Number Representation and Binary Arithmetic
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1 Number Representation and Binary Arithmetic Page 1
2 Unsigned Addition Page 2
3 Binary Addition = = = = 0 and carry 1 to the next column Examples: 0101 ( ( ( ( ( (8 10 Carries Page 3
4 Binary Addition w/overflow Add 6-bit numbers and in binary ( ( (89 10 Carries If the operands are unsigned, you can use the final carry-out as the MSB of the result. Adding 2 k-bit numbers k+1 bit result Page 4
5 More Binary Addition w/overflow ( ( (0 10 If you don t want a 5-bit result, just keep the lower 4 bits. Here, 4 bits is insufficient to hold the result (16. It rolls over back to 0. Page 5
6 Signed Numbers Page 6
7 Negative Binary Numbers Several ways of representing negative numbers Most obvious is to add a sign (+ or - to the binary integer Page 7
8 Sign-Magnitude 0 is + 1 is - Number Sign Magnitude Full Number Easy to interpret number value Page 8
9 Sign-Magnitude Examples ( ( ( ( ( (-7 10 Signs are the same, just add the magnitudes Page 9
10 Another Sign-Magnitude Example Signs are different: determine which has larger magnitude ( (-3 10 Put larger magnitude number on top Subtract ( ( (+2 10 Result has sign of larger magnitude number Page 10
11 Yet Another Sign-Magnitude Example Signs are different: determine which has larger magnitude ( (-5 10 Put larger magnitude number on top Subtract ( ( (-3 10 Result has sign of larger magnitude number Page 11
12 Sign-Magnitude Addition requires two separate operations addition subtraction Several decisions: Signs same or different? Which operand is larger? What is sign of final result? Two zeroes (+0, -0 Page 12
13 Sign-Magnitude Advantages Easy to understand Disadvantages Two different 0s Hard to implement in logic Page 13
14 One s Complement Positive numbers are the same as signmagnitude -N is represented as the complement of N: -N = N Number 1's Complement Page 14
15 One s Complement Examples ( ( ( ( ( (+2 10 If there is a carry out on the left, it must be wrapped around and added back in on the right Page 15
16 One s Complement Addition complicated by end around carry No decisions (unlike sign-magnitude Still two zeroes (+0, -0 Page 16
17 One s Complement Advantages Easy to generate N Only one addition process Disadvantages End around carry Two different 0s Page 17
18 Two s Complement Treat positional digits differently C = -0 x x x x 2 0 = C = -1 x x x x 2 0 = Most significant bit (MSB given negative weight Other bits same as in unsigned Page 18
19 Two s Complement Number 2's Complement none Page 19
20 Sign-Extension in 2 s Complement To make a k-bit number wider replicate sign bit 110 2C = -1 x x 2 1 = C = -1 x x x 2 1 = C = -1 x x x x 2 1 = Page 20
21 More Sign-Extension 010 2C = 1 x 2 1 = C = 1 x 2 1 = C = 1 x 2 1 = 2 10 Works for both positive and negative numbers Page 21
22 Negating a 2 s Complement Number 1. Invert all the bits 2. Add = C = C = = C = C = Page 22
23 Two s Complement Addition ( ( ( ( ( ( ( ( (+5 10 Operation is same as for unsigned. Same rules, same procedure. Interpretation of operands and results are different. Page 23
24 Two s Complement Addition always the same Only 1 zero Negation somewhat complicated The representation of choice Page 24
25 Two s Complement Overflow All representations can overflow Focus on 2 s complement here ( ( (+7 10?? The correct answer (-9 cannot be represented by 4 bits The correct answer is Page 25
26 Overflow Can you use leftmost carry-out as new MSB? ( ( (-9 10 Works here ( ( (+15 10?? Does NOT work here The answer is no, in general Page 26
27 Handling Overflow 1. Sign-extend the operands 2. Do the addition ( ( (-1 10 Page 27
