Sign & Magnitude vs 2 s Complement ( Signed Integer Representa:ons)
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1 Sign & Magnitude vs 2 s Complement ( Signed Integer Representa:ons)
2 Overview Unsigned Integers Representa:on (review) binary Signed Integers Representa:on sign & magnitude (S&M) 2 s complement (2 sc) arithme:c ops (+, -, x, /) overflow rules S&M vs 2 sc (pros/cons) Summary
3 Integer Representa0ons Desirable proper:es Only one bit pajern per value Equal number of posi:ve and nega:ve values Maximum range or values No gaps in the range Fast/economic hardware implementa:on of arithme:c ops
4 Integer Representa0ons Bit Pa4ern Unsigned Sign & magnitude 2 s complement
5 Unsigned Integer Representa0ons Binary (review)
6 Rules of Binary Arithme0c Ops Addi0on Subtrac0on = = = = 1 (borrow 1) = = = 0 (carry 1) 1 1 = 0 Mul0plica0on (repeated addi0on) Division (repeated subtrac0on) 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1 (no carry/borrow) (19) Ex: / (59/3 = 19 r 2) 11 ) (2) remaninder
7 Signed Integer Representa0ons Sign & magnitude (S&M) 2 s complement (2 sc) 1 s complement Excess- 2^n BCD
8 Sign & Magnitude (S&M)
9 sign & magnitude MSB sign bit 0 : posi:ve 1 : nega:ve Remaining bits as in binary rep (magnitude) (- 5) sign magnitude Bit Pa4ern Sign & magnitude
10 sign & magnitude Pros Simple Easy to understand (humans) Range: (- 7, +7) symmetric Cons +0, - 0 Arithme:c Ops difficult ( 2 n 1 1,+2 n 1 1) Signs need to be taken into account explicitly(adders & subtractors) Bit Pa4ern Sign & magnitude
11 sign & magnitude (cons) Arithme:c is difficult/inconsistent (+2) (+2) 1100 (- 4) 1010 (- 2) (- 6) NG 1100 (- 4) NG
12 2 s Complement (2 sc)
13 2 s complement (rep) MSB sign bit 0 : posi:ve / 1 : nega:ve Nega:ve of an integer 2 s complement of its absolute value (2^n x) M1 Take the posi:ve num Flip bits (complement) Add 1 M2 Take the posi:ve num Find first right- most 1 Flip the remaining bits to the lee M1 Ex: (+5) 1010 (flip) 1 (add) 1011 (- 5) (original) 1011 (2 s complement) (2^4)
14 2 s complement MSB sign bit 0 : posi:ve / 1 : nega:ve Nega:ve of an integer 2 s complement of its absolute value (2^n x) M1 Take the posi:ve num Flip bits (complement) Add 1 M2 Take the posi:ve num Find first right- most 1 Flip the remaining bits to the lee M2 Ex: (+5) 0101 (right- most 1) 1011 (flip / to lee) 1011 (- 5) (original) 1011 (2 s complement) (2^4)
15 2 s complement Pros +0 (just one!) Conversions easy/consistent Arithme:c Ops easy X Y = X + (- Y) Simple/fast implementa:on Cons Difficult to read Range: (- 8, +7) asymmetric ( 2 n 1, +2 n 1 1) Bit Pa4ern 2 s complement
16 2 s complement (pros) Conversion posi:ve to/from nega:ve is consistent +2 = 0010 flip the bits à 1101 and add 1 makes 1110 = = 1110 flip the bits à 0001 and add 1 makes 0010 = +2 Arithme:c is easy (same as unsigned binary) no need to keep track of sign (+2) (+2) 1100 (- 4) 1110 (- 2) (- 2) OK 0000 ( 0) OK (last carry on bit ignored)
17 2 s C Arithme0c Ops
18 2 s C Addi0on Follow the same rules as binary addi:on Discard any carry- out bit (last carry) Ex: (- 2) + (- 5) = (- 2) 1011 (- 5) (- 7) Overflow if operands have same signs and they are different from results sign (+A) + (+B) = - C Ex: (- 6) = (- 7) (- A) + (- B) = +C 1010 (- 6) 0011 (+3) ß overflow Overflow never occurs when adding operands with different signs
