OnaT- Nurmbes Syslarns A Number Syples ae a cablectton of dent, rohtch fn gente! has ‘2! Poul — Ioteges & fracional paste, Set ape by a wade pe « (Np = day dygee dr Aides Aye dp den HO Inhejen “Portion Peat frackonod potion where, N = a number b = vadtx a base of he rumor Sle N= no. of digl: in 5 portion ™ = no: of arg Tin -frackonal Potion dns met Sx *fteareke ae (ms4) Ayo = Least Sg feat nik (hs) BO < (Ar way Se b-I Chorocleristes of Cormmenty Used Nurobes Siplemm $ Example, / is Wetghk ovat Number Sylém | Basel) | usd Hi * Rosny! us it toni [Erna Belal 0)1,213/415,6/7 3567-23 Dectrnal 0213, 41516 34415 18,4 Hexadectmal 041) 21314157) BpAg NS, AA,/B,C DEFBinasy Number Systemes Ac) “he numbex spl softy base (or valtx) Le & known ab “Binony Only 2 Symbols ase wed I5 represent nds io ths Syplein they ane 0 and 1 these ae Knaon as ble sens os he Maat +) Siatlan 40 dectmal nor cyplem ) The Soft mest BIE & kno Stgpificent: bet (Mss) x Me wept amett bit & Known as heat sofort bet (LsB) 4 Ina brnany nurnbex Siplem 1 grep of yi bil Is Known o% te bile & Known A ae ‘itbble? & qr of 2 Fy (eto, Octal Number Syslan ¢ tt Known 08 Otte ‘the no Splem with base (ox rodix) £1gh No- ples Yo The Soles seqnt Sposa O41, BASLE UME Used te represent nurnbers: Sinslan te bt no. splem He aloa portional amd has in qe ‘roel pea integer fractional eek opert ‘4 a jadix Coctad) potot &) er (6237- 4051), Vecirral Nurmbest Splems “the no: syslem with base (eo vadtx) len is ne Ney sipleisy Ta Tee Splem Nem Syrbols 0111213446 611814 aE used won a Dectrnol te vepresent nuseben$:2 T te aleo postrondl No sples x has hwo pote * dale % Fractional Bek apart is aradt (decimal point (>: | Eqs AS-D HexaDectmnal Number Sle 5 ro: Sylar iotts bare @ yadtx) 1G & Known oF Hexadectnret No. Spm St vequive 16 Atstinet tata 40 wepren te nowbey “these ase number O trreug> gq b adphabele A Troodn f- Since numeste ataile k alphalbele” bath aye used IF dae aegis, thie & am ‘alghanumentte nurenboort splem! ert “the Se GFA S)ie Deetenel | Binary octal Hexadecimal Oo] 0000 © © | “1 [ [oct | i ! 2 0010 a 2 Po ool) en 3 a | 4] O10 4 4 hs: o101 a S [_é] | ono ae | 6 7] ony ( a & 1000, 10 & F 9 [= 100! = 9 lo 1010 12 A 0 WON a ‘8 | 8S 12 N00 4 ce ae No) r saiegetecae D | 4 Wo 16 e 1s ua 7 F Relation Yio “Vectra, Frrasy cotol 1 Hexacectersd nurobers| Base Conversron Mathiods $ | = Pinang - -Dectenal Convenston s fied dectmal Equivalent of Hinany ember (rood, Gono), = xa oxrtg rx pix24ox2! 41K 2 324+0+34+440tl = (45) 10 ~ Find decimal number of the fellaietng bereoy no: | Coot: 0101), = gl ce ip allah 2 1x 224ox 27+ Ox2! 412? 40639 FIND FORD FE? $4040 4140+0:2540+0-0625 -G “3125 } Dectenal ta Binasy Conver Sion 8 for tolejeut the Conversion ik clbained Cominvoys dkKsion by! X Reeps Brack of tre wemotnders , tchile foe aaa aun Conveaston & affeclad by Continvous muttipicatan by ee 4 Haupiog track of wheges eet Wp Comet (is}_-to am Sqaivalent base-2. nvehen Bucher’ — Remainder wm 6 1 air Ses tie } aie a t 1 z i. | R ~ Comet (5 65625), Cod, te am Spovelent base 2 Nombey | ©:65625x2 = 131256 > | O-31250%2 = 06250070 © 6aso0x2 = 1*2500— | 2 @)0+ @500%2 = 6-50000-40 J 050000 x2 = |' 00000 - } | Cortoier), | Enpeens Vr pleating decinal Numbes in en foro te-6a5), —1 © do ° | | | ° ! ie) Ge),5= Clo1od 2 fractional peo © 6asxo= | 250) © Qb0K2 = 0'S00-4O 4 p-S00x2 = 1:000-4 | 66-625), C10!) (16625). = Cloto-101)., Octal te te Dectenal Conversion § tonrt C63a7- 2051), Inte TE Equivalent decirnds ne foie (632-4051), = 6xé Aone tangle TRE 448 +OKE + 6xe 4x9 + = Bor taa4 lett s +04 B+ 2 4096 6287- 