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0AA u]@83PTʚ;ʚ;g4dddd6  0rppp@ <4dddd@w 0tP*  80___PPT10 ''4XData Representation ,COE 308 Computer Architecture Prof. Muhamed Mudawar College of Computer Sciences and Engineering King Fahd University of Petroleum and MineralsB4Z-0Zd/Z ,\~Presentation OutlinePositional Number Systems Binary and Hexadecimal Numbers Base Conversions Integer Storage Sizes Binary and Hexadecimal Addition Signed Integers and 2's Complement Notation Sign Extension Binary and Hexadecimal subtraction Carry and Overflow Character StoragecPositional Number Systems Binary Numbers0Each binary digit (called bit) is either 1 or 0 Bits have no inherent meaning, can represent Unsigned and signed integers Characters Floating-point numbers Images, sound, etc. Bit Numbering Least significant bit (LSB) is rightmost (bit 0) Most significant bit (MSB) is leftmost (bit 7 in an 8-bit number)T] FS F ds F]SsConverting Binary to DecimalrEach bit represents a power of 2 Every binary number is a sum of powers of 2 Decimal Value = (dn-1 2n-1) + ... + (d1 21) + (d0 20) Binary (10011101)2 = 27 + 24 + 23 + 22 + 1 = 157 <M          "Convert Unsigned Decimal to BinarycRepeatedly divide the decimal integer by 2 Each remainder is a binary digit in the translated valuedHexadecimal Integers16 Hexadecimal Digits: 0  9, A  F More convenient to use than binary numbers  Converting Binary to Hexadecimal!!Converting Hexadecimal to DecimalMultiply each digit by its corresponding power of 16 Value = (dn-1 16n-1) + (dn-2 16n-2) + ... + (d1 16) + d0 Examples: (1234)16 = (1 163) + (2 162) + (3 16) + 4 = Decimal Value 4660 (3BA4)16 = (3 163) + (11 162) + (10 16) + 4 = Decimal Value 152685 P@ P `P`P5                            &                #!Converting Decimal to HexadecimalInteger Storage Sizes Binary AdditionxStart with the least significant bit (rightmost bit) Add each pair of bits Include the carry in the addition, if presenty <y$Hexadecimal AdditionJStart with the least significant hexadecimal digits Let Sum = summation of two hex digits If Sum is greater than or equal to 16 Sum = Sum  16 and Carry = 1 Example:< < < < SSigned IntegersSeveral ways to represent a signed number Sign-Magnitude Biased 1's complement 2's complement Divide the range of values into 2 equal parts First part corresponds to the positive numbers (e" 0) Second part correspond to the negative numbers (< 0) Focus will be on the 2's complement representation Has many advantages over other representations Used widely in processors to represent signed integerst*4.j3f*4.j3f TTwo's Complement Representation'Forming the Two's Complement &Sign Bit8Highest bit indicates the sign 1 = negative 0 = positive9 F9RSign ExtensionnStep 1: Move the number into the lower-significant bits Step 2: Fill all the remaining higher bits with the sign bit This will ensure that both magnitude and sign are correct Examples Sign-Extend 10110011 to 16 bits Sign-Extend 01100010 to 16 bits Infinite 0s can be added to the left of a positive number Infinite 1s can be added to the left of a negative number zuCB: <:uCB:: W!Two's Complement of a Hexadecimal!To form the two's complement of a hexadecimal Subtract each hexadecimal digit from 15 Add 1 Examples: 2's complement of 6A3D = 95C2 + 1 = 95C3 2's complement of 92F15AC0 = 6D0EA53F + 1 = 6D0EA540 2's complement of FFFFFFFF = 00000000 + 1 = 00000001 No need to convert hexadecimal to binary. P. P P P) P.. a #a#a)(Binary Subtraction(When subtracting A  B, convert B to its 2's complement Add A to ( B) 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 1 1 0 (2's complement) 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 (same result) Final carry is ignored, because Negative number is sign-extended with 1's You can imagine infinite 1's to the left of a negative number Adding the carry to the extended 1's produces extended zerosxG P}# %Hexadecimal SubtractionWhen a borrow is required from the digit to the left, then Add 16 (decimal) to the current digit's value Last Carry is