Yintoni a lencopho inombolo ingongoma?

Le ntetho yamanani yokwenene (okanye wokwenene), apho zigcinwa njenge mantissa kunye isalathi ukuba lencopho amanani ngongoma (mhlawumbi ngongoma, njengoko kulisiko kumazwe IsiNgesi abathetha). Nakuba kunjalo, inani inesixhobo ngokuchanekileyo isalamane esisigxina kwaye ukutshintsha elipheleleyo. Ukumelwa esetyenziselwa kakhulu rhoqo, evunyiweyo IEEE 754. imisebenzi eqhelekileyo Mathematical ukuba basebenzise iinombolo lencopho-yokudada ziphunyezwa kwiinkqubo computing - zombini hardware ne-software.

Point okanye isiphumlisi

Uluhlu oluneenkcukacha lwe umahluli Decimal chonga ezo mazwe IsiNgesi abathetha kunye anglofitsirovannye, apho iingxelo zamanani ezahlulwe sisithuba yinxalenye iqhezu yonke indawo, kuba isigama kula mazwe bamkela igama lencopho ngongoma - "lencopho equkuqelayo". Kwi-Russian Federation, inxalenye iqhezu yonke yesithethe, ezahlulwe ikoma, ngoko imela ingqiqo efanayo ngokwembali wayeyiqonda elithi "ezidadayo indawo". Noko ke, namhlanje namaxwebhu yobugcisa yaye iincwadi Russian uvunyelwe zombini iinketho.

Igama elithi "lencopho equkuqelayo 'isuka into yokuba imbonakaliso inani ngokokuma kukho isiphumlisi (lokugqibela eqhelekileyo okanye gocr - ngekhompyutha) ukuba ungalunga naphi phakathi amanani imigca. Lo msebenzi osemqoka qiniseka ukuba imisela kuyo ngokwahlukileyo. Oku kuthetha ukuba imbonakaliso ezidadayo amanani indawo lungathathwa ngozalisekiso computer kwe ngokwe bomtya. Uncedo lokusebenzisa ukumelwa enjalo ngendlela ukumelwa esisigxina-point kunye neenombolo elipheleleyo ukuba isukela lwamaxabiso ekhula kakhulu xa loo ukuchaneka isalamane kuhlala ingatshintshanga.

umzekelo

Ukuba isiphumlisi inani amisiweyo, ngoko awutshise lo format enye kuphela. Umzekelo, anikwe isuntswana abathandathu zamanani digits ezimbini inxalenye iqhezu. Oku kunokwenziwa kuphela ngale ndlela: 123456,78. Ifomathi ye lencopho amanani indawo enika umda ngokupheleleyo ukubonakaliswa. Umzekelo, kunikwa le mivo ezisibhozo efanayo. iinketho Ukurekhoda kungenzeka nayiphi na ukuba wenkqubo akenzi nemivo emibini sinayo umsebenzi intsimi olongezelelweyo, apho iya babhale izalathi ukuba kuvame-10, yaye ukususela-0 ukuya ku-16, kwaye khupha ngelixa inani lilonke iza kuba lishumi 8 + 2.

Ezinye embodiments kwee-recording, evumela ukuba ukufomata amanani kunye lencopho equkuqelayo: 12345678000000000000; 0,0000012345678; 123,45678; 1.2345678 njalo njalo. Kule fomati, kukho nkqu iyunithi yokulinganisa ngesantya! Kunoko, ukusebenza kwenkqubo yekhompyutha esirekhodisha nesantya apho computer owenza imisebenzi apho kukho imbonakaliso lencopho amanani ngongoma. Le yokusebenza ulinganiswa ngokwemiqathango FLOPS (imisebenzi lencopho-yokudada yesibini nganye, nto leyo ethetha ukuba inani iintengiselwano ngomzuzwana ngamnye kunye lencopho equkuqelayo). Le yunithi esisiseko kwinkqubo yekhompyutha womlinganiselo Isantya.

isakhiwo

inombolo irekhodi lencopho ifomati ingongoma kuyimfuneko ngolu hlobo lulandelayo, ukugcina ukulandelelana iinxalenye mandatory, ngenxa yokuba lo irekhodi okucacileyo, nto leyo ibonisa amanani wokwenene njenge mantissa nocwangco. Kuyimfuneko ukuba ukumela amanani ezinkulu kakhulu nezincinane kakhulu, bona lula ukufunda. iindawo efunekayo: inani abhalwe (N), i-mantissa (M), umyalelo wophawu (p) kunye umyalelo (n). Iimpawu ezimbini zokugqibela mqondiso. Ngenxa yoko, N = ^M. n ^p. Ngoko ebhalwe amanani lencopho-yokudada. Imizekelo ziza kuhluka.