5100098),,,1) Pecimal-ts-oclal Converston $ A) Coment (A41);y into Octal te 4 = oO 3 ; ©, 847}, = BEVg +) Convent ©6815) jo inte cekab © 6815*X&8 ~ 55000 55° O-S00¥8 = 400003 4 L “thus, Co 6871S 9= (0-54, Deleb te “Binary Comesion s | Octal nds cam be Converted tale tasialert btn ony nos +4 pep aang Bach octal ang it hoy We aie Eqowalent ry: Ae (136), = Gn on Ho), Birany —ts- octal Converston é a os node Can be Convesthed 1016 ma a ly geep of 3 bil Stating fren LS moving 12 7 Jon toleqen pa © — then replay gach Joop bik ae Sse oclall represectiation Fon factional parks Ihe ger + shit ~~ an [ory glk *2 av ae made sfeorn the4(a) Convent AFA), tle pina ne: CFM) B= Cove wit rool 1010), Broan leHexadectind) ConvenStan & a Morooiroreitt), = o010 1oo! pe we -GTAF), nh a B) (0- c00;; 101010100), = O- ©00\ HtO Jor} aioe es — —_— y| © a + =(o16845 i te : Ayr Gos p, > Oo Bed), = (0011 C000 OV0+ 1101), _ Sared Binary Naumbeuss Sap Magitude Represenietions Mw 4 cakte cam undexslamd only 2 syvas owt, thane fre We rout use tre Same Symtods Ho tadtcalé the Sqn of tra rurcbey also: “Marnally ain oh Placed ab MS: ‘a nepresnts a +e to ee represert a onal Wt Used o4 the Stqn BH He Ne Ne <4 Find tre decimal éqsralent of the fatowing binary nels tong Sarr rmagthide meprererrtedh an of re, tincoy nots 2Convert (lO OIN10), de tte oclab Spivaledk 4 (woo), = (ot oo No), = (120g eH (613: 194) 6 ra oes (672124), = (ne Wt ei. oot o10 1oo ), | exe Dectonal la —Dectmal converston$ | > Oblain dectenal Equivalent of GAA )ig (BA+2 = BIG A AKIE + Axle 16> Gary ee 2 By = 45 HOF AE + Blase = CB 182410 Peoterad ts —Hexa dectral Conversion % precio? fou | ye aes).7 CH colt Gehan pat Frackonal pant aS: on 1s osxne= SOB Ss - t ig re “thus ,C-59LO°S 14 vihus, C75),- CFP 16 C55), =GFOe Hexadectoned le - bxcany Convenston 5 | | | | i syalent binant | Mexadecima nos Cam be Converted inte Sqeveel wy ‘4 | ros by replacing, ach hex Argit by RE 4-bik binasy to Eauivalent-ie b) Bo1D00 aq bhof tg is 4 -ve No Sip bit of Mas i'd o ave No magystude = 01100 CI2Dj5 magnitade D\000 (8 9 | J C(loii0e), = E12) j0 s(o01000),, = (+210 Max: He number = 127 § OW Max, ve nomben = -127 2 VV ININND = 4 Sr E14) 2 C toto), Complement of Nanket Onds Compleroeat Representation : facksd byt Ina Rina vo, if fach “4 is veplaced 4 ‘Ok the susulking no: fe Known as the Ones Complernent of the nowber Hy (ero, suprusente Dio tohene as d's Complement” = (1010), (5)ie This meted ib icidely used fr reprexzetng signed ae tris 4's Complement. representation alo Msa w'0! for ve no's v4 fre Do's: - find ones lerniert allowing binary urobers Eq. -fi Complerne of fi 4 a) o)o0)11001 b) MoOllo1re an IDITO0OILO fm OD; 0010) 7 Represent the “plowing purken A'S Compement -forntS(@) a) +e -7 Yrs r-k joe eal Sm 3S Complemect veprerentation / (4D, = (OND, 4o=(d1000), GID = (OND, (Die= (i000), 8) g= CONN) (15) 2(10000), be daseaved “Whale fir "0! Nombe the “Prom the above Examples, tt Cam in As representation ie Max te no which cam be repraented GED & maw —ve no fy ee edhe Tise's_Complement Representations Dp ‘d! & added to as Complernent own as 2S Complernent of of binary nvenben, the gretubtant no: ka Ye bina nureben tyr os complernent of AP! & st- ts Comnplernest of. Olo1 =) IDIo eae low) S& ince Otohreprerenit (45)}o, harefore toll re presenta (Dio | 2s Complement veproxntartion- 30 Ie repenontarton abo, if Mss kG the nob +e MO aru reprorent -Ve no’ | 7 fer be fer an nk no, ww a's Complement spon & 3 Note: Tt is Obseswed that The as Complerns of 0 rer is. the Noe thet. the wow we 1 whitch can be rrepraeited sty tte max “ve ne i a ent