ignored.; </ < <*Ranges of Signed IntegersUCarry and Overflow|Carry is important when & Adding or subtracting unsigned integers Indicates that the unsigned sum is out of range Either < 0 or >maximum unsigned n-bit value Overflow is important when & Adding or subtracting signed integers Indicates that the signed sum is out of range Overflow occurs when Adding two positive numbers and the sum is negative Adding two negative numbers and the sum is positive Can happen because of the fixed number of sum bitsT 1     VCarry and Overflow ExamplesfWe can have carry without overflow and vice-versa Four cases are possible (Examples are 8-bit numbers)gZg["Range, Carry, Borrow, and Overflow`Unsigned Integers: n-bit representation Signed Integers: n-bit 2's complement representation6a)#+Character Storage,Character sets Standard ASCII: 7-bit character codes (0  127) Extended ASCII: 8-bit character codes (0  255) Unicode: 16-bit character codes (0  65,535) Unicode standard represents a universal character set Defines codes for characters used in all major languages Used in Windows-XP: each character is encoded as 16 bits UTF-8: variable-length encoding used in HTML Encodes all Unicode characters Uses 1 byte for ASCII, but multiple bytes for other characters Null-terminated String Array of characters followed by a NULL characterr-^1r-^ 0PPrintable ASCII CodesQControl CharactersThe first 32 characters of ASCII table are used for control Control character codes = 00 to 1F (hexadecimal) Not shown in previous slide Examples of Control Characters Character 0 is the NULL character used to terminate a string Character 9 is the Horizontal Tab (HT) character Character 0A (hex) = 10 (decimal) is the Line Feed (LF) Character 0D (hex) = 13 (decimal) is the Carriage Return (CR) The LF and CR characters are used together They advance the cursor to the beginning of next line One control character appears at end of ASCII table Character 7F (hex) is the Delete (DEL) character$m    6 4 1 m /4)+6 4  "Shl   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@$?vdd(@#?  n?" dd@   @@``PR    ?  ` p>>   <4 (    6z   `   T Click to edit Master title style! !  0J   `x   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0P " `* .Data Representation COE 308  Computer Architecture Muhamed Mudawar  slide *0P 24c c NN H  0޽h ? 3380___PPT10.И@r Default Design  0 QI`(    6   `   T Click to edit Master title style! !  0l   `b   W#Click to edit Master subtitle style$ $H  0޽h ? 3380___PPT10.88/k&p  h(  h  h Nttt B    v* p88pp h Ntt  wB  x* p88pp h TЧtt .   v* p88pp h Tptt  w.  x* p88ppH h 0rllC ? 3380___PPT10.Jp: 0  06(  0~ 0 s *  `r   x 0 c $W  `   H 0 0޽h ? 33___PPT10i.И0+D='  = @B +  0 P<(  ~  s *X'  `    ~  s *    H  0޽h ? f3fy___PPT10Y+D='  = @B +   0   P \ (  \ \ 0 "`0Pa j JDifferent Representations of Natural Numbers XXVII Roman numerals (not positional) 27 Radix-10 or decimal number (positional) 110112 Radix-2 or binary number (also positional) Fixed-radix positional representation with k digits Number N in radix r = (dk 1dk 2 . . . d1d0)r Value = dk 1r k 1 + dk 2r k 2 + & + d1r + d0 Examples: (11011)2 = 124 + 123 + 022 + 12 + 1 = 27 (2103)4 = 243 + 142 + 04 + 3 = 147.0@ 0H 0`2 -bb7fffn fffb+fffff ffffnnfnnffnfnfnn fFf    fFf f  n $F$(f((f((n((f((n((f(,F,,N,,F,,N,,F,0f04F44N44F44N44F44N44F48f8 y"4qq<^>x \ c $  `    H \ 0޽h ? ̙33y___PPT10Y+D='  = @B +W@  0 --pCOg-(  ~  s *H  `      s *\ 0,<$@ 0   (z > ^/     ,$D 0h  s *X99?"`> ^/    BC DEF @`~ * q J    BCDEF  @`U q J B  3 ~ U *   6   (  M10ga   BC DEF @`M * 8J    BCDEF  @`8J B  3 M *   60   (  M00ga   BC DEF @`* J    BCDEF  @`J B  3 *   6 b(  M00ga   BC DEF @`* J    BCDEF  @`J B  3 *    6 1y(  M10ga !  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BCDEF  @`J B / 3 &*  0 6h l(  M10ga 1 6`   M20g a 2 6r  #  M70ga 3 6w   M20g a 4 6t   M60ga 5 6@{ r   M20g a 6 6    M50ga 7 6܉ A {  M20g a 8 6 w   M40ga 9 6  ;  M20g a : 6D 8 ^  M30ga ; 6x    M20g a < 6  :  M20ga = 6ħ/    M20g a > 6` /    M10ga ? 