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Representation is a convention defined for expressing the number. A number may have different representations, e.g. BCD, hexadecimal, 2 s complement, etc. 2 s complement: a kind of representation defined as follows ¡¤7P›P<PPPP7fÉþb fÉþ}b@bª  ó’=Ÿ¨ Excercisesª Ÿ \ What is 2 s complement representation of +3 ?óŠ8Ÿ 2 s Complement¡`ª  óŒ9Ÿ >Signed Integers(2 s Complement)¡ b(ª  Ÿª óƒ4Ÿ ,2 s Complement (Cont.)¡b$ª  Ÿ ¢Representing a negative number  x From the representation of x to that of  x ¡ˆ"P+PPP"b+fÉþbâbªQ  ó‘<Ÿ ŒWhy  X can be calculated from X by bit-wise complemeting X and add 1 ?¡GG(ª, Ÿ ² Hint : using x = +4 as an example How is +4 represented ? How to do -4 brute force from +4 s representation ? More cleverly ? (hint: negation of the most significant bit of +4 is what (in binary) ?) Try yourself !¡&#·#·óh"Ÿ¨#Fit a shorter number to longer bits¡$$b(ª $ Ÿ¨ùWhy is it necessary? Compare an integer with a long integer: typecasting Load a byte to a word To add an immediate field to a 32-bit number MIPS ALU only works with values in 32-bit registers How to deal with smaller sizes? What about larger sizes?¡l ‚x ‚4P9P`x`4`9 b  ª ú ó}1Ÿ¨#Fit a shorter number to longer bits¡$$b(ª $ Ÿ¨‹copy the sign bit into the other bits(sign extension) 0010 -> 0000 0010 1010 -> 1111 1010 Proof? (Hint ? 1x(2^(-7) + 111 1010 = ?)¡Da ‚+ ‚7`*a+`ª Œ óH Ÿ¨Addition & Subtraction ¡ `ª  Ÿ¨¨Addition: Just like in grade school 0110 + 0001 Substraction: a-b=a+(-b) Two's complement operations easy 0111-0110= 0111 +1010 01¡V n! n2 n&bc ccbb1bccbbfÌþgÉþª&=   ` ó„5Ÿ¨Addition & Subtraction (Cont.)¡ `ª  Ÿ¨uWait! There is one extra bit lost! 0111-0110= 0111 +1010 10001 Why the 1 can be omitted? Because¡# nK n n#bbccbbfÌþfÉþgÉþbfÌþbbª v ó†6Ÿ¨Addition & Subtraction (Cont.)¡ `ª  Ÿ Questions: Do you have an adder and a  subtractor ? What about unsigned numbers? (Another adder?) Advantage of using 2 s complement Sub can share the same logic as add Sign bit can be treated as a normal number bit in addition. These are the clever points!¡€  nW n" n} n `H`b"b} ` ª&'   Ð ói#Ÿ¨Overflow¡  `ª  Ÿ¨EAddition: 0111 1111 + 0110 +1000 ¡œA n n bccbbcbbª F ó�;Ÿ¨Overflow¡  `ª  Ÿ¨WAddition: 0111 1111 + 0110 + 1000 1101 10111¡œB n n bccbbcbbª X óI Ÿ¨Detecting Overflow¡ `ª  Ÿ¨8What about addition and subtraction of unsigned numbers?¡9 9`ª 9 óJ Ÿ¨"What should CPU do about overflow?