of the 8 Comptement: feed cig 1 I petgned enemy Prem aoe as Cpe | 9 oe Lito | cece £000 L 0000 +4 eeo) CooL Ooot +2. colo colo e010 +3 oo ool Cols +4 O100 100 e100 + clot Ole! elo} +6 oO1loO Ono Ol +7 owl om ol -3 1000 - - —T a0 1000 loo} “6 Io lool to1o =] lol 110d lou —t | 1100 four [ iloo “3 loll 100 tor aaa; loo Ito1 [ Htto =1 lool mr) i ie, -d |_-looo | to, | Es ¥ ~ find Ine als Complement of tre flpong nos Uy ble01110 Mf) ooetro.or Sd No oloolt!o No OEIlOlo}) Moololo i cool 85 cox» Aeris o ; Conny ir 101 }0010 troo1otl ie ea 1 jained Nores We obsenve thar tte ae ement of ano L obt oe, Scamming the no: -from 1S8 Is MSB bit by bit 4 vetatning “the MA! be axe uple” b includto g Ihe orcunence of the first| Rly Ae ° Binary Addition | Rules of “Benaony AAXa A Ory Addend | Sor Nugen: cE yi & leu O18} Cee | [Lenore pol aaa toy oney Binany Subheitlions Rules of Binoy Subbactan When bormo=!,ax in the 24yow, this & [5 be Subheubed fromm ht new ighet becaong lott as ik & done in dectmal Subhacton, s- POV 1 Mined = 011 0 Suttrahend 0) 07 Differenceany Mableton Qe & etmilan (ST dectmal Nubttpltcadttan. Tn benanty, gach pastial -preducle is etlhen %er0 (multiplication - | “by 0) a Eaactly Same ay the roubhiplt amd CemeHtiication by 49. | Agr Holiply loot by Nor looy Moltiplicamd y Tloo Hobtipliey ree! ee evo0o lool “Poa presale ea 111019! -firal product ct: The multiplication operation is produc to dblato \ ao tre -fieal prot: Ina at wepeated additions of all pari Binary Driston’ O} is dblained using TM Same prune as decime division gr Divide 111010) by ool 11017 Quotient’ pier é fool) Wi 1o1e 1 pividen of lool! doll tool 00100) loo} pone Remainder Subbracton bsieg as lomplernent 3 Binary Submaction ca be performed by adding aS Complement=aH of Pe Sabhahend to The minvend: 102) 77 & final Camry ia grossed discon the camy Lathe answer i | ee by lhe pene tits which & positive ( the minvend 16 > the Subleahend). oat te fieal Casey bo, ave (the invent) is Bwrerbay Smalley than {he Sulbtral as Complement representation fours — perform si Subhaction boing the answer & Ne hend) & iS ie als Complement of -ve nombes: iz Ct cy OL 1 + Minund See ee Ce 3-316 caenplemant-af Sabhohend a Tod | o- q. Discard Fina canny wath: answer & CO1O 3 (4216 oF O10] Hinuend -4 - TI DP _ F1OO 1 Bs complement of Subhahend tlio re k ista 1G” | final Camy =o Thenfane, the answer te Covnphernante foro gs Complement of Mtoe &G@ol 0) 62), aig D010 os +1000 een 1o10 Fical casy-o : ‘ e as Nument © yo 2D carnpl f 1olo 110) 5 6),4 =)g Lubhackon bang Onds Ceenplernent 2 Jn th, -ve no b veprorerted ty the I'S Complemen-fory and aca) addttion i& Paformed te a desired rest > hey Subbacton tS performed by adding Is complement of the Subhahend to the rrinvend: : ig qeamnated, the answer i in tue foro 2 Ave ~ aE Final Cars Add cosa so tre siewlt bs {* the final reste ~ of Cowes tu nck qe nest & —ve AS temperment: ool oO — Hinuend ey 4 =) 41010 — Ms comnplernent of Salchranend answer sir -13 +) 1000000 psec — ©0000! ose oe dw i (000001), = (V0 +10 eae etic ae 2 4+ 1001 +4 Jooll ' Soon 0100 = » Awe uo (eree), =U+) 105lodest 8(@) | Conpadars & other drgttel one proves data tn Ine bit forvrel: | Vawous inary codes a used to Tepresedt the le iohtch may te Rumeric,aiphabels 6 Spectal chagacles Atthevah In Enany code vthed the toforenedton is represented to binary foun, the inteprectattion f trie binany iwformnation is powele if jhe Code in which the ‘nfoxmtion available ts Known: F-10000 00) represent “G5iL tin Aectenal) to StrughE BP (decimal) In BED Code “AY (abphatack) to ASCH code. 4 hecho