
napier kii haddiyaan

napier kii haddiyaan ack manual roop seey sanchaalit ganana upkaran haye jise merchiston, scotland key johnn napier dwara utpaadon aur sankhyaon key anshon kii ganana key liye banaaya gayaa haye. yeh vidhi jaali gunan parr adharit thee, aur ise "rabdology" bhee kahaa jaataa thaa, ack shabd jiska aavishkaar napier ney kiya thaa. napier ney 1617 mein apana sanskaran prakaashit kiya. yeh edinberg mein mudrit kiya gayaa thaa aur apane sanrakshak alekjendar setton koo samarpit kiya gayaa thaa.
chhad mein antarnihit gunan taalikaon kaa upayog karke, gunan koo jodd sanchaalan aur ghataav mein vibhajan mein cumm kiya jaa sakata haye. chhad kaa unnat upayog vargamool nikaal sakata haye. napier kii haddiyaan laghuganak key samaan naheen hain, jinake saath napier kaa naam bhee judaa huaa haye, lekin vichhedit gunan taalikaon parr adharit hain.
poorna upkaran mein aamtaur parr ack rima key saath ack aadhar board shamil hotaa haye; upayogkarta gunaa yaa vibhajan karane key liye rima key ander napier kii chhad rakhta haye. board key baaen kinare koo nau vargon mein vibhajit kiya gayaa haye, jisamen sankhya 1 seey 9 taka hoti haye. napier key mool design mein, chhaden dhaatu, lakadi yaa hathidant seey bunny hoti hain aur ack vargakar cross-section hotaa haye. pratyek chhad koo chaar cheharon mein seey pratyek parr ack gunan talika key saath utkeern kiya gayaa haye. kuchh baad key designs mein, chhaden sapaat hoti hain aur una parr dou mez yaa kewal ack utkeern hoti haye, aur plastic yaa bhaari cardboard seey bunny hoti hain. aisee haddiyon kaa ack samooh ack leey jaane vaale mamle mein sanlagn hoe sakata haye.
ack chhad kaa chehra nau vargon seey chihnit hotaa haye. sheersh koo chhodkar pratyek varg koo neeche baaen koney seey oopar daaen taka ack vikarn rekha dwara dou hisson mein vibhajit kiya jaataa haye. varg mein ack saral gunan talika hoti haye. pahale mein ack hii ank hotaa haye, jise napier 'ekal' kahata haye. anya ekal key gunakon koo pakadate hain, arthaat ekal key dou baar, ekal key teen baar aur nauvein varg taka jisamen sheersh varg mein nau gunaa sankhya hoti haye. ekal-ankiya sankhyaaen nichale daaen trikon mein likhi jaati hain aur dusare trikon koo khaali chhod dete hain, jabaki dou ankon kii sankhyaaen vikarn key donon or ack ank key saath likhi jaati hain.
yadi taalikaon koo ack tarfa chhad parr rakhaa jaataa haye, tou 4-ankon kii sankhyaon koo gunaa karane key liye 40 chhad kii aavashyakta hoti haye-kyonki sankhyaon mein baar-baar ank hoe sakate hain, 0 seey 9 ankon mein seey pratyek key liye gunan talika kii chaar pratiyon kii aavashyakta hoti haye. yadi vargakar chhad kaa upayog kiya jaataa haye, tou 40 gunan taalikaon koo 10 chhad parr utkeern kiya jaa sakata haye. napier ney taalikaon kii vyavastha karane key liye ack yojana kaa vivaran diyaa taaki kissy bhee chhad mein ack hii talika kii dou pratiyaan naa houn, jisase harr sambhav chaar ankon kii sankhya koo 10 mein seey 4 chhad dwara darshaaya jaa sakey. 20 chhadon kaa ack samooh, jisamen napier kii 10 chhad kii dou samaan pratiyaan hoti hain, aathh ankon taka kii sankhya key saath ganana kii anumati detaa haye, aur 30 chhad key ack sett kaa upayog 12 ankon kii sankhya key liye kiya jaa sakata haye.
gunaa.
