digtial electronics
DESCRIPTION
Representation of negative numbersTRANSCRIPT
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ECE 301 – Digital Electronics
Representation of Negative Numbers,Binary Arithmetic of Negative Numbers,
andBinary Codes
(Lecture #11)
The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
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Spring 2011 ECE 301 - Digital Electronics 2
Representation of Negative Numbers
(continued)
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Spring 2011 ECE 301 - Digital Electronics 3
Signed Binary Numbers
Representations for signed binary numbers:
1. Sign and Magnitude2. 1's Complement3. 2's Complement
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Spring 2011 ECE 301 - Digital Electronics 4
Signed Binary Numbers
Arithmetic circuits are difficult to design for Sign and Magnitude binary numbers.
Consequently, this number system is not typically used in digital (computer) systems.
Instead other number systems, namely the 1's and 2's Complements, are more commonly used.
As we will see, it is rather easy to design arithmetic circuits for binary numbers represented in these number systems.
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Spring 2011 ECE 301 - Digital Electronics 5
1's Complement
A positive number, N, is represented in the same way as in the Sign and Magnitude representation.
For an n-bit number, The leftmost bit (sign bit) = 0.
Indicating a positive number. The remaining n-1 bits represent the
magnitude.
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Spring 2011 ECE 301 - Digital Electronics 6
1's Complement
A negative number, -N, is represented by the “1's complement” of the positive number, N.
N' = 1's complement representation for -N. For an n-bit signed binary number,
The leftmost bit (sign bit) = 1 for all negative numbers in the 1's Complement system.
N' = (2n – 1) – N
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Spring 2011 ECE 301 - Digital Electronics 7
1's Complement: Examples
Using 8 bits, determine the 1's Complement representation for the following negative
numbers:
-15-102
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Spring 2011 ECE 301 - Digital Electronics 8
1's Complement
The 1's Complement representation for -N can also be determined by taking the bit-wise complement of N.
N' = 1's Complement representation for -N. For an n-bit signed binary number,
i.e. complement N, bit-by-bit
N' = bit-wise complement of N
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Spring 2011 ECE 301 - Digital Electronics 9
1's Complement: Examples
Using 8 bits, determine the 1's Complement representation for the following negative
numbers:
-15-102
(Use the bit-wise complement)
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Spring 2011 ECE 301 - Digital Electronics 10
1's Complement
For an n-bit 1's Complement binary number,
Includes a representation for +0 and -0. Represents an equal number of positive and
negative values.
- (2n-1 – 1) <= N <= + (2n-1 – 1)
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Spring 2011 ECE 301 - Digital Electronics 11
1's Complement
To determine the magnitude of a negative number, -N, that is represented by its 1's Complement, N', simply take the “1's complement” of the 1's Complement.
N = (2n – 1) – N'
N = bit-wise complement of N'orpositive #
1's complement rep.for negative #
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Spring 2011 ECE 301 - Digital Electronics 12
2's Complement
A positive number, N, is represented in the same way as in the Sign and Magnitude representation.
For an n-bit number, The leftmost bit (sign bit) = 0.
Indicating a positive number. The remaining n-1 bits represent the
magnitude.
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Spring 2011 ECE 301 - Digital Electronics 13
2's Complement
A negative number, -N, is represented by the “2's complement” of the positive number, N.
N* = 2's complement representation for -N. For an n-bit signed binary number,
The leftmost bit (sign bit) = 1 for all negative numbers in the 2's Complement system.
N* = (2n) – N
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Spring 2011 ECE 301 - Digital Electronics 14
2's Complement: Examples
Using 8 bits, determine the 2's Complement representation for the following negative
numbers:
-12-95
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Spring 2011 ECE 301 - Digital Electronics 15
2's Complement
The 1's and 2's Complement representations for a negative number, -N, are related as follows:
N* = (2n) – N
N' = (2n - 1) – N
N* = N' +1
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Spring 2011 ECE 301 - Digital Electronics 16
2's Complement
Thus, the 2's Complement representation for -N can also be determined by adding 1 to the 1's Complement representation for -N.
N' = 1's Complement representation for -N. N* = 2's Complement representation for -N.
For an n-bit signed binary number,
N* = N' + 1
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Spring 2011 ECE 301 - Digital Electronics 17
2's Complement: Examples
Using 8 bits, determine the 2's Complement representation for the following negative
numbers:
-12-95
(Use the 1's complement)
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Spring 2011 ECE 301 - Digital Electronics 18
2's Complement
For an n-bit 2's Complement binary number,
Includes only one representation for 0. Represents an additional negative value.
- (2n-1) <= N <= + (2n-1 – 1)
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Spring 2011 ECE 301 - Digital Electronics 19
2's Complement
To determine the magnitude of a negative number, -N, that is represented by its 2's Complement, N*, simply take the “2's complement” of the 2's Complement.
N = (2n) – N*
N = (N*)' + 1orpositive #
2's complement rep. for negative #
bit-wise complement of 2's complement
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Spring 2011 ECE 301 - Digital Electronics 20
Signed Binary Numbers
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Spring 2011 ECE 301 - Digital Electronics 21
Binary Arithmeticof
Signed Binary Numbers
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Spring 2011 ECE 301 - Digital Electronics 22
2's Complement Addition
Addition of n-bit signed binary numbers is straightforward using the 2's Complement number system.