6 / |   M20g a @ 64/    M00ga A 6|/ jp O02g c  B 6/ p O12g c  C 6/ p O22g c  D 6+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =-o6Bdissolve*<3<*ND' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-o6Bdissolve*<3<*1D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =-o6Bdissolve*<3<*O+`1  0 ,,BW,(  ~  s */  `  /    c $/ 0@b /  ,VH"#$~(F > ^/  B  j h C s *X99?"`> ^/  D  BC DEF @`~ * q J  E  BCDEF  @`U q J B F 3 ~ U *  G 6   (  M10ga H  BC DEF @`M * 8J  I  BCDEF  @`8J B J 3 M *  K 6|   (  M00ga L  BC DEF @`* J  M  BCDEF  @`J B N 3 *  O 6 b(  M00ga P  BC DEF @`* J  Q  BCDEF  @`J B R 3 *  S 6/ 1y(  M10ga T  BC DEF @`* J  U  BCDEF  @`J B V 3 *  W 6 / F(  M10ga X  BC DEF @`* rJ  Y  BCDEF  @`XrJ B Z 3 X*  [ 64&/ (  M10ga \  BC DEF @`X* AJ  ]  BCDEF  @`&AJ B ^ 3 X&*  _ 6+/ (  M00ga `  BC DEF @`&* J  a  BCDEF  @`J B b 3 &*  c 61/ l(  M10ga d 6T-/   M20g a e 6R/  #  M70ga f 6L/   M20g a g 6/   M60ga h 6e/ r   M20g a i 6i/    M50ga j 6\/ A {  M20g a k 6m/ w   M40ga l 6G/  ;  M20g a m 6>/ 8 ^  M30ga n 65/    M20g a o 6+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+  0  w (  ~  s *M3  `    ~  s *M3  ` 3  48 @D   r  BA?D r  BA?@  <3 J w ,$ 0 T37 = (100101)2*0(2  l <  )< ,$D 0`B   0DjJ< ))  03  Kleast significant bit(2l <   h < O ,$D 0`B  0DjJ< h h   0h3    Jmost significant bit(2Bl  d  R,$D 0   603  } d Pstop when quotient is zero(2  c BCDE FAjJm@S"  H  0޽h ? f3f; 3 ___PPT10 +#D '  = @B Dz ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+0 +/  0 F>` `(  `r ` S 3  `  3  r ` S `i3  ` 3  8 RA `RA ` B ,RT f,Binary, Decimal, and Hexadecimal Equivalents-0(2-n ` S 0APicture1"`AH ` 0޽h ? 33___PPT10i.pIpݭ+D='  = @B +H  0 x(  ~  s *  `     < 0P  FEach hexadecimal digit corresponds to 4 binary bits Example: Convert the 32-bit binary number to hexadecimal 1110 1011 0001 0110 1010 0111 1001 0100 Solution:x= vFZ vF vF4B;(c B UU   ZL jJ"`C }e ,$D 0 >0100 2c  Z0 jJ"`! }C ,$D 0 ;4 2c   ZK jJ"`C \|e ,$D 0 >1001 2c  Zx jJ"`! \|C ,$D 0 ;9 2c    Z jJ"`C < \e ,$D 0 >0111 2c   Z\ jJ"`! < \C ,$D 0 ;7 2c    ZT jJ"`C  < e ,$D 0 >1010 2c   Z jJ"`!  < C ,$D  0 ;A 2c    Z jJ"`C  e ,$D  0 >0110 2c  Z5 jJ"`!  C ,$D  0 ;6 2c   ZDejJ"`C e ,$D  0 >0001 2c  ZejJ"`! C ,$D  0 ;1 2c   ZpejJ"`D f ,$D 0 >1011 2c  ZejJ"`" D ,$D 0 ;B 2c   Z ejJ"`C e ,$D 0 >1110 2c  ZejJ"`! C ,$D 0 ;E 2cH  0޽h ? f3fJ4B4___PPT10"4+%D0'  = @B DA0' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +y  0 0(  ~  s *p8e0P> e r  S H9e `  e H  0޽h ? f3fy___PPT10Y+D='  = @B +  0 um(  ~  s *?e `  e j  BA?5.   <Ae A,$ 0 WDecimal 422 = 1A6 hexadecimal0(2l      ,$D 0  0Je :   Pstop when quotient is zero(2`B  0DjJ l  YP@  Y P@,$D 0`B  0DjJ \  0MeYP@ Mleast significant digit(2l  z Pa   z Pa ,$D 0`B   0DjJ \    0,Tez Pa  Lmost significant digit(2   0We  ` aRepeatedly divide the decimal integer by 16 Each remainder is a hex digit in the translated valueb <bH  0޽h ? f3f; 3 ___PPT10 +xbD '  = @B Dz ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+0 +{&  0 ia(v (  ~  s *ce `  e   <4eeC"?h Y,$@ 0 What is the largest 20-bit unsigned integer? Answer: 220  1 = 1,048,575>I0 26/t r  g #"&SY  = <de?6   z0 to (264  1), #? ; <De? 6  g0 to 18,446,744,073,709,551,615  #? 9 <pe?   S Double Word  #? # <\we?6$   z0 to (232  1), #? " <e?$ 6  Z0 to 4,294,967,295#? ! <Xe?$   LWord#?   <e?6> $  z0 to (216  1), #?  <e?> 6$  S 0 to 65,535  #?  <e?> $  Q Half Word  #?  <De?6X>  x0 to (28  1), #?  <e?X6>  P0 to 255  #?  < ne?X>  LByte#?  <pe?6rX S Powers of 2  #?  <i?r6X VUnsigned Range#?  <i?rX T Storage Type  #?`B $ 0jJ ?rrZB % s *jJ ?XXZB & s *jJ ?> > ZB ' s *jJ ?$ $ `B ( 0jJ ?  `B ) 0jJ ?r ZB * s *jJ ?r ZB + s *jJ ?6r6 `B , 0jJ ?r ZB : s *jJ ?  m8  V u Tn   0X99?"` V   <i"`  4 LByte,F B   Hi"`  / G8*F B  o BGi"` x  H16*F B  p Bi"` R2 H32*F B  q BTi"` ,o H64*F B  r 6@H "` s  Q Half Word,  F B  s 68H "` M  LWord,F B  t 64bH "` '  S Double Word,  F B   H0 H "`z`   G Storage Sizes0(2H  0޽h ? f3f  ___PPT10 +٬5D> '  = @B D' = @BA?%,( < +O%,( < +D8' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-o6Bdissolve*<3<*-D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-I%(D' =-o6Bdissolve*<3<*-I+8+0+0 +$[  0 ,,JO@/,(  ~  s *|ne `  e ~  s *Te `r e f  0X99?"