¡# #b(ª # Ÿ¨5Ignore it? Don't always want to detect overflow Addu, addiu, subu (MIPS: do not generate a overflow) Generate a trap so that the programmer can try to deal with it? An exception (interrupt) occurs Control jumps to predefined address for exception Interrupted address is saved for possible resumption Add, sub¡ˆ  ZZ Z@ Z� Z bZb@b� b  c ªZ1      ô ók$Ÿ¨ Instructions¡ `ª  Ÿ¨­Comparison Unsigned numbers sltu: set on less than unsigned sltiu: set on less than immediate unsigned Signed numbers slt: set on less than slti: set on less than immediate ¡” ZZKZZ8Z ``K``6 `  b ªt    5     óŽ:Ÿ  Example (cont d)¡`ª  Ÿ¨ What are the values of $s0 and $s1? $s0= 1111 1111 1111 1111 1111 1111 1111 1111 $s1= 0000 0000 0000 0000 0000 0000 0000 0001 Answer: slt $t0, $s0, $s1 -- the result of $t0 is 1 if both are signed sltu $t1, $s0, $s1 -- the result of $t1 is 0 if both are unsigned ¡J„›%`.b`Êbªž .                             F  C ól%Ÿ¨Instructions (Cont.)¡`ª  Ÿ¨'Load/Store lb: load byte lbu: load byte unsigned Example: lb $s1, 100($s2) What is the values of $s1 When memory[$s2+100] = 0000 0000? When memory[$s2+100] = 0000 0001? When memory[$s2+100] = 1111 1111? 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 ðXŸ¨0110¡$ ª  ð¾ ð “ ð6€ г�‚ƒ„…¿¿ÿðç Ì Ä m  ðXŸ¨0101¡$ ª  ð¾ ð “ ð6€L׳�‚ƒ„…¿¿ÿðû  �  ðXŸ¨0100¡$ ª  ð¾ ð “ ð6€”á³�‚ƒ„…¿¿ÿðç´ ¬ m ðXŸ¨0010¡$ ª  ð¾ ð “ ð6€ç³�‚ƒ„…¿¿ÿð üô � ðXŸ¨0001¡$ ª  ð¼ ð “ ð6€ìë³�‚ƒ„…¿¿ÿð~³a 4 ðVŸ¨+0¡$ ª  ð¼ ð “ ð6€”ï³�‚ƒ„…¿¿ÿðD Ä ú ðVŸ¨+1¡$ ª  ð¼ ð “ ð6€¤ô³�‚ƒ„…¿¿ÿðaÿ ­  ðVŸ¨+2¡$ ª  ð¼ ð “ ð6€üù³�‚ƒ„…¿¿ÿð–U  L  ðVŸ¨+3¡$ ª  ð¼ ð “ ð6€Ì �‚ƒ„…¿¿ÿðð a  ¦  ðVŸ¨+4¡$ ª  ð¼ ð “ ð6€h �‚ƒ„…¿¿ÿð ÿ ­   ðVŸ¨+5¡$ ª  ð¼ ð “ ð6€  �‚ƒ„…¿¿ÿð / Ý Ò  ðVŸ¨+6¡$ ª  ð¼ ð “ ð6€ �‚ƒ„…¿¿ÿð Ž< Ê  ðVŸ¨+7¡$ ª  ð¼ ð “ ð6€ø �‚ƒ„…¿¿ÿð fîÖ  ðVŸ¨-8¡$ ª  ð¼ ð “ ð6€\  �‚ƒ„…¿¿ÿðA ßg÷  ðVŸ¨-7¡$ ª  ð¼ ð “ ð6€D �‚ƒ„…¿¿ÿðó êr©  ðVŸ¨-6¡$ ª  ð¼ ð “ ð6€T �‚ƒ„…¿¿ÿðÊ ”€  ðVŸ¨-5¡$ ª  ð¼ ð “ ð6€° �‚ƒ„…¿¿ÿðŠ”@  ðVŸ¨-4¡$ ª  ð¼ ð  “ ð6€H( �‚ƒ„…¿¿ÿðH¢þ ðVŸ¨-3¡$ ª  ð¼ ð! “ ð6€¸, �‚ƒ„…¿¿ÿð]ßg ðVŸ¨-2¡$ ª  ð¼ ð" “ ð6€P1 �‚ƒ„…¿¿ÿð~4 ðVŸ¨-1¡$ ª  𸢠ð# ƒ ð0€`6 ¿�ƒ¿Àÿð  P @ ðXŸ¨ (For 4 bits)¡  2 aª  ðߢ ð$ ƒ ð0€ì: ¿�ƒ¿Àÿð ÀŽ ðŸ¨1Reference: Katz: Contemporary Logic Design, p243¡2 22cª 2 ðŽ¢ ð% ƒ ð0€4? ¿�ƒ¿ÀÿðT Ÿ ç ð,$ñDÿÿÿ  0 ðúŸ¨ŽWhy 1111 and 0000 are adjacent? What is the meaning of bit 3? What is the sign of 0? What is the smallest number? 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ˆðY€ �ðà    ð žð¤ ðì c ð$€¨O ¿ÿ@ ˆðÏ›ÿ'ð<$ñ 0à   ð žð6 ð  ðÈó&-  ð=ì ð#"ñŸ Ã(ˆð@»›z ð,$ñD 0ð` ðì £ ð<€P ¿�ƒ¿Àÿ?ð©9&¥ ð쟠 bn-1 bn-2 & & b1b0¡†b j çÿb j çÿb j çÿb j çÿª  ¦ø @`€ðÕ ðì £ ð<€àQ ¿�ƒ¿Àÿ?ðÈ9©¥ ðaŸ¨x¡`ª  ¦ø @`€ð¦ ðì £ ð<€ w ¿�ƒ¿Àÿ?ð©¥&-  ð2Ÿ &b¢ðn-1b¢ðn-2 & & b¢ð1b¢ð0¡Æb â j çÿb â j çÿb â j çÿb â j çÿª  ¦ø @`€ðÖ ðì £ ð<€\j ¿�ƒ¿Àÿ?ðÈ¥©-  ðbŸ¨-x¡`ª  ¦ø @`€ðâ ðì £ ð<€LT ¿�ƒ¿Àÿ?ð©ó&9 ðnŸ¨Representation¡`ª  ¦ø @`€ðÛ ðì £ ð<€`“ ¿�ƒ¿Àÿ?ðÈó©9 ðgŸ¨Number ¡`ª  ¦ø @`€ðfB ðì “ ð6…¿ÀËŸo×ÿ ?