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C <$e2"H/ 0@Pp// 7 **& 9999PKKTBKKSB 2+KdRBKRXYKTXYKSXY++++++++sEEha#DsEEha#DEEha#DEh#Dt++tu++++u+++++++++++++s+++sssstsssss+ss+s+u+us+u+s+++++++++++++++++++s+stssPQn:e1375008:Tahoma BoldBNon:e1375008:Tahoma BoldR2S i@D# %5:IEJ / / 0@ !""" 7/+]3/993//]?9]?3/]10]!!56676654&#"#6$3 !Tmb|o9e43H? | |Z&wnXs\ho0 E?քYs*S*(V;9x@22::332LCD2 !@ 7@ 7+4!!(? ? 4@ 744;+p/]3/3/+9/?3/?3/9++10]"&'332654&'.'.5467>32#.'.#"|H@91{HJq4K(z/ECC|vDN%,m9Li4W-u6y}JDH6$(.!.2'( )K4332>32?9(N.?8+R'^V_g)e\N|+//1Sr$$!# 1Rs$$!(c|IRba^e226x%b¦SE 9@%44 4CC CB @  qn22????10]4&#"326%#"&'!!>32mfj/]-#G#}fT|F^Mi: &(8BRo¬SR4 #%@@ @ q%?%p$]??10! ! 4&'&&#"32676610!N2/L $"T3*W 1?=d(*!)'gg&'$$$+ylFSFDO%@ 70@]9/+??10!!!^TLk 2 simboli 4*N32n:e1371008:TahomaBNoӧLk(alfabeto binario)+*-+---+, BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      n:e1371008:TahomaR9 S C :f@I 818)%==5 MKE TX%Z8[9k8l9255'++/' 2!!!*!* !p/]33//]22?2/?9/910]4&'.#"3267>%467>32#"&'53327#"&'.@81,Z6{06&`7@7KA>\g=MUSQV(Z' j7PUSEPRm40$^z,-%1sEBJEAR˹mtz 62$4=TCLk9n:e1371008:TahomaBNoiLk" ӮLk%Le informazioni vengono rappresentate%(+.,D+$,-(+-.,-,*.-+$+-+n:e1371008:TahomaRq²ST;d| A@(;=0LKF@ B o"p!22???99?10]10##"467>3273.#"326R_OA>]V~B Gp@OdKE1KIP('0|#DTӮuLkmediante sequenze di simboliD+-+-+$+--+-%*-$D-,iLk" ӮLkNel caso dei simboli binari, le6+&*%,-+$D--..*ӮLLk&informazioni (numeri, oggetti, parole)&-,D*%,..-D+,--+-+,*ӮLk"sono rappresentate da sequenze dei"$,.,*--+%*.++-+$+-.+-$+-+ӮLk due simboli --+$D-,iLk" ӮLk#Servono regole di manipolazione dei#-+),.,+--+-D+--,*$,.+-+ӮLksimboli$D-,n:e1371008:TahomaBNoӫLkRappresentazioneE;>>(;2:?%;1=>[Lkdell informazione>;?#=(^;11&;^#>?^:)3n:e1371008:TahomaBNoi%Lk" Ӯ%Lk#Per determinare un sistema numerico#-+-+*D-++-.$$+D*-.D+%ӮLkserve:$+)+n:e1371008:TahomaŢBNoLk Lk#un insieme limitato di simboli (le #('(&<%;&&' <''&n:e1371008:TahomaŢBNoe">f??ed Lkcifre!n:e1371008:TahomaŢBNof??edӓLk), che!'ELk*rappresentano quantit prestabilite (1, 2,*%('% %(%(''(&'&'% %(&'&n:e1371008:TahomaRV¶S9@<7Wv7WvLFICVY@! 7$ ?$/]]]3/]+99??9910]]]#3/,XS; @< 69 Wfi     7 7 7@+ 7   h   @ 7  -  "/]3/]]+9??910.+}ć.+}++++ććć]!##335di.TI"TLkV, X, M)+*7Lk Lk!le regole per costruire i numeri:!&%''&'&!& '&((;&n:e1371008:TahomařqBNo ZLk" NZLksistemi numerici posizionali 3""3 "!!"  Lk" NLk sistemi numerici non posizionali  3""3 "!""!!" ?BSb,{?Lk MP,TGLk MP,TBSb c B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk11n:e1371008:TahomaBNo LkSistemi numerici>1&;^#>?