Erinevus lehekülje "Faktoriaal" redaktsioonide vahel

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loetelud muudetud
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{| class="wikitable" style="margin:0 0 0 1em; text-align:right; float:right;"
{{ToimetaAeg|kuu=veebruar|aasta=2008}}
|+ Valitud liikmed faktoriaalide [[jada]]st {{OEIS|id=A000142}}; [[arvu standardkuju|standardkujul]] esitatud väärtused on [[ümardamine|ümardatud]] antud täpsuseni
[[Naturaalarv]]u ''n'' '''faktoriaal''' (tähistus ''n''!) on ''n'' esimese positiivse täisarvu [[korrutis]].
|-
! ''n''
! ''n''!
|-
| 0 || 1
|-
| 1 || 1
|-
| 2 || 2
|-
| 3 || 6
|-
| 4 || 24
|-
| 5 || 120
|-
| 6 || 720
|-
| 7 || {{gaps|5|040}}
|-
| 8 || {{gaps|40|320}}
|-
| 9 || {{gaps|362|880}}
|-
| 10 || {{gaps|3|628|800}}
|-
| 11 || {{gaps|39|916|800}}
|-
| 12 || {{gaps|479|001|600}}
|-
| 13 || {{gaps|6|227|020|800}}
|-
| 14 || {{gaps|87|178|291|200}}
|-
| 15 || {{gaps|1|307|674|368|000}}
|-
| 16 || {{gaps|20|922|789|888|000}}
|-
| 17 || {{gaps|355|687|428|096|000}}
|-
| 18 || {{gaps|6|402|373|705|728|000}}
|-
| 19 || {{gaps|121|645|100|408|832|000}}
|-
| 20 || {{gaps|2|432|902|008|176|640|000}}
|-
| 25 || {{val|1.5511210043|e=25}}
|-
| 42 || {{val|1.4050061178|e=51}}
|-
| 50 || {{val|3.0414093202|e=64}}
|-
| 70 || {{val|1.1978571670|e=100}}
|-
| 100 || {{val|9.3326215444|e=157}}
|-
| 450 || {{val|1.7333687331|e=1000}}
|-
| {{gaps|1|000}} || {{val|4.0238726008|e=2567}}
|-
| {{gaps|3|249}} || {{val|6.4123376883|e=10000}}
|-
| {{gaps|10|000}} || {{val|2.8462596809|e=35659}}
|-
| {{gaps|25|206}} || {{val|1.2057034382|e=100000}}
|-
| {{gaps|100|000}} || {{val|2.8242294080|e=456573}}
|-
| {{gaps|205|023}} || {{val|2.5038989317|e=1000004}}
|-
| {{gaps|1|000|000}} || {{val|8.2639316883|e=5565708}}
|-
| {{val|1.0248383838|e=98}} || [[googolplex|10<sup>{{val|e=100}}</sup>]]
|-
| [[googol|{{val|e=100}}]] || 10<sup>{{val|9.9565705518|e=101}}</sup>
|}
 
[[Naturaalarv]]u ''n'' '''faktoriaal''' (tähistus ''n''!) on ''n'' esimese positiivse täisarvu [[korrutis]].<ref name="Kaasik2002"> Kaasik, Ü. (2002). ''Matemaatikaleksikon''. Tartu. </ref>
 
== Definitsioon ==
Kui ''n'' on [[positiivne]] [[täisarv]], siis
 
:<math>n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n = \prod_{i=1}^n i</math>,
 
On kokku lepitud, et
 
Negatiivsete arvude jaoks pole faktoriaal defineeritud.
 