`58 &   L&  H  C ?l  F   <e "  A0$GH   C  l F    <$ec "  A0$GH   C  l F    <`e<  "  A0$GH   C  l F   <ge X "  A1$GH  C  l | F   <C= 1 "  A1$GH  C | l U F   < <=a   E0(G  ? <= a  E1(G  @ <ܷ= a  E2(G  A <= a 0  E3(G  B <= a J  E4(G  C <=a /  Q bit position:( G  D <=< a q  E5(G  E <=c a  E6(G  F < =a   E7(G  H <=B w  ,$  0 E1(G  I <=h  ,$  0 E1(G  J < = Q  ,$  0 E1(G  * <=E  ,$ 0 A0$G , <=E c ,$ 0 A1$G . <~=E <  ,$  0 A0$G 0 <\+bE  X ,$  0 A1$G 2 </bE 1 ,$ 0 A0$G 4 <D4bE ,$ 0 A0$G 6 <8bE ,$ 0 A1$G 8 <,=bE y ,$ 0 A1$GH  0޽h ? f3f. .___PPT10-+&yD*'  = @B D|*' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-o6Bdissolve*<3<*:D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-o6Bdissolve*<3<*;D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bdissolve*<3<*8D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bdissolve*<3<*6D+' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-o6Bdissolve*<3<*4D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =-o6Bdissolve*<3<*OD8' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bdissolve*<3<*2D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =-o6Bdissolve*<3<*JD8' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bdissolve*<3<*0D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*HD8' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bdissolve*<3<*.D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =-o6Bdissolve*<3<*ID' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-o6Bdissolve*<3<**D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<++0+:0 ++0+;0 ++0+<0 ++0+H0 ++0+I0 ++0+J0 ++0+*0 ++0+,0 ++0+.0 ++0+00 ++0+20 ++0+40 ++0+60 ++0+80 +{>  0   *3@p (  ~  s *_b `    ~  s *p`b0uH  b  ! TLcb"` rp,$  0 AA$0 c  " Tgb"` pn,$  0 AF$0 c  # Tdkb"` nl,$ 0 AC$0 c  $ TN ` NPp   I k   N@ "`` N^p A9$0 c   N,@ "`^N\p A3$0 c   N@ "`\NZp A9$0 c   NT@ "`ZNXp A5$0 c   N @ "`XNVp AE$0 c   N̽@ "`VNTp A8$0 c   Nx@ "`TNRp A4$0 c   Nh@ "`RNPp AB$0 c  + N0@ "`  A+$0 c p   B$@ S w>,$D 0 |A + B = 10 + 11 = 21 Since 21 e" 16 Sum = 21  16 = 5 Carry = 12?0   Bl .  3 . ,$D 0 ( T@ "`d b  A5$0 c  @ . A  0. A   T(@ "` A  A1$0 2c - N @ "`.   Fcarry:$0 2c`B .B 0Db @H  0޽h ? f3f+#___PPT10+HD'  = @B DR' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-o6Bdissolve*<3<*3D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bdissolve*<3<*'D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-o6Bdissolve*<3<*1D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bdissolve*<3<*2D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-o6Bdissolve*<3<*#D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-o6Bdissolve*<3<*"D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-o6Bdissolve*<3<*!+P+0+!0 ++0+"0 ++0+#0 ++0+$0 ++0+'0 ++0+ 0 +}  0 p $(  r  S x&@  `  @  r  S P'@  `x @  H  0޽h ? 33___PPT10i.)+D='  = @B +SG  0 jFbF gE(  r  S -@  `  @  % E 18  #"B6 [981] P @  i <;@  ?"`68/  N+126#? g <`D@  ?"`@6/  M126#? e <N@  ?"`E 6@/  R01111110  #? a <|?@  ?"`D8= L+2#? _ <_@  ?"`@D= K2#? ] <\i@  ?"`E D@= R00000010  #? Z <Z@ ?=86 O. . .#? X <z@ ?@=6 O. . .#? V <@ ?E =@6 O. . .#? L <v@ ? 8 L-1#? J <0 <gZ? 8  O. . .#? < <=Z?@   O. . .#? : <D(Z?E  @  O. . .#? 7 <p1Z?! 8  N-127#? 5 <$Z?@!   M129#? 3 <Z?E ! @  R10000001  #? 0 <lZ?( 8!  N-128#? . <DZ?@( !  M128#? , <Z?E ( @!  R10000000  #? ) <(Z?/ 8(  N+127#? ' <Z?@/ (  M127#? % <Z?E / @(  R01111111  #?   <djZ?q8D L+1#?   <hsZ?@qD K1#?   <؝Z?E q@D R00000001  #?   <Z?8q K0#?   <?@q K0#?  < ?E @q R00000000  #?  <?18 X Signed value ( #?  <D ?@1 ZUnsigned value(#?  <?E 1@ ^8-bit Binary value(#?`B  0o ?E 181ZB  s *1 ?E 8ZB  s *1 ?E q8qZB  s *1 ?E D8D`B  0o ?E 8ZB  s *1 ?@1@ZB  s *1 ?1ZB - s *Ԕ ?E ( 8( ZB 4 s *1 ?E ! 8! ZB ; s *1 ?E  8 ZB B s *1 ?E  8 ZB I s *1 ?E 8 ZB W s *1 ?E 686ZB ^ s *1 ?E =8=ZB f s *1 ?E / 8/ ZB  s *o ?E / E ! `B  0o ?E 1E / ZB  s *o ?8/ 8! `B  0o ?818/ `B  0o ?E ! E `B  0o ?8! 8P r <, "` <  Positive numbers Signed value = Unsigned value Negative numbers Signed value = Unsigned value  2n n = number of bits Negative weight for MSB Another way to obtain the signed value is to assign a negative weight to most-significant bit = -128 + 32 + 16 + 4 = -766_!