^:)3n:e1371008:TahomaBNoi Lk" Ӯ Lk!Sistemi numerici non posizionali:!-$+D-.D+%.,--,$$,-+n:e1371008:TahomaŢBNo Lk  Lk'valore delle cifre indipendente dalla'$%'%(%%!&&('(%('&'&'&D Lk posizione ''  ''n:e1371008:TahomaBNoi Lk" Ӯ LkSistemi numerici posizionali:-$+D-.D+%.,$$,-+n:e1371008:TahomaŢBNo Lk  Lk(il valore delle cifre dipende dalla loro(#&'&'&&!&'(%('&'&&'uLk&posizione all interno del numero (ogni&''  ''&&(%(&(%'(<%'''(Lkposizione ha un peso)''  ''&(%'((% & BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk12n:e1371008:TahomaBNoӾLkSistemi numerici posizionali>1&;^#>?^:)3#><21<>;n:e1371008:TahomaBNoiLk" ӮLkEsempio:.$+D-,iLk" ӮLkSistemi a base fissa:-$+D*-+$+$$+n:e1371008:TahomaŢBNoLk n:e1371008:TahomaR=RSRP j#@ /3]@*,,-4:CIUZb VV -/3/]3/?2/10]#3#B蒰@qT[Lk ; V(N) = d $(,n:e1371008:TahomaŘ&BNo?[Lk1n:e1371008:TahomařqBNon:e1371008:TahomaR*S pv@lla a{{s sO@ 7@7 7  O       (  /]392/99/99]?3]3/]39/3/310++]]%#'-73%A}BbBF}HAwnlnnmnTU[Lk*p"n:e1371008:TahomaŘ&BNoӘ[Lk1n:e1371008:TahomařqBNon:e1371008:TahomaR+tStrC5 =@( 0 @  p* 0@P/]<]<]/]<<10!#!5!3!5<Tӯ[Lk + d,n:e1371008:TahomaŘ&BNo#[Lk2n:e1371008:TahomařqBNo9[Lk*p!n:e1371008:TahomaŘ&BNo|[Lk2n:e1371008:TahomařqBNoӒ[Lk + d-n:e1371008:TahomaŘ&BNo[Lk3n:e1371008:TahomařqBNo[Lk*p!n:e1371008:TahomaŘ&BNo`[Lk3n:e1371008:TahomařqBNov[Lk + d,n:e1371008:TahomaŘ&BNo[Lk4n:e1371008:TahomařqBNo[Lk*p"n:e1371008:TahomaŘ&BNoC[Lk4n:e1371008:TahomaCU7BNonLkiLki\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cB c,{n:e1371008:TahomaBNoӟLk5,{ c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk13n:e1371008:TahomaBNoӴWLkSistema decimale>1&;^:#>;3^;n:e1371008:TahomaBNoi%Lk" Ӯ%LkIl sistema decimale utilizza:$$+D+-+&E*+.$$+n:e1371008:TahomaŢBNoLk Lkr = 104'Lk Lkd = 0,1,2,3,4,5,6,7,8,9'3&''&''&''n:e1371008:TahomaBNoiLk" ӮLk! importante notare che qualsiasi!.D-,*.+.,++&-+.-+$+$Ӯ/Lk"sistema posizionale a base fissa "$$+E*-,$$,-++*-*%*$$+n:e1375008:Tahoma BoldBNon:e1375008:Tahoma BoldRr SUxgN@ 7 @ 7D  @ 7  n2]3/+???+2/910+#.#"!!>7>32xL.7y9^U#&e-* cC!d¶SG6#=@);; ;;#JL LL#A!B o% p$22????10]!!5#"4676632!&&#"326"W!*R9LC=aXqK^T~eq.^u@> PIU%'"' $n†SQ3@4 E  D o  n2?3 !Yn5.3US.`'$90/YPd^_>O4v" % A6O;*0Ș}-.$ ]G-#)t¶SWJX@  ? ?  @ 7 ?O  /]3]32/]3/+3?2/?323/10]"&5#53!!!326734Æ^QGD[9 Ar;X##) eSH4%\@!% 4 =AOA J%<`  ">`@ 7@  q''% p&2]3/?3/+]?9/]10] ! !32676673&&#"<8.