19 2 s C Subtrac0on Same as the binary addi0on of the minuend to the 2 sc of the the subtrahend (adding a nega:ve number is the same as subtrac:ng a posi:ve one) Discard any carry- out bit (last carry) Ex: 5 3 = (+5) 1101 (- 3) ß 2 sc of 3 = (+2) Overflow if operands have different signs and the results sign is the same as that of the subtrahend (+A) - (- B) = - C Ex: +7 (- 6) = (+7) (- A) - (+B) = +C 1010 (- 6) (+6) ß 2 sc of (- 6)= (- 3) ß overflow Overflow never occurs when subtrac:ng operands with same signs
20 2 s C Mul0plica0on Follow the same rules as binary mul:plica:on First convert the 2 sc numbers to their absolute values and negate the result if the operands sign are different Ex: +3 x - 2 = x (+3) 0010 (+2) ß abs(- 2) (+6) 1010 (- 6) negated (2 sc) Similar rules for Division!
21 Summary Signed Integer Representa:ons Sign & Magnitude 2 s complement 2 s complement is most commonly used in computers Arithme:c ops are easy and consistent Subtrac:on can be done with addi:on (one operator!) Just one 0 Fast implementa:on
22 Integer Representa0ons Bit Pa4ern Unsigned Sign & magnitude 2 s complement
23 End
24 2 s complement Quiz What range of integers can be represented using 16- bit 2 s complement representa:on? A. - 2^15 to 2^15-1 B. - 2^16 to 2^16-1 C. - 16^2 to 16^2 What is the decimal equivalent of the 8- bit 2 s complement number ? A B C What is the 8- bit 2 s representa:on of the decimal number - 18? A B C
25 2 s C Mul0plica0on (Notes) The product of two N- bit numbers requires 2N bits to contain all possible values Do sign extension to 2N- bit number Wasteful! Adapt algorithms to work with 2 sc numbers If the mul:plier is nega:ve à negate (i.e. take the 2 sc of) both operands The mul:plier will be now posi:ve (good!) Because both operands we negated the s:ll have the correct sign
26 2 s C Division Repeated 2 sc subtrac:on. Calculate the 2 s C of the divisor then add to dividend. Next replace the dividend with the quo:ent. Repeat un:l quo:ent is too small for subtrac:on or is zero. Answer is the total of subtrac:on cycles plus the reminder. Ex: 7/ 3 = 2 r (+7) 1101 (- 3) (+4) 1101 (- 3) (+4) 0001 (+1) reminder
27 2 s C <- > decimal 2 sc - > decimal M1: negate and add one Ex: 1011 to decimal à - ( ) = = - 5 M2: For n bits, treat MSB as nega:ve number and add it to the remaining bits (as posi:ve) Ex: 1011 = = = - 5 decimal - > 2 sc M1: subtract one and negate Ex: - 3 = - (0011) à 0010 à 1101 M2: For n bits, treat MSB as nega:ve number and add it to the remaining bits (as posi:ve) Ex:- 3 = à à 1101
28 2 s C Sign Extension To add two numbers, we must represent them with the same number of bits. Replicate the MSB (sign bit) to the lee 4- bit 8- bit 0100 (+4) (+4) 1100 (- 4) (- 4) If we just pad with zeroes on the lee (NG!!!) 4- bit 8- bit 0100 (+4) (+4) 1100 (- 4) (12 not - 4) Similarly, when shieed to right the MSB must be maintained However, when shieed to the lee, a 0 is shieed in! These rules preserve the seman:cs that lee shies mul:ply by two, while right shie divide by two.
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