{  8 , L s  ]  @n  0X99?"`, L s    BC DEF @`ln _    BCDEF  @`C _ H  C l Cn   <0= l  A1$G   BC DEF @`;n &    BCDEF  @`  & H  C ; n   <C l  A0$G   BC DEF @` n     BCDEF  @`  H  C   n   <HP l  A1$G   BC DEF @`n     BCDEF  @`  H  C  n   < O bl  A1$G   BC DEF @`n     BCDEF  @`w  H  C  wn   <(U /l  A0$G   BC DEF @`wn `    BCDEF  @`F ` H  C w Fn   <D[ l  A1$G   BC DEF @`Fn /    BCDEF  @` / H  C F  n   <`a l  A0$G   BC DEF @` n    BCDEF  @` H  C  n   <$pZ l  A0$G  <Ptk *C  D-128$G   <w} C  B64$G   <}K C  B32$G   <䁀 C  B16$G   <H $C  A8$G   < C  A4$G   < C  A2$G   <Dj C  A1$G H  0޽h ? 33___PPT10i.A%+D='  = @B +e/  0 =5@6(  ~  s *,@  `  @    <r@  ,$ 0 0Sum of an integer and its 2's complement must be zero: 00100100 + 11011100 = 00000000 (8-bit sum) Ignore Carry4q0(27+  BZjJ T >,$D 0 Another way to obtain the 2's complement: Start at the least significant 1 Leave all the 0s to its right unchanged Complement all the bits to its left4K0 2L0   tRH  - #"">=>=0P @  # ZdZ_ж_ж?@R \00100100 = +36a#? ! ZZ_ж_ж?tR@ Tstarting value#?  ZZ_ж_ж?@ V11011011  a#?  Z _ж_ж?t@ n(step1: reverse the bits (1's complement)))#?  Z`_ж_ж?@ H  \11011100 = -36a#?   Zۀ_ж_ж?t @H  i#sum = 2's complement representation$$#?   Z_ж_ж?@  V+ 1  a#?  Z_ж_ж?t@  l&step 2: add 1 to the value from step 1''#?`B  0o ?tRRZB  s *1 ?t  `B  0o ?tH H `B  0o ?tRtH ZB  s *1 ?@R@H `B  0o ?RH ZB  s *1 ?tZB " s *1 ?t . <LjJ :P?,$@ 0 )5Binary Value = 00100 1 00 2's Complement = 11011 1 006 2ccg g cgggg c " 2 6jJ`),$D  0l ( ,  6 (, ,$D 0f" 1 6jJ(   4 N<"` ,  Kleast significant 1  5  BkC$DE FAjJk%$@S"  0 H  0޽h ? f3f___PPT10+9D'  = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D(' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-o6Bdissolve*<3<*.D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*. %(D' =-o6Bdissolve*<3<*. D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*. %(D' =-o6Bdissolve*<3<*. D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bdissolve*<3<*6Ds' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.)%(D' =-o6Bdissolve*<3<*.)D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.)6%(D' =-o6Bdissolve*<3<*.)6D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bdissolve*<3<*2++0+0 ++0+0 ++0+.0 +2  0 110KPV1(  ~  s *Z `  Z   c $T 00 Z *?@ABd  <@) T,T For Hexadecimal Numbers, check most significant digit If highest digit is > 7, then value is negative Examples: 8A and C5 are negative bytes B1C42A00 is a negative word (32-bit signed integer)0 2.8 x  P/ n  0X99?"`x    B>C?DEF@?>? @`* i    B@C=DEF@=@?@= @` , iH   C , *   <., gj  A1$G    B>C?DEF@?>? @` * i    B@C=DEF@=@?@= @` , iH  C  , *  <D3* gh  A1$G   B>C?DEF@?>? @` * i   B@C=DEF@=@?@= @` , iH  C  , *  <`9) gg  A1$G   B>C?DEF@?>? @` * i   B@C=DEF@=@?@= @` , iH  C  , *  <|?' ge  A1$G   B>C?DEF@?>? @` *i   B@C=DEF@=@?@= @` ,iH  C  , *  <E% gc  A0$G   B>C?DEF@?>? @` *i   B@C=DEF@=@?@= @`,iH  C  ,*  <K#ga A1$G    B>C?DEF@?>? @`*i !  B@C=DEF@=@?@= @`,iH " C ,* # <Q!g_ A1$G $  B>C?DEF@?>? @`*i %  BAC=DEFA=A?A= @`,iH & C ,* ' <Wg] A0$G (  B>C?DEF@?>? @`' f  )  B@C=DEF@=@>@= @` ) f H * C ) '  + <^, dj  A0$G ,  B>C?DEF@?>? @` ' f  -  B@C=DEF@=@>@= @` ) f H . C  ) '  / <$d* dh  A0$G 0  B>C?DEF@?>? @` '  f  1  B@C=DEF@=@>@= @` ) f H 2 C  ) '  3 <@j) dg  A0$G 4  B>C?DEF@?>? @` '  f  5  B@C=DEF@=@>@= @` ) f H 6 C  ) '  7 <\p' de  A0$G 8  B>C?DEF@?>? @` ' f  9  B@C=DEF@=@>@= @` )f H : C  ) '  ; <xv% dc  A1$G <  B>C?DEF@?>? @` ' f  =  B@C=DEF@=@>@= @`)f H > C  )'  ? <|#da A0$G @  B>C?DEF@?>? @`' f  A  B@C=DEF@=@>@= @`)f H B C )'  C <!d_ A1$G D  B>C?DEF@?>? @`' f  E  BAC=DEFA=A>A= @`)f H F C )'  G <̈d] A0$G J 6XFg v FSign bit" D@ K <9 HNegative$ G L <Xc  HPositive$ GZB N s *DD D (ZB O s *DD D +H  0޽h ? f3fy___PPT10Y+D='  = @B +u   0   P  (  r  S T  `    r  S p  `x   _8 0H   P   B "` HH  H10110011 = -77c  H "`  0H  Q11111111 10110011 = -77cl2  <jJ %  l"  <jJ % 2   # BޢENG HôIPJQjJ 0`Tޢ0`TޢPޢP  B `   0J  UF 0H     P   B "` HH  H01100010 = +98c  Bh "`  0H  Q00000000 01100010 = +98cf2  6jJ %  f"  6jJ % 2   BޢENG HôIPJQjJ 0`Tޢ0`TޢPޢP  B `  0J  H  0޽h ?