Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic

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1 Today Binary addition Representing negative numbers 2

2 Binary Addition Consider the following binary numbers: How do we add these numbers? 3

3 Binary Addition

4 Binary Addition And we have a carry now! 5

5 Binary Addition And we have a carry again! 6

6 Binary Addition and again! 7

7 Binary Addition

8 Binary Addition One more carry! 9

9 Binary Addition

10 Binary Addition Behaves just like addition in decimal, but: We carry to the next digit any time the sum of the digits is 2 (decimal) or greater 11

11 Negative Numbers So far we have only talked about representing non-negative integers What can we add to our binary representation that will allow this? 21

12 Representing Negative Numbers One possibility: Add an extra bit that indicates the sign of the number We call this the sign-magnitude representation 22

13 Sign Magnitude Representation

14 Sign Magnitude Representation

15 Sign Magnitude Representation What is the problem with this approach? 25

16 Sign Magnitude Representation What is the problem with this approach? Some of the arithmetic operators that we have already developed do not do the right thing 26

17 Sign Magnitude Representation Operator problems: For example, we have already designed a counter (that implements an increment operation)

18 Sign Magnitude Representation Operator problems: Increment 28

19 Sign Magnitude Representation Operator problems: Increment

20 Sign Magnitude Representation Operator problems: Increment !!!! 30

21 Representing Negative Numbers An alternative: When taking the additive inverse of a number, invert all of the individual bits The leftmost bit still determines the sign of the number 31

22 One s Complement Representation Invert

23 One s Complement Representation Invert Increment

24 One s Complement Representation Invert Increment

25 One s Complement Representation What problems still exist? 35

26 One s Complement Representation What problems still exist? We have two distinct representations of zero :

27 One s Complement Representation What problems still exist? We can t directly add a positive and a negative number:

28 One s Complement Representation

29 One s Complement Representation !!!! 39

30 Representing Negative Numbers An alternative: (a little intuition first) Decrement 40

31 Representing Negative Numbers An alternative: (a little intuition first) Decrement

32 Representing Negative Numbers An alternative: (a little intuition first) Define this as Decrement

33 Representing Negative Numbers A few more numbers:

34 Two s Complement Representation In general, how do we take the additive inverse of a binary number? 47

35 Two s Complement Representation In general, how do we take the additive inverse of a binary number? Invert each bit and then add 1 48

36 Two s Complement Representation Invert each bit and then add Two s complement 49

37 Two s Complement Representation Now: let s try adding a positive and a negative number:

38 Two s Complement Representation Now: let s try adding a positive and a negative number:

39 Two s Complement Representation Now: let s try adding a positive and a negative number:

40 Two s Complement Representation Two s complement is used for integer representation in today s processors 53

41 Two s Complement Representation Two s complement is used for integer representation in today s processors One oddity: we can represent one more negative number than we can positive numbers 54

42 Implementing Subtraction How do we implement a subtraction operator? (e.g., A B) 55

43 Implementing Subtraction How do we implement a subtraction operator? (e.g., A B) Take the 2s complement of B Then add this number to A 56

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