¿ðÈó&óð`B ðì ƒ ð0…¿ÀËœ1ÿ ?¿ðÈ9&9ðfB ðì “ ð6…¿ÀËŸo×ÿ ?¿ðÈ- &- ðfB ðì “ ð6…¿ÀËŸo×ÿ ?¿ðÈóÈ- ð`B ðì ƒ ð0…¿ÀËœ1ÿ ?¿ð©ó©- ðfB ðì “ ð6…¿ÀËŸo×ÿ ?¿ð&ó&- ð`B ðì ƒ ð0…¿ÀËœ1ÿ ?¿ðÈ¥&¥ðŒÒ ð-ì ƒ ð0…‡�ƒ¿Àÿð pð,$ñD 0ðŒr ð1ì ƒ ð0…‡�ƒ¿ÀÿðU]äê ð,$ñD 0ðŒ" ð3ì ƒ ð0…‡�ƒ¿Àÿð� ªì ð,$ñD 0ð*¢ ð5ì ƒ ð0€˜— ¿�ƒ¿Àÿð Ð ð,$ñ 0 𖟨Invert each bi, then add 1¡L 2 gÉþoÉþçÿ gÉþª  ð” ð?ì “ ðú€¸´ �ƒ¿Àÿ �ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelð~ ¶€®ð,$ñ  0 ð6Ÿ 4Shortcut: Invert every bit and add 1 to the least significant bit Example: - (0100) = 1011 + 1 = 1100 Example: 10010 = 01100100 àð 10011100 = -10010¡¼Cÿ0n<PXþ0n<Paqaiçÿ aáaa aiçÿª › ¦øØÔÐðð ð@ì “ ðú€ » �ƒ¿Àÿ �ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelð� ›fÑ ð,$ñ  0 𿟨Proof of correctness: ¡Vÿ0n<Pÿ0n<Paiª  ¦øØÔÐððH ðì ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ@EŒ·ÁëffìØ‚²²²o‰÷ÏÛýˆ£Š›º___PPT10‹{ê.ë.'ˆÃ ©¢+•“›VDñc'ñ Æ ÿÿÿÿ=ñ @Bñ Dñ'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ì"%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ì"M%ñ(ñóDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*=ìÿÿÿÿÿÿÿÿ%ñ(ñóDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*-ìÿÿÿÿÿÿÿÿ%ñ(ñóDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*1ìÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*5ìÿÿÿÿÿÿÿÿ%ñ(ñóDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*3ìÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*@ì%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*?ìÿÿÿÿÿÿÿÿ%ñ(ñó+à+0+ì0 ++0+5ì0 ++0+?ì0 ++0+@ì0 +î}ï € 0 ”ðŒ ð0ð$ð( ð ð0ðr ð0 S ð€¼è ¿ÿ? ð�ð`ðà    ð žðr ð0 S ð€”é ¿ÿ@ ðÀ˜¸àðà   ð žðH ð0 ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ@EŒ·ÁëffìØ‚²²²o‰÷ÏÛýˆ‘Љº___PPT10‹ië.œÀÅp£> +Dñ='ñ Æ ÿÿÿÿ=ñ @Bñ +î ï € 0 Àð¸àð”ðPð( ð ð”ðr ð” S ð€œÝ ¿ÿ? ðÉ€À™ðà    ð žðž ð” S ð€œÞ ¿ÿ@ ðÐð<$ñ 0à   ð žðH ð” ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ@EŒ·ÁëffìØ‚²²²o‰÷ÏÛýˆòŠêº___PPT10‹Êê.ë.ŒÃÀóØ+¾c‹DñZ'ñ Æ ÿÿÿÿ=ñ @Bñ Dñ'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”Dñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”I%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”IDñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”I_%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”I_Dñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”_�%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”_�Dñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”�Á%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”�ÁDñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”Áá%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”ÁáDñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*”áú%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*”áú+8+0+”0 +îï € 0 ðÐðàð§ð( ð ðàðx ðà c ð$€| ' ¿ÿ? ˆðÉ€À™ðà  '  ð žð¤ ðà c ð$€h ' ¿ÿ@ ˆð¿ƒ{Éð<$ñ  0à '  ð žðC ðà “ ðú€€' �ƒ¿Àÿ �ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelð~h`�ðDˆ<Š4º___PPT9‹¬€ ðÍŸ¨sExample(2-bit): 10= 1 *(-21)+0 *20=-210 Representing -210 in 4-bit word: Can you do that? What is the intuition?