A user musk bbe very Care alcouk the code beng use: ‘interpret Sofermad on avallable tn binasy “jor @ ae abe used | exe Concedtion ke da drgitcl Splems, cod Crh detection « bet ghted codes, dave ee \ “i He pay Binary Bep 7 Hemmin 8421 lective clement Fx) a ahi " Cecles By Greny coe } In leet ghted Cader, Each dtgit poston af no represents a Spee! 7 Non saghted codes ant vot aoegned vate ay weight” tz Each digit Position [athe codes vokich Constslé of beth numbers 5 alphabel are called Aphanurnesie Caces fie OEWhee akg snfrereon to rang fer 16 chamaited fom one oe anothes, Cel | Syslem® an Seror cus, Te mainlaty the cola tnle- YT between transmilen and vecerven Exlea bile & mere thame one brk one added- the dala along loilts the Exbia BIE[bIGT

aA Hesettes cae ea oo | Bed Addrtion’ Bep addtten Prcedune is as follows? |. Add too BCD Hurdbers Using erdmany binany addition a If 4-bit Sum is Equal to & lon than 7, no Coweekion ib Needed: she -resslt & io peoper BCP fore: SB: Tp the 4-bit Sern qrealai than J 8 tf Comry is gprerated Ife | {he Som t& Tenalid Hence Corsottion is needed, |. To Comet the Inala Sum, add 6 (otto) © -feor- it Sxn-if a Csr TOAUIIE efron ‘This addition, add i to next Higher onder ao | digit: CoO10 O100 ~) BCD for 24 ye 34 tig 5 ©00! 1000 2 Be Ie te ae pelt 1100, loo y4sadd 6 4ik oe +oll Oo Col\ Nolo > if Cary agiravated add Wo rene ae wage oder digi Ooo 80°10 eect4 ? +s ©2100 1000 +58 3 e10 loo + 1 000 ee *Yoat | oo08 cans grate -alC i 41 ol10 toqge ol) o 7 loiwrd then odd & +O110 jjoooo ol1\|o ) ° 6 Bop _Subhactions Using a's Compbnedt Mathod 3 4 wm Bach athe 4S Complement of @ decimal nor is fond by Subsvacting 4° diate fo the ro. “fen 9. ~ Pacedure fo Rep Subhacion tusiog a's complesert $ “find The o's temperment OF negative n0- ~ Ada the too nos using BCP addition | iw 73 conn & od gered mest ts negedive w find FS ae verolt : sthe result, Gireroiie result & pertive } add canny 77 Ege Pao G6),.- G20 i Beo vsng 48 Comnplement: coli ) AS compemedt of 92 & 99-22 = 77 a 46 cloo O11 0 Beef 4b E11 Fy OLE T O11 1 as compement of 22 VOLT [1O! sfentt 74 add 6% toito o1lo ibeaet A Rice tS : : fecien eer Ts day i gennated TN eee ee & aa Coney +e oe1o0 01°00 22 a absense BMar @4),,- C9, obieg ap as lampemet Foy =“ 2D to BD Subhailion Using Ios Complement Method § othe Wes Compleenent ofa Aectenk No bs Spo te a's comperent flus A. 4g- (3491) 188 Coraplernest AL een the wt ig Same oS 4 comglerrant Method: Gees Conky discard comy “for VE ne) Eye (46), 2210 VSG Rep's Ws Cosplarceot sey ws Composers of 22 1S a ad a 78 | 2 46 6100 ono | +16 9D O14 tooo | te, | Toy? tty a POLLO D1 © — carey in gered Toei wel oO. tear | do cee ata cosy | ey) 4 GOI6-CVIg = CDiB- 4) ,~ Gog bsg Reo (68 Compement Soli rl SY 18s compewest of 86 's ane “¥3 =r 2 at ooto dioo ae 444 o100 0100 See eee o;1eo goo? La no camry ne be | Ley’ fake to'S Comglernent | é 8 2) Ws complement of 62s JD 31 44 su * @#) oo 58)io= E3210 Kxeess-3 Codes | the cale fr Each decimal degit ‘a cblained by adding & fo the natural &CD Code of te ang: Hy decimal 2) COLO TOo_I1S O10 J) SacenS code > 94 is met weigitled : a8 + m Welgitied code Qeciawl | Peotode| Exons a the code ba Self Complementary lode ‘ e000 aol Cede which means Complement: ee aa tre elds t's comy O0l0)] o101 | of the coded no: ft eve eld ee Of the no: iGelf- | Ooo} ot td : 010) | 1800 | 44!