sabse saral prakaar kaa gunan, ack ekal ank vaali sankhya dwara kaee ankon vaali sankhya, baain or key kinare key khilaf frame mein bahu-ankon kii sankhya kaa pratinidhitva karane vaali chhad lagaakar kiya jaataa haye. uttar koo ekal-ank sankhya key anuroop pankti seey padhaa jaataa haye joe frame key baain or chihnit haye, jisamen ack chhoti matra mein atirikt kii aavashyakta hoti haye, jaisa kii neeche diye gaye udaharanon mein samajhaaya gayaa haye.
jaba ack bahu-ankiya sankhya koo ack anya bahu-ankiya sankhya seey gunaa kiya jaataa haye, tou badee sankhya frame mein chhad parr sett kii jaati haye. chhoti sankhya key pratyek ank seey gunaa karane key liye upkaran dwara ack madhyavarti parinam utpann kiya jaataa haye. inhen likhaa jaataa haye aur antim parinam kii ganana kalam aur kagaj seey kii jaati haye.
gunan key liye napier kii haddiyon kaa upayog kaise kiya jaae, yeh pradarshit karane key liye, badhati kathinaaee key teen udaaharan neeche bataae gaye hain.
udaaharan 1-ack chhoti ekal-ankiya sankhya seey gunaa.
pahala udaaharan ganana kartaa haye.
4, 2 aur 5 key liye napier kii haddiyon koo crum mein board mein rakhaa jaataa haye. yeye haddiyaan badee aakriti dikhaati hain joe gunaa kii jaayegi. pratyek stambh yaa huddy mein cumm sankhyaaen, ack vikarn rekha key oopar aur neeche sthit, sambandhit purnank key liye sadharan gunan taalikaon dwara paae jaane vaale ank hain. (udaaharan key liye, 4 huddy kii saatvin pankti mein dikhaae gaye ank, pratinidhitva karate hain.) key liye neeche diye gaye udaaharan mein, haddiyon koo yahaan laal (4), peela (2), aur neela (5) key roop mein darshaaya gayaa haye.
sabse baaen stambh, joe rung kii haddiyon seey pahale dikhaaya gayaa haye, 1 huddy kaa pratinidhitva curr sakata haye. (pratyek ank key oopari baain or ack khaali sthaan yaa shoonya, ack vikarn rekha dwara alag kiya gayaa, samajha jaanaa chaahiye, kyonki,, aadi.) ack chhoti sankhya chuni jaati haye, aamtaur parr 2 seey 9, jisase badee sankhya koo gunaa kiya jaataa haye. iss udaaharan mein chhoti sankhya jisase badee koo gunaa karna haye vaha 6 haye. kshaitij pankti jisamen yeh sankhya khadee haye, shesh gananaon koo karane key liye aavashyak ekmatra pankti haye aur abb ise prithak roop seey dekhaa jaa sakata haye.
ganana key liye, urdhvaadhar rekhaaon dwara alag kiye gaye ank (i.ee. vikarn rekhaaon key beach jodee gaee, ack huddy seey doosari huddy taka paar karane key liye) utpaad key ankon koo banaane key liye ack saath jodaa jaataa haye. usa pankti parr antim (sabse daaen) sankhya koo kabhi bhee jodane kii aavashyakta naheen hogi, kyonki yeh hamesha antim vikarn rekha dwara alag kiya jaataa haye, aur hamesha utpaad kaa antim ank hoga. iss udaaharan mein, chaar ank hain, kyonki vikarn rekhaaon key beach huddy key moolyon key chaar samooh hain. utpaad key ank ganana key anusaar baaen seey daaen crum mein khade honge. pahale aur antim ank key alawa, utpaad key pratyek ank dou alag-alag haddiyon seey liye gaye dou moolyon kaa yog hoga.
utpaad key ankon koo khojane key liye, jaisa kii oopar varnit haye, huddy key moolyon koo ack saath jodaa jaataa haye. iss aarekh mein, peele aur neele rung kii haddiyon key teesare utpaad ank key praasangik moolya harey rung key hotey hain. pratyek raashi neeche diye gaye sthaan mein likhi gaee haye. baaen seey daaen taka yogon kaa crum 2550 kaa aankada utpann kartaa haye. isaliye, 425 koo 6 seey gunaa karane kaa samadhan 2550 haye.