-xG=v1+I%+G>9yadfs0H~Ni,otziTӮLk irridondante $#424305!?BSb,{?Lk MP,TGLk MP,TBSb c B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk14n:e1371008:TahomaBNo LkSistema binario>1&;^:#>>;(n:e1371008:TahomaBNoi Lk" Ӯ LkIl sistema binario utilizza:$$+D+--+,.%$*n:e1371008:TahomaŢBNo Lk  Lkr = 24V Lk V Lkd = 0,1'3&n:e1371008:TahomaBNoiLk" ӮLkOgni cifra detta 9.-&**-++n:e1375008:Tahoma BoldBNo8Lkbit 4"n:e1371008:TahomaBNoӾLk(da-4Lk n:e1375008:Tahoma BoldBNon:e1375008:Tahoma BoldRBS%A$3V@,24-4,?O%@ 755 5@5,d42]3/+9/]9??9/9910#!!24&'&&##3267664&'&&##326766ATGTɛJPOm`!%'%ed/`M)-"^88"wdEp:3/o7B8(+^m($LJEN UIrSrp,^ >@ 3 3 Z Z@ 7  0 @ ]9/+22?2?210)3#!#3"7TNLkBI8n:e1371008:TahomaBNon:e1371008:TahomaRyªS\d]u@=&9Yi6Vf)& @ 7 / /]]3/+9??910]]]]#3j10]';TӭLk nary digi .*)--n:e1375008:Tahoma BoldBNon:e1375008:Tahoma BoldRT^S^\71@5  [[p @ ]9/]]??210!!!!HHKTLkTn:e1371008:TahomaBNoLk) BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk15n:e1371008:TahomaBNokLkAltri sistemi utilizzatiC&(#12%;^#>&21;%n:e1371008:TahomaBNoiLk" ӮLkSistema-$+DLk ottale,*ӬLk:n:e1371008:TahomaŢBNoOLk OLkr = 84Lk Lkd = 0,1,2,3,4,5,6,7'3&''&''&n:e1371008:TahomaBNoiLk" ӮLkSistema esadecimale:-$+D*+$*-+&D*+n:e1371008:TahomaŢBNoLk Lkr = 164'lLk lLk#d = 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F#'3&''&''&''&***0(\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cB c,{n:e1371008:TahomaBNoӟLk6,{ c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk16n:e1371008:TahomaBNoӃWLkConversioni di baseC=>7;)1#>;2n:e1371008:TahomaBNoi%Lk" Ӯ%LkUtilizzando la definizione:5$$+--,*-*.%,.+n:e1371008:TahomaŢBNoLk Lk1010'&'n:e1371008:TahomaŘ&BNoәLk2n:e1371008:TahomaŢBNoӯLk = (1*8 + 0*4 + 1*2 + 0*1)4&''4''&3&''4'&'n:e1371008:TahomaŘ&BNo8Lk10n:e1371008:TahomaŢBNodLk =Lk= (8+2)4'4'n:e1371008:TahomaŘ&BNoLk10n:e1371008:TahomaŢBNo,Lk = 103&n:e1371008:TahomaŘ&BNoLk10n:e1371008:TahomaBNoiLk" n:e1371008:TahomaR2S20z96RC@ &%+5+5TӮLk$Oppure si pu utilizzare il seguente$9.--+$----$$++$+-.+-ӮLkformato:,E*,n:e1371008:TahomaŢBNo[Lk [LkN = ((04[Lkdn:e1371008:TahomaŘ&BNo[Lkn[Lk-1n:e1371008:TahomaŢBNo([Lk*r +&Ӳ[Lk dn:e1371008:TahomaŘ&BNo[Lkn[Lk-2n:e1371008:TahomaŢBNo+[Lk)*r +&[Lk dn:e1371008:TahomaŘ&BNo[Lkn%[Lk-3n:e1371008:TahomaŢBNon:e1371008:TahomaR& ZSZX %@ U +++ /3/3/?910!#53#53#53TJ[Lk ) & )*r + d :'4n:e1371008:TahomaŘ&BNoә[Lk0?BSb,{?Lk MP,TGLk MP,TBSb c B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk17n:e1371008:TahomaBNoӃ LkConversioni di baseC=>7;)1#>;2n:e1371008:TahomaBNoi Lk" Ӯ LkEsempio: 115.