1. Complement Numbers

2. Outline  Negative Numbers Representation  Sign-and-magnitude  1s Complement  2s Complement  Comparison of Sign-and-Magnitude and Complements  Complements  Diminished-Radix Complements  Radix Complements  2’s Complement Addition and Subtraction  1’s Complement Addition and Subtraction

3. Outline  Negative Numbers Representation  Sign-and-magnitude  1s Complement  2s Complement  Comparison of Sign-and-Magnitude and Complements  Complements  Diminished-Radix Complements  Radix Complements  2’s Complement Addition and Subtraction  1’s Complement Addition and Subtraction

4. Negative Numbers Representation  Till now, we have only considered how unsigned (non- negative) numbers can be represented. There are three common ways of representing signed numbers (positive and negative numbers) for binary numbers:  Sign-and-Magnitude  1s Complement  2s Complement

5. Negative Numbers: Sign-and-Magnitude  Negative numbers are usually written by writing a minus sign in front.  Example: - (12) 10, - (1100) 2  In computer memory of fixed width, this sign is usually represented by a bit: 0 for + 1 for -

6. Negative Numbers: Sign-and-Magnitude  Example: an 8-bit number can have 1-bit sign and 7-bits magnitude. sign magnitude

7. Negative Numbers: Sign-and-Magnitude  Largest Positive Number: 0 1111111 +(127) 10  Largest Negative Number: 1 1111111 - (127) 10  Zeroes: 0 0000000 +(0) 10 1 0000000 - (0) 10  Range: - (127) 10 to +(127) 10

8. Negative Numbers: Sign-and-Magnitude  To negate a number, just invert the sign bit.  Examples: - (0 0100001) sm  (1 0100001) sm - (1 0000101) sm  (0 0000101) sm

9. 1s and 2s Complement  Two other ways of representing signed numbers for binary numbers are:  1s-complement  2s-complement

10. 1s Complement  Given a number x which can be expressed as an n-bit binary number, its negative value can be obtained in 1s-complement representation using: - x = 2 n - x - 1 Example: With an 8-bit number 00001100, its negative value, expressed in 1s complement, is obtained as follows: -(00001100) 2 = - (12) 10 = (2 8 - 12 - 1) 10 = (243) 10 = (11110011) 1s

11. 1s Complement  Essential technique: invert all the bits. Examples: 1s complement of (00000001) 1s = (11111110) 1s 1s complement of (01111111) 1s = (10000000) 1s  Largest Positive Number: 0 1111111 +(127) 10  Largest Negative Number: 1 0000000 -(127) 10  Zeroes: 0 0000000 1 1111111  Range: -(127) 10 to +(127) 10  The most significant bit still represents the sign: 0 = +ve; 1 = -ve.

12. 1s Complement  Examples (assuming 8-bit binary numbers): (14) 10 = (00001110) 2 = (00001110) 1s -(14) 10 = -(00001110) 2 = (11110001) 1s -(80) 10 = -( ? ) 2 = ( ? ) 1s

13. 2s Complement  Given a number x which can be expressed as an n-bit binary number, its negative number can be obtained in 2s-complement representation using: - x = 2 n - x Example: With an 8-bit number 00001100, its negative value in 2s complement is thus: -(00001100) 2 = - (12) 10 = (2 8 - 12) 10 = (244) 10 = (11110100) 2s

14. 2s Complement  Essential technique: invert all the bits and add 1. Examples: 2s complement of (00000001) 2s = (11111110) 1s (invert) = (11111111) 2s (add 1) 2s complement of (01111110) 2s = (10000001) 1s (invert) = (10000010) 2s (add 1)

15. 2s Complement  Largest Positive Number: 0 1111111 +(127) 10  Largest Negative Number: 1 0000000 -(128) 10  Zero: 0 0000000  Range: -(128) 10 to +(127) 10  The most significant bit still represents the sign: 0 = +ve; 1 = -ve.

16. 2s Complement  Examples (assuming 8-bit binary numbers): (14) 10 = (00001110) 2 = (00001110) 2s -(14) 10 = -(00001110) 2 = (11110010) 2s -(80) 10 = -( ? ) 2 = ( ? ) 2s

17. Outline  Negative Numbers Representation  Sign-and-magnitude  1s Complement  2s Complement  Comparison of Sign-and-Magnitude and Complements  Complements  Diminished-Radix Complements  Radix Complements  2’s Complement Addition and Subtraction  1’s Complement Addition and Subtraction

18. Comparisons of Sign-and-Magnitude and Complements  Example: 4-bit signed number ()  Example: 4-bit signed number (positive values)

19. Comparisons of Sign-and-Magnitude and Complements  Example: 4-bit signed number ()  Example: 4-bit signed number (negative values)

