1. Representing Negative Numbers Noah Mendelsohn Tufts University Email: noah@cs.tufts.edunoah@cs.tufts.edu Web: http://www.cs.tufts.edu/~noah COMP 40: Machine Structure and Assembly Language Programming (Fall 2014)

2. © 2010 Noah Mendelsohn Goals for this presentation  Discuss the representation of variables that can take negative or positive values (or zero, of course!)  Develop your intuition about why 2’s complement arithmetic works 2

3. © 2010 Noah Mendelsohn Representing Negative Numbers

4. © 2010 Noah Mendelsohn Bits mean nothing on their own…  What's this: 1111 0100 ?  Ideas? –Positive integer 244 –HALT instruction (CPU idle) –Negative integer -12 –Small o with hat (circumflex) –0.957 pixel brightness (scaled by 255) –bit vector  With 8 bits we can distinguish 256 choices –If we use some bit patterns for negative numbers… –…then those same patterns can no longer stand for positives! 4 Question…what’s the best representation for negative numbers?

5. © 2010 Noah Mendelsohn At least three ways to represent negative numbers  Use the high order bit as a sign (sign and magnitude) – 1000 0101  Flip all the bits for negative (ones’ complement) – 1111 1010  Flip all bits then add one (two’s complement) – 1111 1011 5 We usually represent the number 5 as 0000 0101 How should we represent -5?

6. © 2010 Noah Mendelsohn At least three ways to represent negative numbers  Use the high order bit as a sign (sign and magnitude) – 1000 0101  Flip all the bits for negative (ones’ complement) – 1111 0101  Flip all bits then add one (two’s complement) – 1111 1011 6 We usually represent the number 5 as 0000 0101 How should we represent -5? These seem obvious and simple… …but there are problems For example, both representations have separate representations of +0 and -0, which makes it hard to implement sequences like: x = (-3); x = x + 5;

7. © 2010 Noah Mendelsohn At least three ways to represent negative numbers  Use the high order bit as a sign (sign and magnitude) – 1000 0101  Flip all the bits for negative (ones’ complement) – 1111 0101  Flip all bits then add one (two’s complement) – 1111 1011 7 We usually represent the number 5 as 0000 0101 How should we represent -5? Two’s complement is trickier to learn… … but it doesn’t have these problems. In fact, two’s complement has many advantages, and almost all modern computers use it!

8. © 2010 Noah Mendelsohn Odometer Arithmetic Modular Number Systems

9. © 2010 Noah Mendelsohn The problem:  I get why: 2 is encoded as 00000010  I have no clue why: -2 is encoded as 11111110 9

10. © 2010 Noah Mendelsohn Modular arithmetic 10 What happens when this goes past 999999? 999999 + 1 = 0 So, what should 0-1 be?

11. © 2010 Noah Mendelsohn Modular arithmetic – mod 100 11 1 0 2 50 98 99 75

12. © 2010 Noah Mendelsohn Modular arithmetic – mod 256 12 1 0 2 128 254 255 192

13. © 2010 Noah Mendelsohn Unsigned modular arithmetic – mod 256 13 1 0 2 128 254 255 192 00000001 00000010 10000000 11000000 11111110 11111111 00000000

14. © 2010 Noah Mendelsohn Signed modular arithmetic – mod 256 14 1 0 2 -128 -2 -64 00000001 00000010 10000000 11000000 11111110 11111111 00000000

15. © 2010 Noah Mendelsohn The 2s complement magic 15 The hardware does the same thing, only the interpretatation is different!

16. © 2010 Noah Mendelsohn Polynomials

17. © 2010 Noah Mendelsohn Binary numbers encode coefficients of polynomials 17 0 x 2 7 + 0 x 2 6 + 0 x 2 5 + 0 x 2 4 + 0 x 2 4 + 1 x 2 2 + 0 x 2 1 + 1 x 2 0 = 5 00000101

18. © 2010 Noah Mendelsohn Two’s complement numbers as polynomial coefficients 18 -1 x 2 7 + 1 x 2 6 + 1 x 2 5 + 1 x 2 4 + 1 x 2 3 + 0 x 2 2 + 1 x 2 1 + 1 x 2 0 = -5 11111011 -128 + 64 + 32 + 16 + 8 + 0 + 2 + 1 = -5 Hi order digit subtracted, all others added!

19. © 2010 Noah Mendelsohn Try it with -1 in two’s complement 19 11111111 + 127 = -1 -128