$+D-,-,n:e1371008:TahomaCU7BNo Lk10n:e1371008:TahomaBNo Lk = 1110011 <-,-,-,-n:e1371008:TahomaCU7BNo Lk2n:e1371008:TahomaBNo Lk:n:e1371008:TahomařqBNo Lk115!"{ Lk2D Lk1{D Lk57!2D Lk2{ Lk12 Lk28! Lk22 Lk0 ! Lk14!Ӡ Lk2Lk0ӠLk7WLk2ӠhLk1WhLk3 hLk2WLk1 Lk1Lk2 Lk1Lk0,{K LkMPMPK LkMPMP LkMPMPM LkMPMPӚN LkMPMPӚ LkMPMPa LkMPMPa LkMPMP LkMPMP2LkMPMP3LkMPMPoLkMPMPӅpLkMPMPӅLkMPMP,{Y Lkd0" Lkd1"DLkd2!~Lkd3!5Lkd4!Lkd5!ӢhLkd6",{Ӣ LkMPMP$Fnt LkMP FnY LkMPMP$Fnӄ LkMP Fn"LkMPMP$Fn; LkMP FnoLkMPMP$FnJLkMP Fn~LkMPMP$FnөLkMP Fn5LkMPMP$Fn`LkMP FnSLkMPMP$Fn.LkMP Fn BSb Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk18n:e1371008:TahomaBNoӸLkNumeri frazionariK>^;(#$(;1^;(=#=#3#(;#>;4:22;(n:e1371008:TahomaBNoi%Lk" Ӯ%Lk#Le macchine hanno vincoli spaziali:#(+D*&%.-+.*.-,(-&,$-+$*n:e1371008:TahomaŢBNoLk Lk( necessario conoscere il massimo valore(%'&!% %'!&(' &&;& ;'$%&Lkrappresentabile:%('% %(%(&JLk JLkcon !&(n:e1371008:TahomaŢBNoe">f??edӚJLknn:e1371008:TahomaŢBNof??edJLk% bit si pu rappresentare al massimo%' '(&&'(&&(&%&<% <Lkil numero 2 -1'(<%''Lk Lk facile determinare che%& %'%&<(%&!'[Lk n = INT( log'30)'n:e1371008:TahomaŘ&BNo[Lk2n:e1371008:TahomaŢBNo[Lk (X+1) ) *3'n:e1371008:Tahomař BNo`uLkn?BSb,{?Lk MP,TGLk MP,TBSb c B c,{ӐLkFondamenti di Informatica      /Lk20n:e1371008:TahomaBNo6 Lk Operazioni O>;(;1n:e1371008:TahomaBNoi Lk" Ӯ Lk&Per effettuare operazioni necessario&-+++-++,-++$,-*-+&+$$*Ӯ Lkconoscere la definizione del%--,$&++*-+-$,-+.*ӮX Lk comportamento per ogni coppia di %-D-,*D+.,-+,-.%---+.Ӯ Lksimboli$D-,iLk" ӮLk&Per ogni operazione esiste una tabella&-+,-.,-++$,-++$%+.-++-+ BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk21n:e1371008:TahomaBNoLk Somma binaria >=^^;#=>;(n:e1371008:TahomaBNoiLk" ӮLkLa tabella di definizione :(+*-+*--+.$,-++n:e1371008:TahomaŢBNoOLk OLk 0 + 0'4OLk = 04Lk Lk 0 + 1'4Lk = 14Lk Lk 1 + 0'4Lk = 14Lk Lk 1 + 1'4Lk = 04ELkcon riporto di 1!&((&''Lk Lk 1 + 1 + 1 = 1'4'4'4ELkcon riporto di 1!&((&''n:e1371008:TahomaBNoiLk" ӮLkEsempi.$+D-\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cB c,{n:e1371008:TahomaBNoӟLk8,{ c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk22n:e1371008:TahomaBNoӓWLkSottrazione binaria>=%&(;1>;(n:e1371008:TahomaBNoi%Lk" Ӯ%LkLa tabella di definizione :(+*-+*--+.$,-++n:e1371008:TahomaŢBNoLk Lk 0 - 0'Lk = 04Lk Lk 1 - 0'Lk = 14[Lk [Lk 1 - 1'[Lk = 04Lk Lk 0 - 1'Lk = 14ELkcon prestito di 1 dal!