1. Kuyimfuneko ukuba kurekhodwe inani million, ukwenzela ukuba ulahlekile amaqanda. 1000000 - yinto yokurekhoda eqhelekileyo, arithmetic. A computer na ngolu hlobo lulandelayo: ^1.0. October ^6. Oko kukuthi, neshumi amandla yesithandathu - iimpawu ezintathu, eziyifaneleyo ngo ezininzi kangangoko ezintandathu amaqanda. Ngaloo ndlela kwenzeka ukumelwa kwamanani indawo esisigxina nezinejika apho nangoko uyakwazi ukubona umahluko upelo.

2. Kwaye inani nzima enjalo 1,435,000,000 (enye gidi elinamanci amane anamanci mathathu anesihlanu amawaka) nazo kufuneka zibhalwe nje: ^1.435. September ^10, kuphela. Ngoko ke umqondiso unako ukubhala naliphi na inani thabatha. Yiloo nto, yaye zahluke enye kwenye kunye nenani indawo esisigxina nezinejika.

Kodwa kakhulu ukuba phantsi njani? Ewe, ngokulula kakhulu.

3. Umzekelo, njengoko uphawu olunye sesigidi? = 0.000001 ^1.0. 10 ^-6. Kakhulu lula kunye namanani nokubhala, nokuyifunda.

4. A nzima ngakumbi? Amahlanu anamashumi anesithandathu amane sewaka: 0.000000546 = ^546. 10 ^-9. Apha. Uluhlu lwe lencopho equkuqelayo libanzi kakhulu.

imilo

inombolo zefom eqhelekileyo okanye nganye ezinga malungu abantu. Normal - bawahloniphe ikwazi ezidadayo amanani ngongoma. Kufuneka kuqatshelwe ukuba mantissa kule fomu, ngaphandle kokuthatha ingqalelo sophawu, isiqingatha lokuphumla 0 1, ngoko 0 ⩽ a <1. Hayi ngendlela eqhelekileyo inani uphelelwa ichanile. Engxakini uhlobo oluqhelekileyo kukuba amanani amaninzi angabhalwa ngeendlela ezahlukeneyo, oko luyathandabuzeka. UMZEKELO iirekhodi ezahlukeneyo inani elifanayo: 0 = 0.0001, ^000001. 10 ² = ^0.00001. Januwari ¹⁰ = ^0.0001. 10 ⁰ = ^{0.001. 10 -1} = ^0,01. 10 ^-2, ngoko inokuba kakhulu ngakumbi. Kungenxa yoko le nto computer usebenzisa ubhalo eyahlukileyo nganye ezinga malungu abantu, apho kwesishumi mantissa ithatha ixabiso iiyunithi (qukwanga), yaye ngaloo ndlela ezilishumi (ezingafakwanga), kunye nangendlela efanayo mantissa njengenani eliphindwe kabini unalo ixabiso phakathi omnye (qukwanga) ukuya ezimbini (akukho qukwanga).

Ngoko ke, 1 ⩽ a <10 Le -. Amanani yokubini kunye lencopho equkuqelayo, kunye nale fom zokurekhodisha naliphi na inani (ngaphandle zero) ithumba ngendlela ekhethekileyo. Kodwa kukho ethile - ekungakwazini ngqondweni olu hlobo zero. Ngako oko yokwazisa ubonelela ukusetyenziswa kwamanani ezizodwa 0 uphawu (bit). Inxenye yexabiso (MSB) loMthetho mantissa kwenani kabini ngaphandle zero ifomu nganye ezinga malungu abantu ilingana-1 (unit yeSabelo). Le ngxelo iyasetyenziswa IEEE 754. Inkqubo Inani ngokokuma, apho isiseko ingaphezulu kwe ezimbini (izithathu, koqeqesho kunye nezinye iindlela), le yepropati zithengiwe.