- @)o= (ole. = oleae eie| feet | Wu Exons code fr 7 ey too tchich is o's Complernoot Of 2° tcoo| Weil | toot | 1100 | > this self Geenplernerstng pelt of Wis code helps to parsing Subbeachon opeotn in duped Syplérna .Excea-3 Additions 3 Add +00 sxreaive-s He Hsing > IE comy =1 > aad z(o011) 40 Som =O > Sblmact 3 (0011) fromm ae i 1a aeep oe: oye 46 5B — woor 160° yoo) ener aes 18) 1104 yiegece + tIc2) inany addition of to agit tre Som Sxcom 3 code ris excon-3 code fr oss \ Cony so 084 3 tHlet y110 00°) ey mcenl to oud vo canny Sobback coo lo1o torr ol 90 ia ere encon-s ‘Exon sxce-3 ods fort code for & code far! Exce—3 Subbackon t ~ Complement “the Subrahend: ind > had cornplemented Guidalhend te minve If cawny =) Result is +ve oad zoo) “| =o vent & We Subrvadl By 48-5 ) wot 38 5 1OVN Exepa qt “BF OLN T Campmet ah em q10ee) — ijooro ool) q aad’ O10) 7 cay SEPP fas — ae “O11 8 teens 3 emda end-arourd coves) 2 b a(OON)2 5-% 6 yooe este toro 1 1 OO trecey a ve ne Sea (oo) Gewem 3 of -3| Frey Code § bere oH tba a useful Code in tohich a decimal no- i mopwrented in binary for tm Such a way Jo thal gach gray cade No- fens -frorn tre preceeding ye the Succeeding no E> Single BL A code .'i8 a Spetal care of git Addlance code. , tag he > We Gray code ie a veftected code & Cam be constructed Us 4 eatty a given belovo: g mm 7 code has 2 Code joords Bu reprorentirg dlecioa! Numbeuw 0% 4 respectively GD An n-bIt (nZ2) Hoy code will hove biLT watten to order with a Seadin 4 ° appere + ; uy Ut the fast: or lgrey codes wilt be Equah te tee GY cade 10 orden (arnvenias) 3 & nok a wei fect 2 leader of G19 of am (o-1) bik gay code , woutten 19 rene a mirror placed between first arty Last 2 Fy codes Oily a Leading A-appended B-bit Gray AL Gey Abit g-bit Eadie, code, Fray code spayed at, Gace: lore ro ©00lk (io co =O Ce ath © =(0) 10 oe Wo a. | 2) (2010 eee re at ihe Cte 1 \o = Mio ere. 1o= (Dio ities ele 1o)= (we !160 Hoos Chen | (hor wae . ILL 1 EOP 1110 = G0 101 O= 036 lO 1S 3d too12 (Mo >This code & used Extensively FA shaft freodes became af ib propel:srenult sat- Eq Th ww i aut ey bet eg 3) sleps {6 Coowesting § the MB af the binmy no & f he Code ne 3 To oblain next binasy Ane bE Just tovitten down athe net YAY code bils have been exclusive ws, ° | In gets Beles) = Bast Gray te Broasy Conversion 3 12() The Same as MSBbIE 7 Repeat slips until gy banamy aighlé- Eqi- Convert gay cot 10101] dete o 14 (4) a A f ©. AAAS \ dagt epafene Exch usive oR oprcdion bho code Br walt don The ~peed Pt w binary Spivalent ja 8 | 98 Gn (me) = BoOSo-1 Bar = Bot © Ge-2 wee \ 41 aunt a On! Binary to Gray Code Converston’ Supe for Corwentings ~pthe M88 of qm code ig tre 3 1To obtain “the next Fy digs ious bk Cunnent binasy — Repeat sleps> brit all, Einasy, previous ores Convert Gott tort), ile Some as “SB A ° ° t [1 B= 8.86. -6 - of w| o ' ' ° _--- 8 =BO4 8 - 8,040 of binary ne ve -OR- opeattion behioesn peje Exclus won tha vesult > pit toate do Liles have been Exclus bs yee ored) ith Hedy AE Squvalere Gray lode foo)}10) Ga (mse) =Bn (mse ) Sn-1 in ® Bar ‘ 1 td fat’ feito ® t 4-808, S0 = B® BoSey | fall sep jaxcenas | 4ryp $ Bs Pr B 2 pend fe GE] BaS MO ° | 0000 e000 aa ) | © 004 | 20%! a os gaoro;ecole Hiei aa o ers |e! po oto | 4 p 100] e168° ose | 5 eo toy | ete) eS ie é otto] ote elo) 2 ori ary ort © 10° - foo 8 pooo [100 Sue Lua oos tool Self-complementing Codes § A if cede word of coda & Said te elf Complewentingy a's Comglement of NO Cede word of N 4 gnterchamang Ain a Sexy corplemereq code. code for fi eof a & $2 OO qn) Can be obtained fires Te all os & A's. 6 cornplemert of % code of gis Coenplarren 4 Sel HO areal, 5 Qi, 642-7) aa Qed amd ¥S 3 OM \ complementing, Codes: [> In j2+ 3-4 dreotlonky as ef- 4 q-1=8 wo ool yoo gli tt %o NG) ome of te tocigitect codes 6 % Dectonal =i Rene! | $324 