1. ECE 331 – Digital System Design
Representation of Negative Numbers,
Binary Arithmetic of Negative Numbers,
and
Binary 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.

2. Representation of Negative Numbers
(continued)
Spring 2011
ECE Digital System Design

3. Signed Binary Numbers Representations for signed binary numbers:
1. Sign and Magnitude
2. 1's Complement
3. 2's Complement
Spring 2011
ECE Digital System Design

4. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

5. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

6. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

7. 1's Complement: Examples
Using 8 bits, determine the 1's Complement representation for the following negative numbers:
-15
-102
Spring 2011
ECE Digital System Design

8. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

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)
Spring 2011
ECE Digital System Design

10. ECE 331 - Digital System Design
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)
Spring 2011
ECE Digital System Design

11. ECE 331 - Digital System Design
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'
1's complement rep.
for negative # or
positive #
N = bit-wise complement of N'
Spring 2011
ECE Digital System Design

12. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

13. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

14. 2's Complement: Examples
Using 8 bits, determine the 2's Complement representation for the following negative numbers:
-12
-95
Spring 2011
ECE Digital System Design

15. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

16. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

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)
Spring 2011
ECE Digital System Design

18. ECE 331 - Digital System Design
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)
Spring 2011
ECE Digital System Design

19. ECE 331 - Digital System Design
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* or
2's complement rep. for negative #
positive #
N = (N*)' + 1
bit-wise complement of 2's complement
Spring 2011
ECE Digital System Design

20. ECE 331 - Digital System Design
Signed Binary Numbers
Spring 2011
ECE Digital System Design

21. ECE 331 - Digital System Design
Binary Arithmetic of
Signed Binary Numbers
Spring 2011
ECE Digital System Design

22. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

23. 2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
Did overflow occur?
Spring 2011
ECE Digital System Design

24. 2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
Did overflow occur?
Spring 2011
ECE Digital System Design

25. 2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
Did overflow occur?
Spring 2011
ECE Digital System Design

26. 2's Complement Addition: Example
Using 2's Complement addition and 8-bit representation, add the following numbers:
Did overflow occur?
Spring 2011
ECE Digital System Design

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)
Spring 2011
ECE Digital System Design

28. 2's Complement Subtraction: Example
Subtract the following numbers, using 2's Complement addition and 8-bit representation:
64 – 78
Did overflow occur?
Spring 2011
ECE Digital System Design

29. 2's Complement Subtraction: Example
Subtract the following numbers, using 2's Complement addition and 8-bit representation:
-35 – 62
Did overflow occur?
Spring 2011
ECE Digital System Design

30. 2's Complement Subtraction: Example
Subtract the following numbers, using 2's Complement addition and 8-bit representation:
14 – (-59)
Did overflow occur?
Spring 2011
ECE Digital System Design

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?
Spring 2011
ECE Digital System Design

32. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

33. 1's Complement Addition: Example
Using 1's Complement addition and 8-bit representation, add the following numbers:
Did overflow occur?
Spring 2011
ECE Digital System Design

34. 1's Complement Addition: Example
Using 1's Complement addition and 8-bit representation, add the following numbers:
Did overflow occur?
Spring 2011
ECE Digital System Design

35. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

36. ECE 331 - Digital System Design
Binary Codes
Spring 2011
ECE Digital System Design

37. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

38. ECE 331 - Digital System Design
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
Spring 2011
ECE Digital System Design

39. ECE 331 - Digital System Design
Binary Codes
Examples of 4-bit weighted codes
4 bits → 16 code words
Only 10 code words required to represent decimal digits
Excess-3 (obtained from )
Spring 2011
ECE Digital System Design

40. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

41. ECE 331 - Digital System Design
Binary Codes
Spring 2011
ECE Digital System Design

42. Binary Coded Decimal (BCD)
4-bit binary number used to represent each decimal digit.
Weighted code:
Binary values 0000 … 1001 used to represent decimal values 0 … 9.
Binary values 1010 … 1111 not used.
Very different from binary representation.
Spring 2011
ECE Digital System Design

43. ECE 331 - Digital System Design
Binary Coded Decimal
In BCD, each decimal digit is replaced by its binary equivalent value.
Example:
Binary: =
Spring 2011
ECE Digital System Design

44. ECE 331 - Digital System Design
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.
Spring 2011
ECE Digital System Design

45. ASCII Code (incomplete)
Spring 2011
ECE Digital System Design

46. ECE 331 - Digital System Design
Questions?
Spring 2011
ECE Digital System Design