¡(ÿ0 wn‚Lþ0 wn‚c k çÿc k c k çÿc k c k çÿckçÿ8cª t ¦øØÔÐððH ðà ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ@EŒ·ÁëffìØ‚²²²o‰÷ÏÛýˆ¤ Šœ º___PPT10‹| ê.ë.ŒÃÀóØ+³Ø�Dñ 'ñ ' ÿÿÿÿ=ñ @Bñ DñÇ 'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*àa%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*àaDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*àaDñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*àaŒ%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*àaŒDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*àaŒ+8+0+à0 +î3*ï € 0ùпL0 ã ðÛ Àð ðs ð( ð ðð^ ð “ ð6…‡�ƒ¿Àÿ?ðÅŽèòðæ" ð  ðf€8' �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆð~SçŸ ð<$ñ 0à '  ð žð´ ð  ð`€ü' �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  '  ð žðx ð  ð$€lÿ �xa‚¢­ƒxa„¢­¿G‡ �ƒ¿Àÿ ?�ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelðè /Þð,$ñDÿÿÿ 0 ððŸ¨<So 0111-0110= 0001 Have you specially handled the sign bits?¡n=ÿ0 n<na cgÉþeÉþ*aª = ¦øØÔÐððå¢ ð ƒ ð0€84' ¿�ƒ¿Àÿð@  ó ð,$ñDÿÿÿ 0 ðQŸ¨1¡  2fÉþª  ðå¢ ð ƒ ð0€„1' ¿�ƒ¿Àÿð7 ¦~ ÷ ð,$ñDÿÿÿ 0 ðQŸ¨1¡  2fÉþª  ðå¢ ð ƒ ð0€@B' ¿�ƒ¿ÀÿðÐ y� ð ð,$ñDÿÿÿ 0 ðQŸ¨0¡  2eÉþª  ðå¢ ð  ƒ ð0€¼5' ¿�ƒ¿ÀÿðÐ Ï ð ð,$ñDÿÿÿ 0 ðQŸ¨0¡  2eÉþª  ð÷¢ ð  ƒ ð0€8J' ¿�ƒ¿ÀÿðÙ û ù ð,$ñDÿÿÿ 0 ðcŸ¨1¡2 2eÌþeÉþª  ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆàŠØº___PPT10‹¸ê.ë.$ˆÃ`õ½+îoZDñø'ñ Æ ÿÿÿÿ=ñ @Bñ Dñ³'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*=%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*=V%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Vw%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*wœ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*œ©%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ÿÿÿÿÿÿÿÿ%ñ(ñóDñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*Dñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*=%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*=+ˆ+0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 +î8(ï €  0ùпL0 fð^°ðôðöð( ð ðôð^ ðô “ ð6…‡�ƒ¿Àÿ?ðÅŽèòðº" ðô  ðf€èp' �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆð~Sçd ðà '  ð žð´ ðô  ð`€Dq' �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  '  ð žð· ðô  ð$€@‚' �xa‚¢­ƒxa„¢­¿G‡ �ƒ¿Àÿ ?�ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelðñ ðnkð,$ñDÿÿÿ 0 ð/Ÿ¨1-1=1+1=0 from the viewpoint of 1-bit. So as long as you do not consider the carry brought by 1+1 (the green 1), you are right!