wokuba

amanani Real ne lencopho point kwaye zikholisa kanye njengokuba kunjalo nje kuphela, kodwa ngendlela elula kakhulu ukumela inani lokwenyani, njengokuba kunjalo, kwizibhambathiso phakathi uluhlu lwamaxabiso kunye nokuchaneka. Oku fanekiso ukuba ubhalo bomtya, kuphela lwenziwa kwi computer. lwenani lencopho-yokudada - iseti ziikhowudi ngamnye lwahlulwe lwaba uphawu (uphawu), ukuze (isalathi) kunye mantissa (uMntanezulu). Ifomati iqhelekileyo i IEEE inombolo 754 lencopho-yokudada njenge iseti ziikhowudi ukuba encoder yinxalenye mantissa wayo, enye inxalenye - iqondo kunye bit enye ibonisa umqondiso inani: zero - ukuba umntu lowo unesi sifo, iyunithi - ukuba inani lelikhabayo. Inkqubo yonke irekhodwa yi inombolo (ikhowudi-Shift), kwaye mantissa - ngendlela nganye ezinga malungu abantu, inxalenye yayo iqhezu - kwinkqubo yokubini.

uphawu nganye - yinto bit eyodwa esibonakalisa umqondiso kuzo zonke amanani lencopho-yokudada. Mantissa nocwangco - ke integers, bona, kunye umqondiso uze wenze imbonakaliso ezidadayo amanani ngongoma. Le nkqubo kuthiwa okucacileyo okanye exponent. Ayizizo zonke amanani lokwenene emelwe kwikhompyutha in intsingiselo yazo ngqo, ezinye thaca amaxabiso luqikelelo. A ukhetho ilula kakhulu - ukuba bazithobe inani lokwenyani kunye kwindawo esisigxina, apho ngokwenene yaye inxalenye lonke iya kugcinwa yodwa. Kungenzeka, kangangokuba inxalenye elipheleleyo lisoloko eyabelwa X amasuntswana, kunye iqhezu - Y ziintanda. Kodwa sesakhiwo iziqhubekisi Aniyiva na indlela enjalo, kodwa ngenxa yokuba kukhethwa inani lencopho equkuqelayo.

Ngaphezu

Ukudibanisa lencopho amanani ngongoma ilula. Ngokuphathelele ne IEEE 754 inombolo ngendlela enye oqhelekileyo na inani elikhulu ziikhowudi, ngoko ke kungcono ukuba ugqithele imizekelo, kunye nembono ukuthatha elona nani lincinane lencopho-yokudada. Umzekelo, amanani amabini - X kunye Y.

Amanyathelo zezi zilandelayo:

a) Amanani kufuneka amelweyo ifomu nganye. Kuyinto ngokucacileyo lowo efihlakeleyo. X = ^1,110. 2 ^2, kunye Y = ^1000. 2 ^0.

b) Qhuba inkqubo ubume nga alinganayo kuphela umbukiso, kodwa kufuneka ubhale ixabiso uY Kuya ilingane nexabiso amanani nganye ezinga malungu abantu, nangona eneneni - unnormalizes.

Bala umahluko phakathi izalathi of 2 - 0 = 2. Ngoku yisa mantissa ukuze abuyekeze ngenxa yokutshintsha, oko kukuthi, dibanisa 2 ukuya isalathisi kwekota yesibini, ngaloo ndlela uhambisa iiyunithi ezifihlakeleyo isiphumlisi kwiindawo ezimbini ukuya ekhohlo. 0,0100 ^ifunyenwe. 2 ^2. Oku kuya kuba ezilingana nexabiso yangaphambili Y, ngoko kukho sele a Y '.

c) Ngoku kufuneka dibanisa inani mantissa X kunye Y. luhlengahlengiswe

1,110 + 0,01 = 10,0

Exhibitor kusekho imelwe X i parameter, leyo ilingana no-2.

g) I-mali efunyenwe inyathelo odlulileyo, wayeka iyunithi ukuzinza, ngoko ke kufuneka siguqule isixa isalathi uphinde. 10.0 iyasebenza ungazama ukunciphisa ezimbini ukuya ekhohlo kwesishumi, inani kuyimfuneko ngoku bunjani, oko kukuthi, hambisa isiphumlisi ukuya ekhohlo ngenqaku elinye, kwaye isalathi, ngokulandelelana, kunyuke ngo 1. Kubonakala ^1,000. Matshi ^2.

e) Lixesha ukuguqula i lencopho inombolo ngongoma inkqubo-byte.

isiphelo

Njengoko ubona, la manani akukho nzima kakhulu, nantoni na ndiyihlanganise ndiyidadise ikoma. Ngaphandle kokuba, kakade, ngaphandle lokuzisa inani isalathi elisezantsi phakathi ngaphezulu (kulo mzekelo ungentla, kwaba Y X), kwakunye kukubuyisela kwayo ngoku, okt umba yembuyekezo - ukuhambisa kwesishumi ukuya ekhohlo mantissa. Xa Ukongeza sele isicelo, kunokwenzeka kakhulu kwaye ingxaki enye - perenormirovanie kunye akhronimi bit ukuba inani labo ayihambelani inani ukumela oko.

