Umthetho kaBenford

Ehlabathini lamanani, uhlala ufumana iipateni ezimangalisayo ezinokumangalisa kwaye zikhanyise. Omnye onjalo ngumthetho kaBenford, owaziwa ngokuba nguMthetho weDijithi yokuQala. Esi siganeko semathematika sichaza ukuhanjiswa rhoqo kwamadijithi okuqala kwiiseti ezininzi zedatha yokwenyani kwaye inika ukuqonda okunomdla kubume bamanani njengoko eyenzeka kwindawo yethu.


UMthetho kaBenford, obizwa ngokuba yi-physicist uFrank Benford owaphinda wafumanisa kwakhona ngo-1938, yinto ekhangayo yokuqaphela: kwiiseti ezininzi zendalo, ezoqoqosho kunye nezesayensi, idijithi yokuqala yamanani ayisasazwanga ngokulinganayo. Endaweni yoko, idijithi \(1\) ivela njengedijithi yokuqala rhoqo kakhulu kunamanye amanani. Ngokuthe ngqo, ukuba nokwenzeka kokuba inani liqale ngedijithi enikiweyo \(d\) linikwa yifomula

$$P(d) = \log_{10}(1 + \frac{1}{d})$$

apho \(d\) yenye yamanani \(1\) ukuya \(9\) . Le fomula ithi, umzekelo, idijithi \(1\) ibonakala njengedijithi yokuqala malunga ne \(30,1 \%\) yexesha, ngelixa idijithi \(9\) ivela kuphela malunga ne \(4,6 \%\) ixesha liyenzeka.

Umthetho unokuchazwa ngokungafaniyo kokulinganisa kwe-logarithms. Ukuba ujonga amanani ukusuka kwimiyalelo eyahlukeneyo yobukhulu kwaye uwacwangcise kwisikali se-logarithmic, amanani ambalwa okuqala aya kusasazwa njengoko kuxelwe kwangaphambili ngumthetho kaBenford. Oku kungenxa yokuba isithuba se-logarithmic phakathi kwamagunya amabini alandelelanayo ka \(10\) (umz. phakathi kwe-10 kunye \(100\) okanye phakathi \(100\) kunye \(1000\) ) iba nkulu ngokuba Amanani angawo. Oku kuthetha ukuba amanani amancinci okuqala athatha "isithuba" esingaphezulu kwaye ke ngoko kunokwenzeka ukuba kwenzeke.

UMthetho kaBenford unezicelo kwiinkalo ezahlukeneyo, ukusuka kwi-forensics ukuya kwisayensi yedatha:

  • Ukubhaqwa kobuqhophololo: Abaphicothi-zincwadi basebenzisa umthetho ukukhangela izitenxo kwiinkcukacha zemali. Ukuba ukuhanjiswa kwamadijithi okuqala kumaphepha ebhalansi enkampani kuphambuka kakhulu kuMthetho kaBenford, oku kungabonisa ukukhwabanisa okanye ubuqhophololo.
  • Uhlalutyo lwedatha yenzululwazi: Abaphandi basebenzisa umthetho ukuvavanya ukuthembeka kweeseti zedatha. Ukutenxa kulwabiwo olulindelekileyo kunokubonisa iimpazamo ekuqokeleleni idatha.

Ngaphandle kokusebenza kwawo ngokubanzi, umthetho kaBenford awusebenzi jikelele. Isebenza ngokuyinhloko kwiiseti zedatha eziqulethe amanani obungakanani obahlukeneyo kwaye zisasazwa ngokwemvelo. Uthotho lwamanani aphakathi koluhlu oluncinci okanye aqingqwe ngokuzenzela (ezifana neekhowudi ze-zip okanye iinombolo zokhuseleko loluntu) aziwulandeli lo mthetho.

UMthetho kaBenford uhlala ungomnye wemizekelo enomdla kakhulu yendlela imigaqo yemathematika enokuthi ibonakale ngayo kwihlabathi lenene ngeendlela ezingalindelekanga kunye nengqiqo. Ukusetyenziswa kwayo ekusebenzeni kubonisa ukuba imathematika ayisiyonzululwazi engabonakaliyo, kodwa sisixhobo esiluncedo sokuhlalutya inyaniso. Nokuba kukufumanisa ubuqhophololo okanye ukuqinisekisa idatha yenzululwazi, uMthetho kaBenford unika umbono okhethekileyo kumanani abumba ihlabathi lethu.

Emva