¡j€ÿ0 n<n aeÉþHaeÌþaª € ¦øØÔÐððå¢ ðô ƒ ð0€øw' ¿�ƒ¿Àÿð“ ó Sð,$ñÿÿÿ 0 ðQŸ¨1¡  2fÉþª  ðå¢ ðô ƒ ð0€¼…' ¿�ƒ¿ÀÿðЦ~ Jð,$ñÿÿÿ 0 ðQŸ¨1¡  2fÉþª  ðŒ2 ð ô ƒ ð0…‡�ƒ¿Àÿÿÿðõ­H #ð,$ñDÿÿÿ 0ðä¢ ð ô ƒ ð0€ø‰' ¿�ƒ¿Àÿð÷ 2  ð,$ñDÿÿÿ 0 ðPŸ¨is¡ 2dÉþª  ðô ð ô@ 3 𘅇B6CŽDEÁFÁ�ƒ†A¿ÀÿÿÄÑÔÕÿðÿ6O žŽ@­ ¬€ðî- ð,$ñDÿÿÿ 0ðÁ¢ ðô ƒ ð0€€Ž' ¿�ƒ¿Àÿðè îM o ð,$ñDÿÿÿ 0 ð-Ÿ¨% 0*23 + 1*(-23) 1*23 0¡Ì&bjbfÉþnÉþfÉþb jjnÉþfÉþª $ ðú ðô@ C ðž…‡B6CŽDEÁFÁ�ƒ†Á¿ÀÿÿÄÑÔÕÿˆðÿ6O žŽ@­ ¬€ð$ è ð,$ñDÿÿÿ 0ð¢ ðô ƒ ð0€\™' ¿�ƒ¿Àÿð Ì  ð,$ñDÿÿÿ 0 ðnŸ¨But treated in the addition as¡  2fÉþª  ðð¢ ðô ƒ ð0€0ž' ¿�ƒ¿Àÿð + #¥ ð,$ñDÿÿÿ 0 ð\Ÿ¨* 0*23 + 1*23 1*23 10¡ö+bjbfÉþnÉþfÉþb jjnÌþnÌþnÉþfÉþª ' ðH ðô ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆbŠZº___PPT10‹:ë.$ˆÃ`õ½+F[·Dñö'ñ Æ ÿÿÿÿ=ñ @Bñ Dñ±'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ôÿÿÿÿÿÿÿÿ%ñ(ñóDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ôÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ôÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ôÿÿÿÿÿÿÿÿ%ñ(ñóDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ôÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ôÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ôÿÿÿÿÿÿÿÿ%ñ(ñóDñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ô'%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*ô'Dñ'ñ =ñ%ñ(ñÿÿÿÿDñ¬'ñ =ñ%ñ(ñDñT'ñ =ñA@Bñ�Bñ Bñ °Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ô'€%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*ô'€++0+ô0 ++0+ ô0 ++0+ô0 ++0+ô0 ++0+ô0 +îS$ï €  0ùпL0 °ð¨ ðüð@ð( ð ðüð^ ðü “ ð6…‡�ƒ¿Àÿ?ðÅŽèòðæ" ðü  ðf€É' �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆð™Sç®ð<$ñ  0à '  ð žð´ ðü  ð`€ðÉ' �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  '  ð žðH ðü ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆ3!Š+!º___PPT10‹ !ê.ë.$ˆÃ`õ½+³Ø�Dñ› 'ñ Ë' ÿÿÿÿ=ñ @Bñ DñV 'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ü %ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*ü Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*ü Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ü 4%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*ü 4Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*ü 4Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ü4b%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*ü4bDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*ü4bDñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*üb„%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*üb„Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*üb„Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ü„¨%