rg Bt) ot | do | eoee ooo | Gono | 9200 ) ooo! ooo! eco! ool 2 oolo oot See oolo 3 ooll olol ce Joo!) a e1e° ow 6100 0100 e e110) yooo | ©!9) jrowl 6 D110 Joo) ono ptlo oy oll [Joo oy | {104 & {ooo 1110 got) 1olo 9 lool Feat isch _t \ | ~AlghaNumesite codes § chase Cecles Uses betty Numbers Os rack os alpha - bet to represertt — The Mest Comment wed a, AscTT Codes GS Chrmertcam tlancand Code §% Srgertion Antecchorge ) is a abit code gla 128 Syrbals = rnong joe Symbols we have 26 alphates from A te % 26. alphabeli fern @ to 2) , Mornboss fiom o te a, Serta Symbels, elery oljhanumete codes OS—> Fh 9 e Asosa Codes of © to q ae 30 +039 (ate a) we bl to 1k (Ate 2) ar 41 te 5A Bg fecle aede of Woe is ba vy wee 6800 fr joo! oleae! Set poate ‘CoO! A Aeorg todo. Of eninge ne "8 ea Se 2 roo elo 162 ‘or Bente tose! 12 aol yo) fo] we og AO oe eee eee 4) 42 8 ay ys He AT ya 44 SN Ye we 4D KE uf £0 gy 2 es pv x YS sl SA os 5 si 8% gy Bc OF ¢ code 8 G Extended Breany Cade dectrnad dnteckonge code a aa oi ah sy 8) Ie ee = 28h eo. — 3 Tha Code uses Bed to rep gech EBCOIC codew/a) Frew alelechog b Crnecting Colot sehen bt Lite ate bamarmitted fron one Yocectiory fo amolhey) due te clectsical netee to tranumixston chomnd occun, Due to the dsaramiion excre & bik may esnor ee ke Vice versa be ‘received a8 A insted of ‘ot 5 he Code hich aetects a erhor & a | Ceede- aa AL a Code tohtch dete Bb called Exo detedling Kanes electing Code * pool bite fused oy fr OU qh tk Simph wlechrdque fer am Sxtca bt called A pay Biky de chomarnitked. shee aa thee of panies t ct. as well as corre ty the ERE ® Covsecting codes. le ererdlekson) nh — 9 the Libis Set to odd pal 7 at ae 44 i Aodloding yas " am edd No: 5) Even ga AY ths bit ) i cjotal neq 2's aecluding pe -eo--09-00- o \ ' oO | oO (3) \ ' a| Hesomning Code 8 1s | % th am eMor Correcting Cece: hel ws anvme “that K pail bite Ps Pp-~-Pe ave added to n-bit | Tevsage 4o -form am (tie) bit code. “he | @ vont So a4 "b be able 40 descabe the of amy of NTE porte enor bit Locations kalo no codition: Consequently must Sabisfy (he | aay toidhin a Code word Bo the Location of Each of tre ork bile | decimal nos Sarhing from 4 fe fre Mea ntk fo Tre K pany ~~ Ane Paformed en Selected bE of Sach Code. Each pony check includes one of The poly blr: “The rewtt of Bach posi aletected a8 0 | tok the AVsulle of poly she decimal value of the binay wed formed 6112 —— ~ Ce” then Aecienal value WU be > Aecimal no: tthe bile py aw placed to docedion 1,2)4,-- value of K be chesen in Such aX a ntkt) ly check ‘u vecorded as A tf envy hos Gp (nee ik no €a1or) | position & ocetion no the pawl [ae Values (0 erAN are amtqnedd te the peoily bil Se ante he code have btithen Even pay a edd poly 4 then envi 2d te the beahon ta fare on TRE | cerntaned’ |S whan Hes [exon panrton | Bene | hamming Sccws Ine postion ne bf the erroneous bit SA tre case of BCP Co 3 pouly pik thee ay Seven ene pores de with an yeer-we obsewe tok if am erDr excess 10 porhons \s(a) WaT den cg=l 2)3)6,7 then cae) Aisig7 Jhen c=! Pr & Selected So os to establih een (or odd) pol in persons Y315,7 Po is selected coasts establish even (A oda) paril7in pesttrons 2,3) 6,7" Ps is Selected “50 aa establish even [ov odd) pay fn posthons 41S 6,7" Hr Conabiuel Harnming code for BCP ONO" pce even pails Selim tre value of K mast PE Konekey (heene4) =) Ma qaet) he mins Value of (6 which cohisty tt Sy ie B ate! portly pile ax alached t6 gach BCD code for consul The ataranring code > St fl be Tbit code wilt bik pesisom 43-0 ‘pany tole Pafar Pa OH appended 19 Jocotions 21 *4 svespechively