
Floyd
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Slide 50
Error Detection and Correction Codes
• Hamming error codes
– Hamming code words
– Hex equivalent of the
data bits
Floyd
Digital Fundamentals, 9/e
0000000
0000111
0011011
0011110
0101010
0101101
0110011
0110100
1001011
1001100
1010010
1010101
1100001
1100110
1111000
1111111
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
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Slide 51