&((& &('(%ELkbit di peso superiore''(% ' '(%'n:e1371008:TahomaBNoiLk" ӮLkEsempi.$+D-?BSb,{?Lk MP,TGLk MP,TBSb c B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk23n:e1371008:TahomaBNo LkMoltiplicazione e divisioneV=&>3;182>;($=n:e1371008:TahomaBNoiLk" ӮLk&Sono condizioni in cui si ha un errore&---,&,.-%,.-%-$-+.-+,ӮHLk$nella rappresentazione del risultato$-+**.-+$+-*$,.*-+$-*i1Lk" Ӯ1Lk"Generalmente la rappresentazione "6+.**D+-+*+--+$+-*$,.*ӮLk)formata da un numero finito di bit: se si),E**.*---.D+,.,-.$+%ӮLksupera tale limite si ha errore$.-+*++D+$-++,\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cB c,{n:e1371008:TahomaBNoӟLk9,{ c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk25n:e1371008:TahomaBNoӓLkRappresentazione dei numeriE;>>(;2:?%;1=>;">;"?>^;( Lknei calcolatori>;#4;3=;&<)n:e1371008:TahomaBNoiSLk" ӮSLk!Esiste un limite al numero di bit!.$$+.-D+*.-D+,--ӮLk%impiegati per rappresentare un numero%D-+-*-+*--*%*.++.-.-E*iLk" ӮLkTale limite dipende da:0*+D+--+-.*-*n:e1371008:TahomaŢBNoLk Lkintervallo di variabilit(%#&'($%&'ZLk ZLkoccupazione di memoria' !('&&(&';&;'?BSb,{?Lk MP,TGLk MP,TBSb c B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk26n:e1371008:TahomaBNo LkNumeri positiviK>^;(#>=2&8n:e1371008:TahomaBNoi Lk" Ӯ Lk&La rappresentazione di numeri positivi&(++--+%*.+$,-+---E+-,$)Ӯ Lknon crea problemi---&+*-,-+Di\ Lk" Ӯ\ Lk Si pu avere ---,*)+n\ Lk n:e1371008:TahomaBNoe">f??edӇ\ Lkoverflow,)+,n:e1371008:TahomaBNof??edӹ\ Lk se il risultato$+$.*Ӯ Lk#delle operazioni richiede un numero#-++,-++$--&-+-+-..-D+ӮLk%maggiore di bit di quanto disponibile%D+--,+----.*.,-%-,--i Lk" Ӯ LkEsempio: somma modulo 16.$+D-,%,DD+D---,- BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk27n:e1371008:TahomaBNoLkNumeri negativiK>^;(#>;>;%7n:e1371008:TahomaBNoiLk" ӮLkEsistono diverse possibilit:.$$,.,-)+$+--$$-+n:e1371008:TahomaŢBNoOLk OLkmodulo e segno:<&((&% &'(&n:e1371008:TahomařqBNo Lk" n:e1371008:TahomaR2S206XC@ & %+5+5TNLk2bit pi significativo: positivo (0) e negativo (1)2""""" !!!!! " " !! Lk" n:e1371008:TahomaR TSTRO2 7 7@ 7c@/]?3]/10+++#3O/TNLk(esistono due rappresentazioni per lo  0 ( !"!""  !"  " !"" ! !n:e1371008:TahomaŢBNoeLk eLkcomplemento a 2:!&<(%<%('%&n:e1371008:TahomařqBNo Lk" NLk/per definizione il complemento a 2 di X 2 -X/" ! "!" !3" 3 "!!!"# ! Lk" NLk unica rappresentazione dello  0 ""  ""  " !" " ! " nLk" n:e1371008:TahomaR<‚SOR@*88/<33//<<</3/993/99.+}ć.