ukuphinda-phinda

inkqubo Binary inikeza iindlela ezimbini eziya gaya amanani lencopho-yokudada. Lo msebenzi uya kwenziwa phindo, oqalisa kunye amasuntswana ubuncinane ezibalulekileyo kwaye uqala amasuntswana umyalelo omkhulu lingabali ezinye. Zombini ezi meko ibe inani imisebenzi ngokulandelelana ingqokelela imveliso buso. Ezi imisebenzi zilawulwa ngokuthi kongezwe ziikhowudi wokuphindaphinda. Ngoko ke, xa elinye wayemana lingabali libe linye, isixa imveliso kwenxenye ye multiplicand ukhule a shift ahambelanayo. Ukuba amivo mi-lingabali ngekroba zero, lo gama i-multiplicand leyo added.

Ukuba phindo lwenziwa amanani nje ezimbini, imveliso yamanani imali yayo ayikwazi ukudlula inani lamasuntswana equlethwe izinto, ngaphezu kabini, kwaye amanani amakhulu ngayo kakhulu, kakhulu kunene. Ukuba liphindaphindwe abathile ngenani, imveliso imingcipheko okungalungelaniyo kwiskrini. Ngenxa inani ziikhowudi nawuphi na umatshini lwedijithali anesiphelo kakhulu, kwaye kunyanzela ukuba bamvalele ezona kabini inani amarhamba amanani. Ukuba inani iindawo lunqongophele, kwimveliso baya- lwazisa iimpazamo. Ukuba isixa ikhompyutha mkhulu, isiphoso ukudibana, yaye ngenxa yoko kwandisa kakhulu ukuchana lulonke. Apha, ekuphela kwendlela - ukurhangqa iziphumo phindo, ngoko imisebenzi impazamo zaye alternating. Xa umsebenzi-phindo, kuba kungenzeka ukuba uhambe ngaphaya kwigridi lemivo, kodwa kuphela kwencinane, ngokuba kukho umda ebekelwa inani apho zimelwe ngohlobo fixed-point.

ezinye iingcaciso

Kulungile ukuqala ukusuka ekuqaleni. Eyona ndlela iqhelekileyo ukumela inani - amanani kumgca njenge inani elipheleleyo, apho isiphumlisi akunto imbi kanye ekupheleni. Le umtya inokuba yiyo nayiphi ubude, kodwa isiphumlisi umi kuloo ndawo unelungelo lokufaka kuyo, yahlukanisa kwinani ukusuka inxalenye iqhezu kuyo. Ifomathi ntetho kwenkqubo fixed-point ngokuyimfuneko ubeka iimeko ezithile kwi indawo kwesishumi. notation Scientific isebenzisa imboniselo umgangatho nganye ye ukumelwa amanani. It aqn {\ displaystyle aq ^ {n }} aq n. Apha a {\ displaystyle ^a} a, yaye ibizwa ngokuba umtya mantissa. Nje ngayo kuye kwathiwa 0 ⩽ a icace: n {/ displaystyle n} n - an isalathi elipheleleyo, kunye ^q {/ q displaystyle} q - nayo elipheleleyo, nto leyo isisiseko radix (ileta isoloko 10). Mantissa shiya isiphumlisi emva yokuqala, nto leyo zero, kodwa ukurekhodwa ngakumbi ikhutshelwa ingcaciso ixabiso elikhoyo inani.

lwenani lencopho-yokudada kubhaliwe efana kakhulu onke amanani gca ungeno, kuphela isalathi kunye mantissa abhalwe ngokwahlukeneyo. Yokugqibela ukuya efanayo yaye ngendlela nganye ezinga - indawo esisigxina, nto leyo zinokuhonjiswa yokuqala ebalulekileyo. Nje lencopho point lisetyenziswa kwi-computer, oko kukuthi, ukumelwa ngekhompyutha apho le nkqubo lesishumi kwaye yokubini, apho de mantissa Denormalize ingongoma ahlenga - ngoku phambi kokuba yokuqala, ngoko ngaphambili, kungekhona emva kwayo, apho inxalenye elipheleleyo simiso, akanakuba. Umzekelo, inkqubo yethu lokugqibela bakhe uya kunikela inkqubo yakhe ezilithoba yokubini ukusetyenziswa okwethutyana. Kwaye ukuba abhale kunye mantissa yayo lencopho-yokudada efana nale: +1001000 ... 0, kwaye kunye isalathiso 0 ... 0100. Kodwa inkqubo wokugqibela ongaphumeleliyo ukuveza izibalo olunzima, olunokuba okubini, usebenzisa uhlobo lencopho equkuqelayo.