udaaharan 2-ack badee ekal-ankiya sankhya seey gunaa.
jaba badey ekal ankon seey gunaa kiya jaataa haye, tou yeh aam baat haye kii ack vikarn stambh jodane parr, sankhyaon kaa yog ack sankhya mein hotaa haye joe 10 yaa usase adhik haye.
doosara udaaharan ganana kartaa haye. 
udaaharan 1 kii tarah, sabse badee sankhya key liye sambandhit haddiyon koo board mein rakhaa jaataa haye. iss udaaharan key liye, haddiyon 6,7,8 aur 5 koo uchit crum mein rakhaa gayaa thaa jaisa kii neeche dikhaaya gayaa haye.
pahale stambh mein, vaha sankhya sthit haye jisase sabse badee sankhya koo gunaa kiya jaataa haye. iss udaaharan mein, sankhya 8 thee. shesh gananaon key liye kewal pankti 8 kaa upayog kiya jaayega, isaliye shesh charanon koo samajhaane mein spashtata key liye board key baaki hisson koo saaf curr diyaa gayaa haye.
pahale kii tarah, pratyek vikarn stambh kaa mulyankan kiya jaataa haye, joe daaeen or seey shuru hotaa haye. yadi ack vikarn stambh kaa yog 10 yaa usase adhik key barabar haye, tou iss yog key "duss" sthaan koo oopar leey jaaya jaanaa chaahiye aur neeche dikhaae gaye anusaar aasann baaen stambh mein sankhyaon key saath jodaa jaanaa chaahiye.
pratyek vikarn stambh kaa mulyankan karane key baad, ganana kii gaee sankhyaon koo antim uttar dene key liye baaen seey daaen padhaa jaataa haye; iss udaaharan mein, 54280 kaa utpaadan kiya gayaa thaa. 
isalieh 6785 koo 8 seey gunaa karane kaa samadhan 54280 haye. 
udaaharan 3-ack bahu-ankiya sankhya seey gunaa.
teesara udaaharan ganana kartaa haye.
pramukh sankhya key anuroop haddiyon koo board mein rakhaa jaataa haye. iss udaaharan key liye, haddiyon 8,2 aur 5 koo uchit crum mein rakhaa gayaa thaa jaisa kii neeche dikhaaya gayaa haye.
ack bahu-ankiya sankhya seey gunaa karane key liye, kaee panktiyon kii sameeksha kii jaati haye. iss udaaharan key liye, 9,1 aur 3 key liye panktiyon koo spashtata key liye board seey hataa diyaa gayaa haye.
pratyek pankti kaa vyaktigat roop seey mulyankan kiya jaataa haye aur pratyek vikarn stambh koo jodaa jaataa haye jaisa kii pichhale udaharanon mein samajhaaya gayaa haye. yog koo baaen seey daaen padhaa jaataa haye, jisase lambi haath jodd ganana key liye aavashyak sankhyaaen utpann hoti hain. iss udaaharan key liye, neeche dikhaae gaye parinamon koo utpann karane key liye pankti 9, pankti 1 aur pankti 3 kaa alag seey mulyankan kiya gayaa thaa.
doosari sankhya key sabse daahine ank seey shuru karate hue, raashi koo panktiyon seey kramik crum mein rakhaa jaataa haye jaisa kii ack sthaan dhaarak key liye 0 kaa upayog karate hue ack dusare key neeche daaen seey baaen dekhaa jaataa haye. 
 2475
 8250
 742500
antim uttar dene key liye panktiyon aur sthaan dharakon koo jodaa jaataa haye.