+}10%5 2AOxuu>„S!O R@*88/<<<3/<3//3/993/99.+}ć.+}10 5 5 A2PTNnLk Esempio: -1 <=> 11111111 " 3!!!,-,!!"!!"!n:e1371008:Tahomař BNoLkn\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cBaBSb c,{n:e1371008:TahomaBNoӟLk106,{BSb c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk28n:e1371008:TahomaBNoWLkUso dei numeri negativiI2=#>:#>>^;(#>;>:&8n:e1371008:TahomaBNoi%Lk" Ӯ%LkModulo e segno:?,--,+$+--,n:e1371008:TahomaŢBNoLk Lk'la somma algebrica di numeri positivi e'& &<<%&(%'!%'((<%(& $Lknegativi pu generare problemi(%(%$'(''&(%%%'&(&;JLk JLk)servono sistemi hardware specifici per la) %$&('  &<(%'5%& '%!!(%Lkgestione corretta del formato'& &(& '%%(%';&n:e1371008:TahomaBNoiLk" ӮLkComplemento a due:1,D-+D+.,+-.+n:e1371008:TahomaŢBNo{Lk {Lk&la somma algebrica non genera problemi&& &<<%&(%'!%''('&(%%(&(%<?BSb,{?Lk MP,TGLk MP,TBSb c B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk29n:e1371008:TahomaBNoӭ LkComplemento a 2C=^>;^:?%=#:#n:e1371008:TahomaBNo, Lk" q Lk Motivazione: ?,(+$,-+n:e1371008:TahomaŢBNoӈ Lk  LkSia dato un numero di bit (%'%'((('<%'''n:e1371008:TahomaŢBNoe">f??ed  Lknn:e1371008:TahomaŢBNof??edӈV Lk V Lk*i numeri che si possono rappresentare sono*((<%!(% (' '''&''&&'&& '' Lk nel(%5 Lk range%('n:e1371008:TahomaR[RSRP>x$@  /]3/<??10!!!!Yx]LSLJ@bxj @  /]<3/??10!5!!5!jYx~T Lk [0 - 2 -1] ''&ӈLk Lksi vuole calcolare A-B $''&!%!'&&+ӈxLk xLksi sostituisce -B con (2 -B)  '( !%* '('*ӈLk Lksi ottiene A+(2 -B) '%(%*4'*ӈELk n:e1371008:TahomaR!RSRP#@U`+`@p/]?3/?10##537/TELk,La sottrazione si esegue mediante una somma!,#& &% '(% % %('&<%(%(%((% '<;&n:e1371008:Tahomař BNoӿz LknӯPLknӒLkn BSb,{ Lk MP,T Lk MP,TBSb c\B c,{ӐLkFondamenti di Informatica      /Lk30n:e1371008:TahomaBNotLkRappresentazione numeri realiE;>>(;2:?%;1=>;"?>^;(#):;n:e1371008:TahomaBNoiLk" ӮLkI numeri reali sono nel-.D+++$,---+Lk range*.-Lk [-Symbol BmoLkn:e1371008:TahomaBNoPLk n:e1371008:TahomaR|S|z5 > @# 7 V@ 7V -@  /]<</]++10#3!5!#3`h+55TiLkӤLk +Symbol BmoLkn:e1371008:TahomaBNo3Lk]i[Lk" Ӯ[Lk Talvolta 0*),*[Lk [Lk necessaria una rappresenta--+&+$$++.-++--+$+.+ӮLkzione$,-iLk estesa sulla retta dei reali+$+$+$.++*-+++n:e1371008:TahomaŢBNo%Lk n:e1371008:TahomaR/PSPN+@u8 /3/]?/10.+}] #DXY’S<Z@9$4Df$+ p 9/]32//]??910] #3  m~SoZŒS=<L X@*:5JEZUje55 @ 7 /33/3/+33/?9?9.+}10]!!5!5!!L:iT%Lkcon 3 simboli [+/-], X, Y, Z !