Representing Negative Numbers Noah Mendelsohn Tufts University Web: COMP 40: Machine Structure and Assembly Language Programming (Fall 2014)

© 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

© 2010 Noah Mendelsohn Representing Negative Numbers

© 2010 Noah Mendelsohn Bits mean nothing on their own…  What's this: ?  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?

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

© 2010 Noah Mendelsohn At least three ways to represent negative numbers  Use the high order bit as a sign (sign and magnitude) –  Flip all the bits for negative (ones’ complement) –  Flip all bits then add one (two’s complement) – We usually represent the number 5 as 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;

© 2010 Noah Mendelsohn At least three ways to represent negative numbers  Use the high order bit as a sign (sign and magnitude) –  Flip all the bits for negative (ones’ complement) –  Flip all bits then add one (two’s complement) – We usually represent the number 5 as 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!

© 2010 Noah Mendelsohn Odometer Arithmetic Modular Number Systems

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

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

© 2010 Noah Mendelsohn Modular arithmetic – mod

© 2010 Noah Mendelsohn Modular arithmetic – mod

© 2010 Noah Mendelsohn Unsigned modular arithmetic – mod

© 2010 Noah Mendelsohn Signed modular arithmetic – mod

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

© 2010 Noah Mendelsohn Polynomials

© 2010 Noah Mendelsohn Binary numbers encode coefficients of polynomials 17 0 x x x x x x x x 2 0 =

© 2010 Noah Mendelsohn Two’s complement numbers as polynomial coefficients x x x x x x x x 2 0 = = -5 Hi order digit subtracted, all others added!

© 2010 Noah Mendelsohn Try it with -1 in two’s complement =