== Näited ==
 
Näiteks 5! = 1 &middot; 2 &middot; 3 &middot; 4 &middot; 5 = 120
 
Esimese 110 positiivse täisarvu faktoriaalid:
*1! = 1
*2! = 2
*3! = 6
*4! = 24
*5! = 120
*6! = 720
*7! = 5040
*8! = 40320
*9! = 362880
*10! = 3628800
* 11! = 39916800
* 12! = 479001600
* 13! = 6227020800
* 14! = 87178291200
* 15! = 1307674368000
* 16! = 20922789888000
* 17! = 355687428096000
* 18! = 6402373705728000
* 19! = 121645100408832000
* 20! = 2432902008176640000
* 21! = 51090942171709440000
* 22! = 1124000727777607680000
* 23! = 25852016738884976640000
* 24! = 620448401733239439360000
* 25! = 15511210043330985984000000
* 26! = 403291461126605635584000000
* 27! = 10888869450418352160768000000
* 28! = 304888344611713860501504000000
* 29! = 9789093635926098592530432000000
* 30! = 265252859812191058636308480000000
* 31! = 8222838654177922817725562880000000
* 32! = 263130836933693530167218012160000000
* 33! = 8683317618811886495518194401280000000
* 34! = 295232799039604140847618609643520000000
* 35! = 10333147966386144929666651337523200000000
* 36! = 371993326789901217467999448150835200000000
* 37! = 13763753091226345046315979581580902400000000
* 38! = 523022617466601111760007224100074291200000000
* 39! = 20397882081197443358640281739902897356800000000
* 40! = 815915283247897734345611269596115894272000000000
* 41! = 33452526613163807108170062053440751665152000000000
* 42! = 1405006117752879898543142606244511569936384000000000
* 43! = 60415263063373835637355132068513997507264512000000000
* 44! = 2658271574788448768043625811014615890319638528000000000
* 45! = 119622220865480194561963161495657715064383733760000000000
* 46! = 5502622159812088949850305428800254892961651752960000000000
* 47! = 258623241511168180642964355153611979969197632389120000000000
* 48! = 12413915592536072670862289047373375038521486354677760000000000
* 49! = 608281864034267560872252163321295376887552831379210240000000000
* 50! = 30414093201713378043612608166064768844377641568960512000000000000
* 51! = 1551118753287382280224243016469303211063259720016986112000000000000
* 52! = 80658175170943878571660636856403766975289505440883277824000000000000
* 53! = 4274883284060025564298013753389399649690343788366813724672000000000000
* 54! = 230843697339241380472092742683027581083278564571807941132288000000000000
* 55! = 12696403353658275925965100847566516959580321051449436762275840000000000000
* 56! = 710998587804863451854045647463724949736497978881168458687447040000000000000
* 57! = 40526919504877216755680601905432322134980384796226602145184481280000000000000
* 58! = 2350561331282878571829474910515074683828862318181142924420699914240000000000000
* 59! = 138683118545689835737939019720389406345902876772687432540821294940160000000000000
* 60! = 8320987112741390144276341183223364380754172606361245952449277696409600000000000000
* 61! = 507580213877224798800856812176625227226004528988036003099405939480985600000000000000
* 62! = 31469973260387937525653122354950764088012280797258232192163168247821107200000000000000
* 63! = 1982608315404440064116146708361898137544773690227268628106279599612729753600000000000000
* 64! = 126886932185884164103433389335161480802865516174545192198801894375214704230400000000000000
* 65! = 8247650592082470666723170306785496252186258551345437492922123134388955774976000000000000000
* 66! = 544344939077443064003729240247842752644293064388798874532860126869671081148416000000000000000
* 67! = 36471110918188685288249859096605464427167635314049524593701628500267962436943872000000000000000
* 68! = 2480035542436830599600990418569171581047399201355367672371710738018221445712183296000000000000000
* 69! = 171122452428141311372468338881272839092270544893520369393648040923257279754140647424000000000000000
* 70! = 11978571669969891796072783721689098736458938142546425857555362864628009582789845319680000000000000000
* 71! = 850478588567862317521167644239926010288584608120796235886430763388588680378079017697280000000000000000
* 72! = 61234458376886086861524070385274672740778091784697328983823014963978384987221689274204160000000000000000
* 73! = 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000000
* 74! = 330788544151938641225953028221253782145683251820934971170611926835411235700971565459250872320000000000000000
* 75! = 24809140811395398091946477116594033660926243886570122837795894512655842677572867409443815424000000000000000000
* 76! = 1885494701666050254987932260861146558230394535379329335672487982961844043495537923117729972224000000000000000000
* 77! = 145183092028285869634070784086308284983740379224208358846781574688061991349156420080065207861248000000000000000000
* 78! = 11324281178206297831457521158732046228731749579488251990048962825668835325234200766245086213177344000000000000000000
* 79! = 894618213078297528685144171539831652069808216779571907213868063227837990693501860533361810841010176000000000000000000
* 80! = 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000
* 81! = 5797126020747367985879734231578109105412357244731625958745865049716390179693892056256184534249745940480000000000000000000
* 82! = 475364333701284174842138206989404946643813294067993328617160934076743994734899148613007131808479167119360000000000000000000