&(' <''4*)(Symbol ŢBmoӐ%Lkn:e1371008:TahomaŢBNon:e1371008:TahomaR{S^=46;#";vt%%wXXw%%˳tP;V &%!j99i"%& W;PxžE223322EŊjMRp1,II,1pRMjT%Lk {0,1,#&'t%Lk& n:e1371008:TahomaR}S`bx7^-@: 7 73%353D3\T3je366 *+ +/  #0 /]33//<<3/]<??9/9910]++#"+53267>=467>75.'.=4&'.+532;Xw%%˳tP;V &%!j99j!%& V;Pt%%wX322EŊjMRp1,II,1pRMjžE223TӮ%Lk9}'{Lk#{Lk" possibile rappresentare -999 "'' '&%('% %(%&''&%{LkY{Lk +9994&'Lk Lkoppure 9 * 10'''(%'''HLk HLkoppure [+/-] 9 * 10'''(%4'''n:e1371008:TahomaŘ&BNoLk[+/-] 99 Lk[+/-] 99 \BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cBaBSb c,{n:e1371008:TahomaBNoӟLk116,{BSb c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk31n:e1371008:TahomaBNoWLkVirgola mobileC(>=;#^=>n:e1371008:TahomaBNoi%Lk" Ӯ%LkE la risposta alla.+$-,%**n:e1371008:TahomaR2S20i:6D@ 8&9:4%+5+5>S><g@ GVf/3//3/10]#gvT%Lk necessit -+&*%$k%Lk di manipolare-D+--,+ӮLk%numeri di ordini di grandezza diversi%-.D+-,---.*.-+$$+-)*$iLk" ӮLkNumeri espressi nella forma:6.D++$-*%$-+*,D+]Lk X.YYY * 10 0///,,n:e1371008:TahomaŢBNoLk LkX: parte intera)'%%(&,Lk ,LkY: parte frazionaria)(%%% ''&Lk n:e1371008:TahomaRW S : @[k +$& 86 HH F   $/ + <47 G Xhgzu     JKDE  @ 7_?/]3/]+?=;#^=>n:e1371008:TahomaBNoi Lk" Ӯ Lk Nomenclatura: 6,E+-%+-* Lk A = M * B 1;?-n:e1371008:TahomaŢBNoP Lk P Lk M: mantissa 7<%(  Lk  LkB: base*(% Lk Lk E: esponente (& '''&(n:e1371008:TahomaBNoiLk" ӮLk'Necessita di un segno per la mantissa e'6+&+$$*-.-%*.-,.**D+-$$+ӮGLkuno per l esponente-.,-++$-,.*.n:e1371008:TahomařqBNoc LkE BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk33n:e1371008:TahomaBNoLkVirgola mobileC(>=;#^=>n:e1371008:TahomaBNoi2Lk" Ӯ2LkForma normalizzata:+,D+.,D+$%*+n:e1371008:TahomaŢBNoLk Lk*si sceglie di avere la seguente relazione:*  !%'%'&#&%& %('&(&%% ''&Lk ӎLk0 'Symbol ŢBmoLkn:e1371008:TahomaŢBNoLk M < 173WLk WLklWLk WLk esponente &(&(&'&yWLkӞWLk espresso in complemento a B% '% '(!&<'%<%('%Lk(talvolta in &#'&(n:e1371008:TahomaŢBNoe">f??edyLk eccesso 127 & !% ''&'n:e1371008:TahomaŢBNof??ed Lk)Lk Lk la mantissa &<%( &vLkӛLk espressa in modulo e segno% '% %(;'(''& %('\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cBaBSb c,{n:e1371008:TahomaBNoӟLk126,{BSb c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk34n:e1371008:TahomaBNoWLkVirgola mobileC(>=;#^=>n:e1371008:TahomaBNoi%Lk" Ӯ%LkEsempi usando: B=10, 2 cifre.$+D-.$*.-,0;-,-&ӮLk all esponente e 8 alla mantissa: ++$-,.