/   33___PPT10i.G+D='  = @B +}  0  $(  r  S t `   r  S L `/  H  0޽h ? 33___PPT10i.+D='  = @B +   0   PG (  ~  s * `     s * `x  "4& ^B  6D Q ^B  6D @    B,vUL 2    B,ƒP o p 1+d   <G-H  K   6ܿx 7borrow:   6ăD   6carry:   B˃"`$ 31  Bσ"`K 31  B̓"`* 31  Bփ"`  31  BTڃ"``  31  Bރ"`  31  B"` d  31H  0޽h ? f3fy___PPT10Y+D='  = @B +  0 DD Su 0 D(   ~   s * `      s * TPc<$@ 0  jB   BDjJ**!   T"`_T,$ 0 [Borrow:80 2Fg   T"`vP G-*0(2g   6f O,$D 0X8 o w / , '  N`"`f  w G5*0(2g (  N`"`  w G7*0(2g )  Nl`"`D  w G6*0(2g *  N`"`h w GC*0(2g +  N4`"`"w GF*0(2g ,  N`"`Fw G4*0(2g -  N`"`w G1*0(2g .  N `"`o$w GB*0(2gfF o w 0  - 1  N`"`f  w G7*0(2g 2  N `"`  w G4*0(2g 3  N\`"`D  w G2*0(2g 4  N`"`h w GA*0(2g 5  N`"`"w GE*0(2g 6  N@!`"`Fw G9*0(2g 7  N$`"`w G3*0(2g 8  N(`"`o$w G8*0(2g  :  Tx,`"`$p,$ 0 GE*0(2g  ;  T0`"`Hp,$ 0 G2*0(2g  <  T3`"`p,$  0 G4*0(2g  =  Td7`"`q&p,$  0 G2*0(2g  >  T;`"`p,$  0 G1*0(2g  ?  T>`"`Op,$  0 GB*0(2g  @  TPB`"`sp,$ 0 GD*0(2g  A  TE`"`-p,$ 0 G2*0(2g Nb  B #  D ,$@ 0rB  B BDT    H(`"`NbT E 16 + 5 = 21 08n2   C  TTN` "`T$,$ 0 G1*0(2g D  NR`"`TN,$  0 G1*0(2g E  NlV`"`TP,$ 0 G1*0(2g  \  TDZ` "`Yp,$ 0 GE*0(2g  a  T`` "` p,$ 0 GB*0(2g  b  T`]` "`> p,$ 0 GD*0(2gFl T7p n T7p,$D 0 ^  T\h` "`7p G4*0(2g e  Tl` "`T G1*0(2gFl Tp o Tp,$D 0 _  TDp` "`p G2*0(2g f  T  ZCarry:80 2Fg@ Tp m Tp ]  Th` "`p G2*0(2g j  N`"`T G1*0(2g@l  Tp p T p,$D 0 `  TP` "``p G1*0(2g k  NH`"` T^ G1*0(2g@l > Tb p q T> b p,$D 0 c  T` "` b p G2*0(2g l  N0`"`> T  G1*0(2gH   0޽h ? f3fQQ___PPT10{Q+aDM'  = @B DM' = @BA?%,( < +O%,( < +D8' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C %(D' =-o6Bdissolve*<3<*C DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*B %(D' =-o6Bdissolve*<3<*B D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* ;%(D' =-o6Bdissolve*<3<* ;D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* ;j%(D' =-o6Bdissolve*<3<* ;jD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*: %(D' =-o6Bdissolve*<3<*: D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*; %(D' =-o6Bdissolve*<3<*; D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*< %(D' =-o6Bdissolve*<3<*< D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*= %(D' =-o6Bdissolve*<3<*= D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*> %(D' =-o6Bdissolve*<3<*> D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =-o6Bdissolve*<3<*D D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*? %(D' =-o6Bdissolve*<3<*? D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*E %(D' =-o6Bdissolve*<3<*E D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@ %(D' =-o6Bdissolve*<3<*@ D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A %(D' =-o6Bdissolve*<3<*A D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*u %(D' =-o6Bdissolve*<3<*u D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\ %(D' =-o6Bdissolve*<3<*\ D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*r %(D' =-o6Bdissolve*<3<*r D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*n %(D' =-o6Bdissolve*<3<*n D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*o %(D' =-o6Bdissolve*<3<*o D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*p %(D' =-o6Bdissolve*<3<*p D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*a %(D' =-o6Bdissolve*<3<*a D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*b %(D' =-o6Bdissolve*<3<*b D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*q %(D' =-o6Bdissolve*<3<*q D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*i %(D' =-o6Bdissolve*<3<*i D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* j%(D' =-o6Bdissolve*<3<* j++0+ 0 ++0+: 0 ++0+; 0 ++0+< 0 ++0+= 0 ++0+> 0 ++0+? 0 ++0+@ 0 ++0+A 0 ++0+C 0 ++0+D 0 ++0+E 0 ++0+\ 0 ++0+a 0 ++0+b 0 ++0+i 0 +  0 .&p! (   ~   s *@ `   ,   <P0 For n-bit signed integers: Range is -2n 1 to (2n 1  1) Positive range: 0 to 2n 1  1 Negative range: -2n 1 to -1r0 2!        %   < x,,$D 0 KPractice: What is the range of signed values that may be stored in 20 bits?L0 2Lt xYP  !  #"&SxP (   Z_ж_ж?m P  " 263 to (263  1)B  #?J   Z_ж_ж?x m  p 9,223,372,036,854,775,808 to +9,223,372,036,854,775,80799#?   Z_ж_ж?x x  S Double Word  #?    <L?m P  " 231 to (231  1)B  #?   <?x m  @ 2,147,483,648 to +2,147,483,647!!#?   <&?x x  LWord#?    <2?m% P  " 215 to (215  1)B  #?   <|<?x% m  l$ 32,768 to +32,767#?   <4E?x% x  Q Half Word  #?   <\Q?m?P%   27 to (27  1)B  #?   <[?x?m%  ` 128 to +127  #?   <c?x?x%  LByte#?   <Pm?mYP? S Powers of 2  #?   <+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +8+0+ 0 +}  0  $(  r  S LR `   r  S  `x  H  0޽h ? 33___PPT10i.KV+D='  = @B +;  0 ݍՍ PQ(  "z  vul  T l 0 b,$D 0h U s *X99?"` vul B V 3 P @)  W 64 f  A0$GB X 3 ) @ Y 6t f  A1$GB Z 3 @ [ 6Mf A0$GB \ 3 @ ] 6(&fi A0$GB ^ 3 @ _ 6nfB A0$GB ` 3 @f a 6tf A0$GB b 3 f@? c 6yf A0$GB d 3 ?@ e 64~f A0$GB f 3 P ,)  g 6̂ R  A0$GB h 3 ) , i 6t R  A1$GB j 3 , k 64MR A0$GB l 3 , m 6&Ri A0$GB n 3 , o 6gRB A1$GB p 3 ,f q 6$R A1$GB r 3 f,? s 6R A1$GB t 3 ?, u 6 R A1$G<B v # 7 KL w 6t  r ?+"EB x 3 P p) J  y 6  &  A1$GB z 3 ) pJ  { 64t  &  A0$GB | 3 pJ  } 6!M&  A0$GB ~ 3 pJ   6&&i&  A0$GB  3 pJ   6+B&  A1$GB  3 pfJ   61&  A1$GB  3 fp?J   66&  A1$GB  3 ?pJ   68&  A1$G  6XAG :   0E"`1P, D79&G  0H"`1f, D64&G  0lM"`1,  L 143 (-113)&  G  0R 1:  ]Carry = 0 Overflow = 1(G  6 3 pJ  ? 6R&i&  A1$GB @ 3 pJ  A 6RB&  A0$GB B 3 pfJ  C 6R&  A1$GB D 3 fp?J  E 6R&  A1$GB F 3 ?pJ  G 6R&  A1$G H 6RG :  I 0((R"`1P, K 218 (-38)&  G J 00R"`1f, K 157 (-99)&  G K 0lkR"`1,&  E119&G L 0B2 1:  ]Carry = 1 Overflow = 1(G M 6D2G :  N 6H2G :  O 6M2(]  E1(G  P 6\L2N  31  Q 60Q2t u  <   R 6Y2   <   S 6]2   31 r  S _2 `  2    <^2 "` `  2 #l  vul  -d P# ,$D 0n  0X99?"` vul H  C P @)   <@c2 f  A1$GH  C ) @  <h2t f  A1$GH  C @  <(n2Mf A1$GH  C @  <o2&fi A1$GH  C @  <x2fB A1$GH  C @f  <x~2f A0$GH  C f@?  <2f A0$GH  C ?@  <82f A0$GH  C P ,)   <2 R  A0$GH  C ) ,  <2t R  A0$GH  C ,  <2MR A0$GH  C ,  <2&Ri A0$GH  C ,  <2RB A1$GH  C ,f  <2R A1$GH  C f,?  <\2R A1$GH  C ?,  <P2R A1$GBB  3 7 KL  <Ķ2  r ?+"EH  C P p) J   <2  &  A0$GH  C ) pJ   <2t  &  A0$GH  C pJ   <`2M&  A0$GH  C pJ   <2&i&  A0$GH  C pJ   <,2B&  A0$GH  C pfJ   <2&  A1$GH  C fp?J   <2&  A1$GH  C ?pJ   < 2&  A1$G  <2G :   62"`1P, D15&G  62"`1f, J248 (-8)& G  602"`1,&  C7&G  6x2 1:  ]Carry = 1 Overflow = 0(G  <T2G :   < 2G :   <#2(]  E1(G   <(2N  31   <`&2t   31   <802   31   <42   31 "z  vul   -0 # ,$D 0h  s *X99?"` vul B  3 P @)   672 f  A0$GB  3 ) @  6<2t f  A0$GB  3 @  6m"Mf A0$GB  3 @  6xr"&fi A0$GB  3 @  6v"fB A1$GB  3 @f  6x|"f A0$GB  3 f@?  6"f A0$GB  3 ?@  6"f A0$GB  3 P ,)   6P" R  A0$GB  3 ) ,  6"t R  A0$GB  3 ,  68"MR A0$GB  3 ,  6"&Ri A0$GB  3 ,  6"RB A1$GB  3 ,f  6T"R A1$GB  3 f,?  6"R A1$GB  3 ?,  6P"R A1$G<B  # 7 KL  6"  r ?+"EB  3 P p) J   6<"  &  A0$GB  3 ) pJ   6i"t  &  A0$GB  3 pJ   6"M&  A0$GB  3 pJ   6O"&i&  A1$GB  3 pJ   6"B&  A0$GB  3 pfJ   6"&  A1$GB  3 fp?J   6"&  A1$GB  3 ?pJ   6"&  A1$G  6 "G :   0""`1P, D15&G  0""`1f, C8&G  0""`1,&  D23&G  0d" 1:  ]Carry = 0 Overflow = 0(G  6#"G :   6&"G :   6`)"(]  E1(G   6."NO  <    6p3"t u  <    66"   <    69"   <  H  0޽h ? 33  ___PPT10 .n#+D '  = @B Dm ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-o6Bdissolve*<3<*TD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+y  0 YY <@Y(  < < <K"  `x<$D 0 " r < S xL" `  "    < T@""`8W,$ 0 Zmax = 2n 1* 2H2 l< T"`,$ 0 7min = 0 2l xr <xr,$D 0 W< Nd"`vr DCarry = 1 Addition( @ &  o<x m< N$"` = Numbers > max 2l n< <1w& l xxr <xxr,$D 0 X< N"`qr FBorrow = 1 Subtraction N &  p< xx q< NC"` = Numbers < min 2l r< <1w& l    <  ,$D  0 v< N H"` v  CPositive Overflow( N &  <    < NtL"` = Numbers > max 2l < <1w& l x   < x ,$D  0 u< N(P"` q  ANegative Overflow N &  < x   < N@T"` = Numbers < min 2l < <1w& 0 z< T["`,$  0 jmax = 2n-1 16 2H2H2,$l 0 , < 0,,$D 0 w< N b"`p   MFinite Set of Signed Integers 2"@ 0B , <0B ,!