* 83! = 39455239697206586511897471180120610571436503407643446275224357528369751562996629334879591940103770870906880000000000000000000
* 84! = 3314240134565353266999387579130131288000666286242049487118846032383059131291716864129885722968716753156177920000000000000000000
* 85! = 281710411438055027694947944226061159480056634330574206405101912752560026159795933451040286452340924018275123200000000000000000000
* 86! = 24227095383672732381765523203441259715284870552429381750838764496720162249742450276789464634901319465571660595200000000000000000000
* 87! = 2107757298379527717213600518699389595229783738061356212322972511214654115727593174080683423236414793504734471782400000000000000000000
* 88! = 185482642257398439114796845645546284380220968949399346684421580986889562184028199319100141244804501828416633516851200000000000000000000
* 89! = 16507955160908461081216919262453619309839666236496541854913520707833171034378509739399912570787600662729080382999756800000000000000000000
* 90! = 1485715964481761497309522733620825737885569961284688766942216863704985393094065876545992131370884059645617234469978112000000000000000000000
* 91! = 135200152767840296255166568759495142147586866476906677791741734597153670771559994765685283954750449427751168336768008192000000000000000000000
* 92! = 12438414054641307255475324325873553077577991715875414356840239582938137710983519518443046123837041347353107486982656753664000000000000000000000
* 93! = 1156772507081641574759205162306240436214753229576413535186142281213246807121467315215203289516844845303838996289387078090752000000000000000000000
* 94! = 108736615665674308027365285256786601004186803580182872307497374434045199869417927630229109214583415458560865651202385340530688000000000000000000000
* 95! = 10329978488239059262599702099394727095397746340117372869212250571234293987594703124871765375385424468563282236864226607350415360000000000000000000000
* 96! = 991677934870949689209571401541893801158183648651267795444376054838492222809091499987689476037000748982075094738965754305639874560000000000000000000000
* 97! = 96192759682482119853328425949563698712343813919172976158104477319333745612481875498805879175589072651261284189679678167647067832320000000000000000000000
* 98! = 9426890448883247745626185743057242473809693764078951663494238777294707070023223798882976159207729119823605850588608460429412647567360000000000000000000000
* 99! = 933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000
* 100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
* 101! = 9425947759838359420851623124482936749562312794702543768327889353416977599316221476503087861591808346911623490003549599583369706302603264000000000000000000000000
* 102! = 961446671503512660926865558697259548455355905059659464369444714048531715130254590603314961882364451384985595980362059157503710042865532928000000000000000000000000
* 103! = 99029007164861804075467152545817733490901658221144924830052805546998766658416222832141441073883538492653516385977292093222882134415149891584000000000000000000000000
* 104! = 10299016745145627623848583864765044283053772454999072182325491776887871732475287174542709871683888003235965704141638377695179741979175588724736000000000000000000000000
* 105! = 1081396758240290900504101305800329649720646107774902579144176636573226531909905153326984536526808240339776398934872029657993872907813436816097280000000000000000000000000
* 106! = 114628056373470835453434738414834942870388487424139673389282723476762012382449946252660360871841673476016298287096435143747350528228224302506311680000000000000000000000000
* 107! = 12265202031961379393517517010387338887131568154382945052653251412013535324922144249034658613287059061933743916719318560380966506520420000368175349760000000000000000000000000
* 108! = 1324641819451828974499891837121832599810209360673358065686551152497461815091591578895743130235002378688844343005686404521144382704205360039762937774080000000000000000000000000
* 109! = 144385958320249358220488210246279753379312820313396029159834075622223337844983482099636001195615259277084033387619818092804737714758384244334160217374720000000000000000000000000
* 110! = 15882455415227429404253703127090772871724410234473563207581748318444567162948183030959960131517678520479243672638179990208521148623422266876757623911219200000000000000000000000000
...
 
(loetelu jada 500'st esimesest elemendist võid leida [[Kasutaja:Margusmartsepp/kasutajaartiklid/Faktoriaal|siit]])
 
== Stirlingi valem ==
Stirlingi valemi abil saab näidata, et
Arvus 10! on 7 numbrit<br>
Arvus 20! on 19 numbrit<br>
Arvus 50! on 65 numbrit<br>
Arvus 100! on 158 numbrit<br>
Arvus 200! on 375 numbrit<br>
Arvus 500! on 1135 numbrit<br>
Arvus 1000! on 2568 numbrit<br>
Arvus 2000! on 5736 numbrit<br>
Arvus 5000! on 16 326 numbrit<br>
Arvus 10 000! on 35 660 numbrit
 
on faktoriaali üldistus kompleksarvude jaoks. Gammafunktsioon on faktoriaaliga seotud kui
:<math> \Gamma(n) = (n-1)!\ . </math>
 
 
==Vaata ka==
 
*[[Kahekordne faktoriaal]]
 
== Viited ==
 
{{viited}}
 
== Välislink ==
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