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*ü„¨Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*ü„¨Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*ü¨ä%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*ü¨äDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*ü¨äDñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*üä%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*üäDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*üä+8+0+ü0 +îŸï €  0ùпL0 ð�ðœðžð( ð ðœð^ 𜠓 ð6…‡�ƒ¿Àÿ?ðÅŽèòðº" ðœ  ðf€Œ×' �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆð«n Óðà '  ð žð´ ðœ  ð`€dØ' �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  '  ð žð‚ ðœ  ð$€ Ú' �xa‚¢­ƒxa„¢­¿G‡ �ƒ¿Àÿ ?�ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelð›¶Tpðˆ$ñDÿÿÿ  0ˆTŠLº___PPT9‹.¬&€ 𞟠˜ What is the sign of the input? What is the sign of the output? Can overflow occur in adding a positive and a negative number? Can overflow occur with 0? Consider the operations A + B, and A  B¡´Jþ0 wnn?ÿ0 wnnÿ0 wnP)ÿ0 n<Pˆccc)aª È ¦øØÔÐððH 𜠃 ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆ!Šº___PPT10‹ùë.$ˆÃ`õ½+©w*àDñ•'ñ Ü' ÿÿÿÿ=ñ @Bñ DñP'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*œ%%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*œ%Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*œ%Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*œ%J%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*œ%JDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*œ%JDñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*œJ‰%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*œJ‰Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*œJ‰Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*œ‰¤%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*œ‰¤Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*œ‰¤Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*œ¤Í%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*œ¤ÍDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*œ¤Í+8+0+œ0 +îrï € 0ùпL0 çðßð(ðwð( ð ð(ð^ ð( “ ð6…‡�ƒ¿Àÿ?ðÅŽèòðº" ð(  ðf€hþ' �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆð«n Óðà '  ð žð´ ð(  ð`€@ÿ' �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  '  ð žð[ ð(  ð$€* �xa‚¢­ƒxa„¢­¿G‡ �ƒ¿Àÿ ?�ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelð›¶Tpð,$ñ  0 ðÓŸ¨G What is the sign of the input? What is the sign of the output?¡,Hþ0 wnnHcª&  B ¦øØÔÐððH ð( ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆŠº___PPT10‹óë.