izibalo elide

Xa iikhompyutha elektroniki ziye eyakhelwe-ngaphakathi iipakethe software, apho ebe zabelwe i-mantissa kunye isalathi le Ubungakanani benkumbulo elikhankanywe software, umda kuphela ubungakanani bememori computer. Kubonakala ngathi izibalo elide, oko kukuthi, imisebenzi elula kumanani eyenza ikhompyutha. Kuxhomekeke kuyafana - ukuthabatha kunye Ukongeza, icandelo kunye phindo, imisebenzi nezakhiwo kunye nokwakhiwa ingcambu. Kodwa ke inani ezahlukeneyo kakhulu, umthamo wabo kuphezulu kunoko ubude igama machine. Ukuphunyezwa kwale misebenzi asiyi hardware ne-software, kodwa lisetyenziswa iintsimbi esisiseko ukusebenza ngamanani amancinci kakhulu kwee-odolo. Kukho ngaphezulu nasekubaleni, apho amanani ubude umda kuphela ngememori - engenasizathu ngendlela arithmetic. A arithmetic elide isetyenziswa kwiinkalo ezininzi.

1. Ukuze ukuqokelela ikhowudi (iziqhubekisi, microcontrollers kunye ubunzulu besuntswana eliphantsi - bit-10 iirejista kunye bit ezisibhozo ubude ilizwi, akwanelanga ukuba aphathe ulwazi ukusuka Analog-to-digital (analog-to-digital converter), kwaye ke ayikwazi ukwenza ngaphandle izibalo elide.

2. Oku kwakhona izibalo elide zisetyenziselwa zokufihlakeleyoComment, apho kuyimfuneko ukuqinisekisa ukuchaneka ngesiphumo exponentiation okanye phinda ukuya ^10.309. Inani elipheleleyo arithmetic iyasetyenziswa modulo m - inani elikhulu zendalo, kwaye ayikho into elula.

3. Software for bezimali kunye zezibalo, kakhulu, ayikho ngaphandle izibalo elide, ngokuba nde kwendlela kuphela ukuqinisekisa iziphumo izibalo ephepheni - ngoncedo ikhompyutha, ukuqinisekisa ukuchaneka aphezulu amanani. Incopho kwinqanaba kunokuquka naliphi na inani yokumpompoza elide. Kodwa ke izibalo zobunjineli umsebenzi izazinzulu zifuna izibalo inkqubo yongenelelo rhoqo kakhulu, kuba kunzima kakhulu ukwenza i data igalelo ngaphandle kokwenza iimpazamo. zisoloko kakhulu voluminous ngaphezu iziphumo ngokusondeza.

Alwe iimpazamo

Xa inani imisebenzi apho lencopho equkuqelayo, kunzima kakhulu ukuvavanya ukufaneleka kweziphumo. Hayi kodwa owavela ngokwanelisa zonke ingcamango zemathematika eziya kunceda ukusombulula le ngxaki. Kodwa elipheleleyo impazamo ukuhlola lula. Amathuba ukubatshabalalisa ezingachanekanga phezu - sebenzisa nje kuphela inani eliqingqiweyo-yokudada. Umzekelo, inkqubo zemali wakha phezu kweli mgaqo. Nangona kunjalo, kukho elula: inani elifunekayo amanani emva kokuba kwesishumi laziwa kwangaphambili.

Ezinye izicelo zingaphelelanga, kuba awukwazi ukusebenza kunye nokuba amanani amancinane kakhulu okanye ebanzi kakhulu. Ngoko xa usebenza usoloko sithabathela ingqalelo ukuba kunokubakho ezingachanekanga, yaye kuba Ukuveliswa kweziphumo kuyimfuneko ukuba ngeenxa zonke. Ngaphezu koko, ukuqoqa ezenzekelayo isoloko zokusilela kwenkqubo, yaye ngoko ke sondeza ichazwa ngokungqalileyo. kuyingozi kakhulu kule nkalo, lo msebenzi uthelekiso. nokuba kukho ukuqikelela ubungakanani iimpazamo elizayo kunzima kakhulu.