Addition is carried out in the same way as for n-bit positive numbers.
Carry from the sign bit (leftmost bit) is ignored. Overflow occurs if the correct result (including
the sign bit) cannot be represented in n bits.
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Spring 2011 ECE 301 - Digital Electronics 23
2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
-47 + 83
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 24
2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
-32 + -105
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 25
2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
19 + 52
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 26
2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
64 + 78
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 27
2's Complement Subtraction
Subtraction can be implemented using addition.
Determine the 2's Complement representation for the negative number -B.
Use 2's Complement addition to add A and -B.
A – B = A + (-B)
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Spring 2011 ECE 301 - Digital Electronics 28
2's Complement Subtraction: Example
Subtract the following numbers, using 2's Complement addition and 8-bit representation:
64 – 78
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 29
2's Complement Subtraction: Example
Subtract the following numbers, using 2's Complement addition and 8-bit representation:
-35 – 62
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 30
2's Complement Subtraction: Example
Subtract the following numbers, using 2's Complement addition and 8-bit representation:
14 – (-59)
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 31
2's Complement Subtraction: Example
Subtract the following numbers, using binary subtraction and 8-bit representation:
27 – 45
Can this subtraction be carried out?
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Spring 2011 ECE 301 - Digital Electronics 32
1's Complement Addition
Similar to 2's Complement Addition of n-bit signed binary numbers.
However, rather than ignore the carry-out from the sign (leftmost) bit, add it to the least significant bit (LSB) of the n-bit sum.
Known as the end-around carry.
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Spring 2011 ECE 301 - Digital Electronics 33
1's Complement Addition: Example
Using 1's Complement addition and 8-bit representation, add the following numbers:
-31 + -84
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 34
1's Complement Addition: Example
Using 1's Complement addition and 8-bit representation, add the following numbers:
52 + 73
Did overflow occur?
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Spring 2011 ECE 301 - Digital Electronics 35
Overflow
The general rule for detecting overflow when performing 2's Complement or 1's Complement Addition:
An overflow occurs when the addition of two positive numbers results in a negative number.
An overflow occurs when the addition of two negative numbers results in a positive number.
Overflow cannot occur when adding a positive number to a negative number.
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Spring 2011 ECE 301 - Digital Electronics 36
Binary Codes
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Spring 2011 ECE 301 - Digital Electronics 37
Binary Codes
Weighted Codes Each position in the code has a specific weight Decimal value of code can be determined
Unweighted Codes Positions of code do not have a specific weight Decimal value assigned to each code
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Spring 2011 ECE 301 - Digital Electronics 38
Binary Codes
n-bit Weighted Codes Code: an-1an-2an-3...a1a0 Weights: wn-1, wn-2, wn-3, ..., w1, w0 Decimal Value: an-1 x wn-1 + an-2 x wn-2 + …
+ a1 x w1 + a0 x w0 4-bit Weighted Code
Code: a3a2a1a0
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Spring 2011 ECE 301 - Digital Electronics 39
Binary Codes
Examples of 4-bit weighted codes 8-4-2-1
4 bits → 16 code words Only 10 code words required to represent
decimal digits 6-3-1-1
4 bits → 16 code words Excess-3 (obtained from 8-4-2-1)
4 bits → 16 code words
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Spring 2011 ECE 301 - Digital Electronics 40
Binary Codes
Examples of unweighted codes 2-out-of-5 Code
Exactly 2 of the 5 bits are “1” for a valid code word. 10 valid code words.
Gray Code Code values for successive decimal digits differ in
exactly one bit. 4 bits → 16 code words.
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Spring 2011 ECE 301 - Digital Electronics 41
Binary Codes
Decimal Digit
8-4-2-1 Code (BCD)
6-3-1-1 Code
Excess-3 Code
2-out-of-5 Code
Gray Code
0 0000 0000 0011 00011 0000
1 0001 0001 0100 00101 0001
2 0010 0011 0101 00110 0011
3 0011 0100 0110 01001 0010
4 0100 0101 0111 01010 0110
5 0101 0111 1000 01100 1110
6 0110 1000 1001 10001 1010
7 0111 1001 1010 10010 1011
8 1000 1011 1011 10100 1001
9 1001 1100 1100 11000 1000
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Spring 2011 ECE 301 - Digital Electronics 42
Binary Coded Decimal (BCD)
4-bit binary number used to represent each decimal digit.
Weighted code: 8-4-2-1 Binary values 0000 … 1001 used to represent
decimal values 0 … 9. Binary values 1010 … 1111 not used. Very different from binary representation.
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Spring 2011 ECE 301 - Digital Electronics 43
Binary Coded Decimal
In BCD, each decimal digit is replaced by its binary equivalent value.
Example:
Binary: 937.2510 = 1110101001.012
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Spring 2011 ECE 301 - Digital Electronics 44
ASCII
American Standard Code for Information Interchange Common code for the storage and transfer of
alphanumeric characters. 7-bit Weighted Code
Can represent 128 characters
Used to represent letters, numbers, and other characters
Any word or number can be represented using its ASCII code.
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Spring 2011 ECE 301 - Digital Electronics 45
ASCII Code (incomplete)
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Spring 2011 ECE 301 - Digital Electronics 46
Questions?