20. Outline  Negative Numbers Representation  Sign-and-magnitude  1s Complement  2s Complement  Comparison of Sign-and-Magnitude and Complements  Complements  Diminished-Radix Complements  Radix Complements  2’s Complement Addition and Subtraction  1’s Complement Addition and Subtraction

21. Complements (General)  Complement numbers can help perform subtraction. With complements, subtraction can be performed by addition. Hence, A – B can be performed by A + (-B) where (-B) is represented as the complement of B.  In general for Base-r number, there are: (i) Diminished Radix (or r-1’s) Complement (ii) Radix (or r’s) Complement  For Base-2 number, we have seen: (i) 1s Complement (ii) 2s Complement

22. Diminished-Radix Complements  Given an n-digit number, x r, its (r-1)’s complement is: (r n - 1) – x (r-1)’s complement, or 9s complement, of (22) 10 is: (10 2 - 1) - 22 = (77) 9s [This means –(22) 10 is (77) 9s ] (r-1)’s complement, or 1s complement, of (0101) 2 is: (2 4 - 1) - 0101 = (1010) 1s [This means –(0101) 2 is (1010) 1s ] Same as inverting all digits: (10 2 - 1) - 22 = 99 - 22 = (77) 9s (2 4 - 1) - 0101 = 1111 - 0101 = (1010) 1s

23. Radix Complements  Given an n-digit number, x r, its r’s-complement is: r n – x r’s-complement, or 10s complement, of (22) 10 is: 10 2 - 22 = (78) 10s [This means –(22) 10 is (78) 10s ] r’s-complement, or 2s complement, of (0101) 2 is: 2 4 - 0101 = (1011) 2s [This means –(0101) 2 is (1011) 2s ] Same as inverting all digits and adding 1: (10 2 ) - 22 = (99+1) - 22 = 77 + 1 = (78) 10s (2 4 ) - 0101 = (1111+1) - 0101 = 1010 +1 = (1011) 2s

24. Outline  Negative Numbers Representation  Sign-and-magnitude  1s Complement  2s Complement  Comparison of Sign-and-Magnitude and Complements  Complements  Diminished-Radix Complements  Radix Complements  2’s Complement Addition and Subtraction  1’s Complement Addition and Subtraction

25. 2s Complement Addition/Subtraction  Algorithm for addition, A + B: 1.Perform binary addition on the two numbers. 2.Ignore the carry out of the MSB (most significant bit). 3.Check for overflow: Overflow occurs if the ‘carry in’ and ‘carry out’ of the MSB are different, or if result is opposite sign of A and B.  Algorithm for subtraction, A – B: A – B = A + (–B) 1.Take 2s complement of B by inverting all the bits and adding 1. 2.Add the 2s complement of B to A.

26. 2s Complement Addition/Subtraction  Examples: 4-bit binary system  Which of the above is/are overflow(s)? +3 0011 + +4 + 0100 ---- ------- +7 0111 ---- ------- -2 1110 + -6 + 1010 ---- ------- -8 11000 ---- ------- +6 0110 + -3 + 1101 ---- ------- +3 10011 ---- ------- +4 0100 + -7 + 1001 ---- ------- -3 1101 ---- -------

27. 2s Complement Addition/Subtraction  More examples: 4-bit binary system  Which of the above is/are overflow(s)? -3 1101 + -6 + 1010 ---- ------- -9 10111 ---- ------- +5 0101 + +6 + 0110 ---- ------- +11 1011 ---- -------

28. Outline  Negative Numbers Representation  Sign-and-magnitude  1s Complement  2s Complement  Comparison of Sign-and-Magnitude and Complements  Complements  Diminished-Radix Complements  Radix Complements  2’s Complement Addition and Subtraction  1’s Complement Addition and Subtraction

29. 1s Complement Addition/Subtraction  Algorithm for addition, A + B: 1.Perform binary addition on the two numbers. 2.If there is a carry out of the MSB, add 1 to the result. 3.Check for overflow: Overflow occurs if result is opposite sign of A and B.  Algorithm for subtraction, A – B: A – B = A + (–B) 1.Take 1s complement of B by inverting all the bits. 2.Add the 1s complement of B to A.

30. 1s Complement Addition/Subtraction  Examples: 4-bit binary system +3 0011 + +4 + 0100 ---- ------- +7 0111 ---- ------- +5 0101 + -5 + 1010 ---- ------- -0 1111 ---- ------- -2 1101 + -5 + 1010 ---- ------ -7 10111 ---- + 1 ------ 1000 -3 1100 + -7 + 1000 ---- ------- -10 10100 ---- + 1 ------- 0101