@ 0B , <0B ,lB y< <DjJ0 , `B {< 0D1  `B |< 0D1  `B }< 0D1  `B ~< 0D1k k `B < 0D1G G `B < 0D1  `B < 0D1i i `B < 0D1E E `B < 0D1! ! `B < 0D1  `B < 0D1B B `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1h h `B < 0D1 `B < 0D1 `B < 0D1e e `B < 0D1A A `B < 0D1  `B < 0D1c c `B < 0D1? ? `B < 0D1 `B < 0D1 `B < 0D1  `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1b b `B < 0D1> > `B < 0D1 `B < 0D1_ _ `B < 0D1; ; `B < 0D1  `B < 0D1 `B < 0D19 9 `B < 0D1  `B < 0D1 `B < 0D1 `B < 0D1 `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1^ ^ `B < 0D1  `B < 0D1\ \ `B < 0D18 8 `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0DjJB `B < 0D1  `B < 0D1  `B < 0D1[ [ `B < 0D17 7 `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1  `B < 0D1n n `B < 0D1J J `B < 0D1& & `B < 0D1$ $ `B < 0D1  `B < 0D1  `B < 0DjJpB p`B < 0DjJ g   < Tt~"` @  10 2# < Tp"`@,$ 0 ] min = -2n-16 2H2H2"8 0, <0, < N"`pr OFinite Set of Unsigned Integers 2 !@ 0, <0,lB < <DjJ0r,r`B < 0D1N`B < 0D1N`B < 0D1N`B < 0D1kNk`B < 0D1GNG`B < 0D1N`B < 0D1iNi`B < 0D1ENE`B < 0D1!N!`B < 0D1N`B < 0D1B NB `B < 0D1 N `B < 0D1N`B  < 0D1N`B !< 0D1N`B "< 0D1 N `B #< 0D1 N `B $< 0D1 N `B %< 0D1 N `B &< 0D1h Nh `B '< 0D1 N `B (< 0D1 N `B )< 0D1e Ne `B *< 0D1A NA `B +< 0D1 N `B ,< 0D1c Nc `B -< 0D1? N? `B .< 0D1 N `B /< 0D1 N `B 0< 0D1 N `B 1< 0D1 N `B 2< 0D1 N `B 3< 0D1 N `B 4< 0D1 N `B 5< 0D1 N `B 6< 0D1 N `B 7< 0D1 N `B 8< 0D1 N `B 9< 0D1b Nb `B :< 0D1> N> `B ;< 0D1 N `B << 0D1_ N_ `B =< 0D1; N; `B >< 0D1 N `B ?< 0D1 N `B @< 0D19N9`B A< 0D1N`B B< 0D1 N `B C< 0D1 N `B D< 0D1 N `B E< 0D1N`B F< 0D1N`B G< 0D1N`B H< 0D1^N^`B I< 0D1N`B J< 0D1\N\`B K< 0D18N8`B L< 0D1N`B M< 0D1N`B N< 0D1N`B O< 0D1N`B P< 0D1N`B Q< 0D1N`B R< 0DjJ`B S< 0D1N`B T< 0D1N`B U< 0D1[N[`B V< 0D17N7`B ^< 0D1N`B _< 0D1N`B `< 0D1N`B a< 0D1N`B c< 0D1N`B d< 0D1N`B e< 0D1nNn`B f< 0D1JNJ`B g< 0D1&N&`B h< 0D1$N$`B i< 0D1N`B j< 0D1N`B k< 0DjJppH < 0޽h ? ̙33___PPT10d+7%&Dh'  = @B D#' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l<%(D' =-o6Bdissolve*<3<*l<D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* <%(D' =-o6Bdissolve*<3<* <D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<,a%(D' =-o6Bdissolve*<3<*<,aD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*z<%(D' =-o6Bdissolve*<3<*z<D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<++0+ <0 ++0+l<0 ++0+z<0 ++0+<0 +  0 $<(  $~ $ s *8 `   ~ $ s *Ȼ Px  H $ 0޽h ? f3fy___PPT10Y+D='  = @B +{  0 zz !zz(  r  S $ `   x  `   #"."$%&%N&$ `    ZΓ?/ `  MDEL#?  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Z ! 0D ! `x Examples: ASCII code for space character = 20 (hex) = 32 (decimal) ASCII code for 'L' = 4C (hex) = 76 (decimal) ASCII code for 'a' = 61 (hex) = 97 (decimal)&  H  0޽h ? 33___PPT10i.Pf+D='  = @B +  0 @ J(  r  S H  `       6`"` `x  H  0޽h ? 33___PPT10i.P"+D='  = @B +r CL~WP+fzrT C;`b>!>^ pY")!&UJZ* OXl%1Oh+'0\ `h   Data RepresentationDr. Muhamed MudawarMuhamed Mudawar288Microsoft Office PowerPoint@" @ И@sG@g  L  y--$xx--'--$<<--'@BComic Sans MS-. $2 *!Data Representationa."System8-@"Arial-. 2 GBCOE 308.-@"Arial-. '2 P2Computer Architecture.-@"Arial-. '2 Z.Prof. Muhamed Mudawar.-@"Arial-. I2 f,College of Computer Sciences and Engineering.-@"Arial-. L2 n.King Fahd University of Petroleum and Minerals.-՜.+,0(    On-screen ShowKFUPMOG Q %ArialComic Sans MS WingdingsTimes New Roman Courier NewSymbol HelveticaDefault DesignData RepresentationPresentation OutlinePositional Number SystemsBinary NumbersConverting Binary to Decimal#Convert Unsigned Decimal to BinaryHexadecimal Integers!Converting Binary to Hexadecimal"Converting Hexadecimal to Decimal"Converting Decimal to HexadecimalInteger Storage SizesBinary AdditionHexadecimal AdditionSigned Integers Two's Complement RepresentationForming the Two's Complement Sign BitSign Extension"Two's Complement of a HexadecimalBinary SubtractionHexadecimal SubtractionRanges of Signed IntegersCarry and OverflowCarry and Overflow Examples#Range, Carry, Borrow, and OverflowCharacter StoragePrintable ASCII CodesControl CharactersShl  Fonts UsedDesign Template Slide Titles Custom Shows'_$&0Muhamed MudawarMuhamed Mudawar  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Pictures!Current UserSummaryInformation(PowerPoint Document(H&DocumentSummaryInformation8