1. ECE 301 – Digital Electronics 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, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2. Spring 2011ECE 301 - Digital Electronics2 Representation of Negative Numbers (continued)

3. Spring 2011ECE 301 - Digital Electronics3 Signed Binary Numbers Representations for signed binary numbers: 1. Sign and Magnitude 2. 1's Complement 3. 2's Complement

4. Spring 2011ECE 301 - Digital Electronics4 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.

5. Spring 2011ECE 301 - Digital Electronics5 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.

6. Spring 2011ECE 301 - Digital Electronics6 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' = (2 n – 1) – N

7. Spring 2011ECE 301 - Digital Electronics7 1's Complement: Examples Using 8 bits, determine the 1's Complement representation for the following negative numbers: -15 -102

8. Spring 2011ECE 301 - Digital Electronics8 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

9. Spring 2011ECE 301 - Digital Electronics9 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)

10. Spring 2011ECE 301 - Digital Electronics10 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. - (2 n-1 – 1) <= N <= + (2 n-1 – 1)

11. Spring 2011ECE 301 - Digital Electronics11 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 = (2 n – 1) – N' N = bit-wise complement of N' or positive # 1's complement rep. for negative #

12. Spring 2011ECE 301 - Digital Electronics12 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.

13. Spring 2011ECE 301 - Digital Electronics13 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* = (2 n ) – N

14. Spring 2011ECE 301 - Digital Electronics14 2's Complement: Examples Using 8 bits, determine the 2's Complement representation for the following negative numbers: -12 -95

15. Spring 2011ECE 301 - Digital Electronics15 2's Complement The 1's and 2's Complement representations for a negative number, -N, are related as follows: N* = (2 n ) – N N' = (2 n - 1) – N N* = N' +1

16. Spring 2011ECE 301 - Digital Electronics16 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

17. Spring 2011ECE 301 - Digital Electronics17 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)

18. Spring 2011ECE 301 - Digital Electronics18 2's Complement For an n-bit 2's Complement binary number,  Includes only one representation for 0.  Represents an additional negative value. - (2 n-1 ) <= N <= + (2 n-1 – 1)

19. Spring 2011ECE 301 - Digital Electronics19 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 = (2 n ) – N* N = (N*)' + 1 or positive # 2's complement rep. for negative # bit-wise complement of 2's complement

20. Spring 2011ECE 301 - Digital Electronics20 Signed Binary Numbers

21. Spring 2011ECE 301 - Digital Electronics21 Binary Arithmetic of Signed Binary Numbers

22. Spring 2011ECE 301 - Digital Electronics22 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.

23. Spring 2011ECE 301 - Digital Electronics23 2's Complement Addition: Example Using 2's Complement addition and 8-bit representation, add the following numbers: -47 + 83 Did overflow occur?

24. Spring 2011ECE 301 - Digital Electronics24 2's Complement Addition: Example Using 2's Complement addition and 8-bit representation, add the following numbers: -32 + -105 Did overflow occur?

25. Spring 2011ECE 301 - Digital Electronics25 2's Complement Addition: Example Using 2's Complement addition and 8-bit representation, add the following numbers: 19 + 52 Did overflow occur?

26. Spring 2011ECE 301 - Digital Electronics26 2's Complement Addition: Example Using 2's Complement addition and 8-bit representation, add the following numbers: 64 + 78 Did overflow occur?

27. Spring 2011ECE 301 - Digital Electronics27 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)

28. Spring 2011ECE 301 - Digital Electronics28 2's Complement Subtraction: Example Subtract the following numbers, using 2's Complement addition and 8-bit representation: 64 – 78 Did overflow occur?

29. Spring 2011ECE 301 - Digital Electronics29 2's Complement Subtraction: Example Subtract the following numbers, using 2's Complement addition and 8-bit representation: -35 – 62 Did overflow occur?

30. Spring 2011ECE 301 - Digital Electronics30 2's Complement Subtraction: Example Subtract the following numbers, using 2's Complement addition and 8-bit representation: 14 – (-59) Did overflow occur?

31. Spring 2011ECE 301 - Digital Electronics31 2's Complement Subtraction: Example Subtract the following numbers, using binary subtraction and 8-bit representation: 27 – 45 Can this subtraction be carried out?

32. Spring 2011ECE 301 - Digital Electronics32 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.

33. Spring 2011ECE 301 - Digital Electronics33 1's Complement Addition: Example Using 1's Complement addition and 8-bit representation, add the following numbers: -31 + -84 Did overflow occur?

34. Spring 2011ECE 301 - Digital Electronics34 1's Complement Addition: Example Using 1's Complement addition and 8-bit representation, add the following numbers: 52 + 73 Did overflow occur?

35. Spring 2011ECE 301 - Digital Electronics35 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.

36. Spring 2011ECE 301 - Digital Electronics36 Binary Codes

37. Spring 2011ECE 301 - Digital Electronics37 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

38. Spring 2011ECE 301 - Digital Electronics38 Binary Codes n-bit Weighted Codes  Code:a n-1 a n-2 a n-3...a 1 a 0  Weights:w n-1, w n-2, w n-3,..., w 1, w 0  Decimal Value:a n-1 x w n-1 + a n-2 x w n-2 + … + a 1 x w 1 + a 0 x w 0 4-bit Weighted Code  Code:a 3 a 2 a 1 a 0

39. Spring 2011ECE 301 - Digital Electronics39 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

40. Spring 2011ECE 301 - Digital Electronics40 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.

41. Spring 2011ECE 301 - Digital Electronics41 Binary Codes

42. Spring 2011ECE 301 - Digital Electronics42 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.

43. Spring 2011ECE 301 - Digital Electronics43 Binary Coded Decimal In BCD, each decimal digit is replaced by its binary equivalent value. Example: Binary:937.25 10 = 1110101001.01 2

44. Spring 2011ECE 301 - Digital Electronics44 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.

45. Spring 2011ECE 301 - Digital Electronics45 ASCII Code (incomplete)

46. Spring 2011ECE 301 - Digital Electronics46 Questions?