28 When Can Overflow Occur? Adding two positive numbers - yes Adding two negative numbers - yes Adding a positive to a negative - no Adding a negative to a positive - no Page 28
29 General Handling of Overflow Adding two k-bit numbers gives k+1 bit result Two options for handling: 1. Keep entire k+1 bit sum 2.Keep only k bits (throw away generated bit i. Detect overflow and signal an error ii. Ignore the overflow The choice is up to the designer Page 29
30 Detecting Overflow Overflow occurs if: Adding two positive numbers gives a negative result Adding two negative numbers gives a positive result Overflow will never occur when adding a positive and a negative number together There are other ways of detecting overflow vs. what is suggested here Page 30
31 Binary Arithmetic Comparison Negative Number Sign Magnitude Easiest to Understand Simple to Compute One's Complement Two's Complement Easy to Compute Hardest to Compute Zeroes 2 Zeroes 2 Zeroes 1 Zero Largest Number Logic Required Overflow Detection Same number of + and - Numbers Requires Adder and Subtracter Extra Logic to Identify Larger Operand, Compute Sign, etc. Carry from High Order Adder Bits Same number of + and - Numbers Only Adder Required One Extra Negative Number Only Adder Required Carry Wraps Around - Sign of Both Operands is the Same and Sign of Sum is Different Sign of Both Operands is the Same and Sign of Sum is Different Page 31
32 Arithmetic Circuits Page 32
33 Binary Adder A B C out Full Adder C in Inputs: Operand A Operand B Carry In Outputs: Sum Carry Out S Page 33
34 Binary Adder An example of an iterative network A n-1 B n-1 A 2 B 2 A 1 B 1 A 0 B 0 C n Full C n-1 Adder... C 3 Full Adder C 2 Full Adder C 1 Full Adder C 0 0 S n-1 S 2 S 1 S 0 This type of adder is often called a ripple-carry adder because the carry ripples through from cell to cell Page 34
35 Full Adder Derivation A i B i C i C i+1 S i A i B i C i+1 Full Adder C i S i Page 35
36 Full Adder Derivation A i B i C i C i+1 S i A i B i C i+1 Full Adder C i S i Page 36
37 Full Adder Derivation A B C in C out S A BC in A 0 1 BC in C out S Page 37
38 Full Adder Derivation A B C in C out S A BC in A C out = AC in + AB + BC in BC in C out S S = AB C in + A B C in + ABC + A BC in Page 38
39 Full Adder Derivation A i B i A i B i C i S i C i+1 Full Adder C i A i B i S i B i C i A i C i C i+1 Page 39
40 Binary Subtracter X n-1 Y n-1 X 2 Y 2 X 1 Y 1 X 0 Y 0 B n Full B n-1 Sub.... B 3 Full Sub. B 2 Full Sub. B 1 Full Sub. B 0 D n-1 D 2 D 1 D 0 D is the difference (X-Y B is the borrow signal Equations generated similarly to full adder Page 40
41 Binary Subtracter Using Adder Recall that A-B is equivalent to A+(-B We can negate an operand by inverting the bits and adding 1 A subtracter can be implemented by a negater followed by an adder Page 41
42 Another Way to Make a Subtracter (2 s complement S 3 S 2 S 1 S 0 C 4 Full Adder C 3 Full Adder C 2 Full Adder C 1 Full Adder C 0 = 0 A 3 B 3 A 2 B 2 A 1 B 1 A 0 B 0 Page 42
43 Another Way to Make a Subtracter (2 s complement S 3 S 2 S 1 S 0 C 4 C 3 C 2 Full Adder Full Adder Full Adder C 1 Full Adder C 0 = 1 B 3 B 2 B 1 B 0 adding 1 inverting A 3 B 3 A 2 B 2 A 1 B 1 A 0 B 0 S = A + (-B -B is generated by inverting and adding 1 Page 43
44 Binary Adder/Subtracter (2 s complement S 3 S 2 S 1 S 0 C 4 C 3 C 2 C 1 Full Adder Full Adder Full Adder Full Adder Add Add Add Add A 3 B 3 A 2 B 2 A 1 B 1 A 0 B 0 The bar over the Add input means that the circuit will add when Add = 0. The notation Add# is also common. Page 44
45 Summary Understand the three main representation of binary integers Sign-Magnitude One s Complement Two s Complement Design binary adders and subtracters for each representation Page 45
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