Show belowo | t 2 3 Berd | \ee3 Po ™ Ps np Ne 94 [ened | prigiel BCP =” eee Ret tven pay a) pares, kaenjrepre acl t ” © 11 0 ‘tren pasilGj in poritiens Gaerne poi 21 to fg {ven pasly in povkon sie. 7) eq" psso >! jo 211 © ie Hararntog code for gep Argit Oo With Exveo p's je WoooFE Constouct wHoseing Ceele O11) Sh- oO ra 5 distance ig 1 Since there is Dnly Lee iffenert « b) Delesrnine Abrnening Codes for oro LOIN > fied the did bus then + USe ey Seb “the Aaeming cockas fr OWve KON 1, ame gpren below 2 aioe 4 Bcp code », Po Dy Ps M2 13 9} pee * rt 008410 o1i\o4 Nas oo 8 fF UA © Thus Maeneaig code differ to pester 12/417 ethus distamnee Io Them 18 F- || Beeleaun Agelnat 11@) a Bostcally a analued boolean algebra 15 defined on a set-ef clement, B= {0,14 , with miles for the two binary operators a? + yes! os shown in Aeliewing epecter alles. ao | He | ale °o | ! e\l ' 1 10 1 io t\) 5 Beclean Expresion ot coratevcted PY connecting qr Conrlamts Vomables with Borteann pperaiv'ane ee (arene whet > & we berkeann fr gsith Yasiabbla A Bie Duality $ be obtained the binary cpa Dual of « beelean algelora may + fdenbla glemenia Lo x4) Bt Inlexchange Rerteam algebsa ls called Duality preciple - TE the cual of on algebraic Expremor * ‘nlerchan ge OR (4 AND ( opodes + vepa as. ay dual of ABTBC je carerl | “Posieher of Redon gens? he postalalés are Lorie anioms of YH algebraic grruche. breed otros % Yura t 2 dorsi pope aaa Bry = yA =A Commblatve prey | ) Tyr ar=nyees ») aya Qty C4 2)9 Diss proyel ) wats} Warso =y Compliment perey_Theorers of Bewtean Abra | The eovems must be peren from pestalala 1) 4x = % Ha =% 2) at = | tb) HoH 3la (a)! 2% 2 tovedukion Ata) X44) Harpe © ly 8)= apa wanmialive per 5@) Gry! = ey us ypiarey 9 & 6a) aint =% ch) KeKeY) = ety =) Absorphio pepe = xe my EU pa% Porhulales | @ Le Pestulate L Peshulati 2 lidentity) Aton Postulate 3 (Coumwletie) AsB=BtA _Poslalates (Ducdibutie), _| AlatedePBtAC perulae S (Cornplewert) reir ana Aa if aeraeen ge “Theorems (tegen AtAaA toe at | “theorem 3 (aovdvhes) @ b= « “Theorem 4{ Abscrphen) | Apapsa | AUIEDEA ee MABH=ATB A+H=AB | ctheorerné(AMaatie), AN (BHC) WIBHC ABey=OHC reat of thucwem 500) Atte Atansss — 51d) * A(A4BD = ANAS = ATB(ATA) = OMB = Ate o AB16(2) Operalsy precedence’ “the operals« precedence for Evaluating Borleam Expression om 4) poverttheses 2) NOT 3) AND 4)OR | Se ayy = ay : Fin Lets, The Expremion nrenvll Complumerted: FD RAS, The Covptervent of ay We towplernert OF fost 4 The result treo fnade The parorthers is evaluted first tecthen The au both evaluated Anoed, Boolean functions (sottehing functions) | Bosteay algebre ia am algebrr drak deals wits Bq vale ke | lege opestions: : J) boekean -fov deeabed by Oo algebres ¢ Espromion — binary vandables, the Constante owt ® ™ legreal operalers Sy . iA a gen value tre Kinaay “to bE Spal elthan 140: er Conside, the beoteam fo faat '3 L A. Peosteom fo Cam be frarformesd eee ie Seprenten 1° a 1 |=_¢_* t alee) My 3°83 Vastiables -fpirning am AND term) vith Zach Vamable being normal A compernent, provide ” persiele Combinations, Calted minterms &) Shorndand prodoct: Pm Vosiables forming am OR lefeo | voit Unpaimed ; prow de oo ponitle Combinat Standard Som: = Berlean funchen Exprened as Sum of minlerma A prcluck- of vroyleriai ax Said -6 be fn canmenical form: Glo Som of, Mie ( Semn=of = product “Tesrd) 8 “the minlerara Lhese Sven defines the boolean cfamahion ane dhese f Tre ofr in a truth dab gach Variable, being priened & song Called Max lems (or which give the #s © . 