$ˆÃ`õ½+΃ÍDñ�'ñ * ÿÿÿÿ=ñ @Bñ DñJ'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*(%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*(Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*(Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*('%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*('Dñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*('Dñ{'ñ =ñ%ñ(ñÿÿÿÿDñ#'ñ =ñ%ñ(ñDñË'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*('H%ñ(ñDñ'ñ ô=ñ+ñÙ4ñ 8?ñ\CñBñ#ppt_xBñCñèBñ#ppt_xBñ*ñY3ñ>ñBñ ppt_x<ñû*('HDñ'ñ ô=ñ+ñá4ñ 8?ñdCñBñ1+#ppt_h/2BñCñèBñ#ppt_yBñ*ñY3ñ>ñBñ ppt_y<ñû*('H+8+0+(0 +îð ï € 0ùпL0 êðâ€ððzð( ð ðð^ ð “ ð6…‡�ƒ¿Àÿ?𵞥âðæ" ð  ðf€ˆ* �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆð– w sð<$ñ 0à *  ð žð´ ð  ð`€`* �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  *  ð žð2 ð  ð$€è* �xa‚¢­ƒxa„¢­¿G‡ �ƒ¿Àÿ ?�ÃÄRectangle: Click to edit Master text styles Second level Third level Fourth level Fifth levelðeù9 ðPˆHŠ@º___PPT9‹"¬€ ð†Ÿ¨ÔOverflow occurs when: add two positives yields a negative add two negatives gives a positive or, subtract a negative from a positive and get a negative or, subtract a positive from a negative and get a positive ¡lÿ0 wnZ¿ÿ0 n<Zc ½cc ª Õ ¦øØÔÐððH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆ–ŠŽº___PPT10‹nê.ë.$ˆÃ`õ½+ÖÖœDñþ'ñ Æ ÿÿÿÿ=ñ @Bñ Dñ¹'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*9%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû*9+8+0+0 +î~ï € 0ùпL0 °ð¨pð ð@ð( ð ð ð^ ð  “ ð6…‡�ƒ¿Àÿ?ðÅŽ•òðæ" ð   ðf€04* �xa‚¢­ƒxa„¢­…‡ˆ¿G‡ ¿ÿ ?@ ˆðê¤\ð<$ñ 0à *  ð žð´ ð   ð`€5* �xa‚¢­ƒxa„¢­…‡ˆ¿¿ÿ ?? ˆðÀ€ �ðà  *  ð žðH ð  ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿU>=UU(ˆ^ŠVº___PPT10‹6ê.ë.$ˆÃ`fÀ+ÖÖœDñÆ'ñ Æ ÿÿÿÿ=ñ @Bñ Dñ�'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*  %ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû*  Dñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*  1%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû*  1Dñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* 1f%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû* 1fDñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* f¦%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû* f¦Dñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ¦Æ%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû* ¦ÆDñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* Æø%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû* ÆøDñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* ø-%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû* ø-Dñð'ñ =ñ%ñ(ñÿÿÿÿDñ˜'ñ =ñ%ñ(ñDñ@'ñ =ñA@Bñ�Bñ  Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû* -6%ñ(ñDñ§'ñ ô=ñ-ño6ñBñdissolve*ñ<3ñ<ñû* -6+8+0+ 0 +îâï € 0 ð`ð¤ð¨ð( ð ð¤ðr ð¤ S ð€HK* ¿ÿ? 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