+-++,**D+-$%*n:e1371008:TahomaŢBNoLk Lk +1 4ELk 0 01 10000000 ''&&'''&''VLk VLk -63517,8''&''EVLk 1 05 63517800 ''&&'''&''Lk Lk -0,00063517,8'&''&'''&ELk 1 97 63517800 ''&&'''&''#Lk #Lk -8,75 * 10'''''E#Lk 1 88 87500000 ''&&'''&''n:e1371008:Tahomař BNoӯLk-13 ?BSb,{?Lk MP,TGLk MP,TBSb c B c,{ӐLkFondamenti di Informatica      /Lk35n:e1371008:TahomaBNo LkVirgola mobileC(>=;#^=>n:e1371008:TahomaBNoih Lk" Ӯh LkMoltiplicazione e divisione:?,.&*$,.*+-)$,.+n:e1371008:TahomaŢBNo Lk  Lk*si moltiplica o si dividono le mantisse in* <'' &' (#('''&<%( %& Lk modo consueto <&(' '( '& Lk  Lk)si sommano o si sottraggono gli esponenti)  ';<&'''  '&'''(''% ''(%( Lk  Lk si normalizza ('<% ZLk ZLkEsempio: 10,4 * 200 =( %<'&&'&&&''Lk"0 02 10400000 * 0 03 20000000 ="''&&'''&'''''''''&'''&'Lk0 05 02080000 = 0 04 20800000''&&'''&'''4''''&'''&' BSb,{ Lk MP,T Lk MP,TBSb c\B c,{n:e1371008:Tahomař BNoӐLkFondamenti di Informatica      /Lk36n:e1371008:TahomaBNoLkVirgola mobileC(>=;#^=>n:e1371008:TahomaBNoiLk" ӮLkSomma e sottrazione:--DD++%,+$,.*n:e1371008:TahomaŢBNo0Lk 0Lksi uguagliano gli esponenti ('(%(&''(& '''&'Lk Lkle mantisse vengono sommate&<%( &$%(''(& '<;&Lk Lkaggiustamento in caso di%('( %<%('(!% '' Lk traboccamento&'' !&;&'dLk dLkEsempio: 10,4 + 2 =( %<'&&'&3&Lk"0 02 10400000 + 0 01 20000000 ="''&&'''&'''4'''''&'''&'Lk"0 02 10400000 + 0 02 02000000 ="''&&'''&'''4'''''&'''&'dLk0 02 12400000 = 12,4''&&'''&'''4''\BSb,{Lk MP,TLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% _ k BSbӥ ` LkMPMP% # / BSbӥ $ LkMPMP%  BSbӥ LkMPMP%  BSbӥ LkMPMP% o{BSbӥ pLkMPMP% 3?BSbӥ 4LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% GSBSbӥ HLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% W\BSbӥ XLkMPMP%1D(%&Cdd*u??+wj,{TBSby cBaBSb c,{n:e1371008:TahomaBNoӟLk136,{BSb c?B c,{n:e1371008:Tahomař BNoӐMLkFondamenti di Informatica      /MLk37n:e1371008:TahomaBNoWLkVirgola mobileC(>=;#^=>n:e1371008:TahomaBNo;Lk" ӀLkApprossimazioni:1--,%$D*%,.n:e1371008:TahomaŢBNoӗoLk oLk 34,56 + 0,005 =''''4'''&Lk0 02 3456 + 0 98 5000 =''''&''4&&'''&'Lk Lk0 02 3456 + 0 02 0000 =''''&''4&&'''&'pLk pLk0 02 3456 = 34,56''''&''4&'&ӗLk Lk.La precisione data dal numero di cifre della.#&'&  '(%&(%%'%('<%''!%'&,Lk mantissa: <%( &n:e1371008:TahomařqBNoLk"  Lk)Doppia precisione: doppia lunghezza della))!"" ! !" !!"" ""!"  "  Lk mantissa ( 3 " 9Lkrange "!Lk# invariato, precisione raddoppiata)#"  !! !"  "!!""  ?BSb,{?Lk MP,TGLk MP,T  BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% JVBSbӥ KLkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP% BSbӥ LkMPMP%1DIB%-12345X@PJL EOJ %-12345X