9 tbe D+ “NY He -{ns tak Con be formed vith nevaxiabls is Tym Papen tre brodlean for f= Atel as Som of smnleim f= atele AwtirB! = AtBed ) +(A4N Jad = AGHA = AB+AB AAR! = M3412 +mMo— Bi ad+a' al 5 (0,23)=a+0! feanracan'e! 19(a) F ABE eo} ove tor in Ey Expen the bovlean fw fea tle a Som of min Lerma ot fa A4e = ABER Cte HAtN) Be = alweandaderge rade thee = pact awelt agcrndc+ Ate sade! = Apetapdendeedde +n = Erm Cups 61> Slanded Feduct off Max lems (product -of ~ Sun len) Each of the 2" functions of bina Variable can be alo Exprersed O8 a parduct of max ler « To Expren a Beelean -frr as £ fate a for of OR lemma daw xtys = Grpcaray- chen ame rating etn 1. ORed with 19» Eq Eeprem The boolean -frv fey as peoducl of monctersar sir Fay = Grdd-lo+yd = Gry! DOG 5: ri ty Coty) Leeper +) = 4G ap Gly = 20,29 product of moxlem, H must be “thes ray be done Using dismbyhve Variable. in Zach oR aig Expren Fauyens as product of max: len i = (uy 109+ (08 #0) Fay bee HRERYS = ayers ee Gay) = YQ ray ar a'yarr'ys = EO 3r6r1) = K(0,2 4,5)20 Converston bu Comonitcal -fovens s “jhe Camglarnent of a fr Expremed as the fam Gf minlertre Eq Sawn of rnin less mabasing fiom the original fn: his ts because the exignal for & Expromed toy there moaterns peaxe re fod, WEN as fe Complemert 4, for thew. few which Ane fr & as 0 B- FcAWBC) = SU 41516079 Complement > Fl 0) = 5:(0,2,3) =mretmn*ms, Complemenk- oF % Dedenganis theoces \ 1 Fe tp tay emg) = mens M3 <= MoMe Ma= Kl0,213) is : = Mj Since we Knew that ny 2 ae ore md = (ABCD = Al 4B + Cl =Ms- Be F Cur aD = 5(1)31 617) 2 Fieangy ad = E (012/415) © marina Fen Cotmtmgms)'= rah em}: wah ml = MoMo MMe Standard ows 2 4(0/214.) we define a lijerd le be a Snge voaiable within a tern, to Comperunted A umcernplemented fm: algebra axe. basic fons gat tabler fore Least no: of eral; because gach ald the Variables, isthe The tho Cannonfeal -fowes of beote: cone oblain fom veading & gwen fr from the one very Seldom the ones voit the rinleam & mor letra must Contain; | Comeplem ented 6 uncomplementid | DE SS20( 3 Analfin way Is Exprew bodlean fr & in Stamdad form: Lo shin 9 Confiqarction lemma ‘that fon Ihe fr> re of (erate, Thee om atyper of Standand -fovns! Som q products (Sop) » Prvctuct of Sum (pend Be Fi= yl aya! is boolean Experian in Glandand Sop farm fo nye Ciiegnal) beeleasn Exprewion in Slandand patie FA bodleamn fy may be Exprened to am Slandand foro» fe Example, the for F- Are (Dee) is neilfan tm Sop 6) pes form: Cobain one hwo A on! ov BY he implemented on of hu Expremion F requires Q AND gla k Q OR qas cthoe are 3 leveb of gerg an thin Crt a Fg 3 level implementabion Th can be chomaed le standard form by vang remove Tre Prentheis ” mon Aistibohve tao = f= AB+cl(DtE) = AB+CD+DE The Sop Sxpreston koe. is implemented aw, * 8 : fo twee oe “3 devels 9 979 2 in tha Or c ao )»~ e qe 2 exe ienplerneat 35 Tn gensial, o 2 -level ienplamentation & perpame d because T probes Ahe Jeast ameunl of delay Aoough “the ee tohon the Signa propaga fom tnps to The out ptproblems on Sep les pos lets 2) Express A(A+B9 (A840) fS Sop pos lérma, Selim pong (let) F= AL ATED CRY B +E ) = (+ BB) CArB4CE) CAEBHE D = (AtB)ATB) (A4B40) CAFBIE ) (A4BAE D = (PrBrcz ) (A+BtCe) (A+ BIO) (At B4z yA4BAZ ) = BHC CAFBAT) (a4 Bre (OVEN) CA BAe) (ABET) 7 (A4B47, = @i@+c) (baz (or BecnlneRaEy (aaron are ) = My: Mir Ma Mas Me: Ms Sp = Alo, 1,213;4,5) s f= p(Ar8) (AaB sz) = A(AA+AB+AETA B+RBAEZ Y = ALN+ her ne+ 8482] HAEERAB EAR +A RN BE = OF040 + AB+ABT = AB(i+e) el = ABD = AB(C+e) -ABc+48 ze = Mm = S/T) Expres f= =mlo,2 10. Sap ge pos forms: slr FH Mo d= MptmAMs = Ar Tyr ay Dpetyrey +e = RgHp+ay = aay = (X44 Grp =O = My = AML)