 2475
 8250
 + 742500
 753225
iss udaaharan mein, antim uttar 753225 thaa. isalieh 825 koo 913 seey gunaa karane kaa samadhan 753225 haye.
vibhajan.
vibhajan isi tarah seey kiya jaataa haye. 46785399 koo 96431 seey vibhajit karane key liye, vibhaajak (96431) key liye pattiyon koo board parr rakhaa jaataa haye, jaisa kii neeche diye gaye chitra mein dikhaaya gayaa haye. abecus kaa upayog karate hue, 1 seey 9 taka key bhaajak key sabhi utpaad pradarshit sankhyaon koo padhkar paae jaate hain. dhyaan dein kii labhansh mein aathh ank hotey hain, jabaki aanshik utpaadon (pahale key liye bachaaen) mein sabhi mein chhah hotey hain. isaliye 46785399 key antim dou ankon, arthaat '99' koo asthaayi roop seey nazarandaj curr diyaa jaataa haye, jisase sankhya 467853 reh jaati haye. phir, sabse badaa aanshik utpaad joe kataa huaa labhansh seey cumm haye, paaya jaataa haye. iss mamle mein, 385724. dou cheejon koo chihnit kiya jaanaa chaahiye, jaisa kii aarekh mein dekhaa gayaa haih choonki 385724 abecus kii '4' pankti mein haye, isaliye ack '4' koo bhagfal key sabse baaen ank key roop mein chihnit kiya gayaa haye; aanshik utpaad, baaen sanrekhit, mool labhansh key tahat, bhee likhaa gayaa haye. dou shabdon koo ghataaya jaataa haye, joe 8212999 chhod detaa haye. vahee kadam dohraaye jaate hainh sankhya koo chhah ankon mein kaat diyaa jaataa haye, aanshik utpaad koo turant kataa huaa sankhya seey cumm chunaa jaataa haye, pankti sankhya koo bhagfal key agale ank key roop mein likhaa jaataa haye, aur aanshik utpaad koo pahali punaravritti mein paae gaye antar seey ghataaya jaataa haye. prakriya koo aarekh mein dikhaaya gayaa haye. chakr taba taka dohraya jaataa haye jaba taka kii ghataav kaa parinam vibhaajak seey cumm naa hoe jaae. bachee hui sankhya shesh haye.
tou iss udaaharan mein, joe shesh rahata haye vaha 16364 key shesh key saath 485 kaa bhagfal haye. prakriya aamtaur parr yahaan rook jaati haye aur uttar aanshik roop kaa upayog kartaa haye.
adhik sateekta key liye, chakr koo aavashyak kaee dashamlav sthaanon koo dhoondhana jaari rakhaa jaataa haye. bhagfal key antim ank key baad ack dashamlav bindu koo chihnit kiya jaataa haye aur shesh key saath ack shoonya jodaa jaataa haye joe 163640 chhod detaa haye. chakr jaari rahata haye, pratyek baar shoonya koo bhagfal key liye jodaa jaataa haye.
ghataav key baad parinam.
vargamool nikaalana.
vargamool nikaalane key liye, ack atirikt huddy kaa upayog kiya jaataa haye joe dusaron seey alag hoti haye kyonki isamen teen stambh hotey hain. pahale collum mein pahali nau varg sankhyaaen hain, dusare mein pahali nau summ sankhyaaen hain, aur antim mein 1 seey 9 sankhyaaen hain.
46785399 key vargamool koo khojane key liye, isake ankon koo daaeen or seey shuru honey vaale dou ankon mein vargeekrit kiya gayaa haye taaki yeh iss tarah dikheh
sabse baaen samooh koo pahale chunaa jaataa haye, iss mamle mein 46. vargamool huddy parr sabse badaa varg 46 seey cumm chunaa jaataa haye, joe chhathi pankti seey 36 haye. samadhan kaa pahala ank 6 haye, kyonki chhathi pankti chuni gaee thee.
phir, vargamool huddy parr chhathi pankti seey dusare stambh mein sankhya, 12, board parr sett kii jaati haye.
chhathi pankti key pahale stambh mein moolya, 36,46 seey ghataaya gayaa haye, joe 10 chhodata haye.
ankon kaa agala samooh, 78,10 key baad jodaa jaataa haye; yeh shesh 1078 chhod detaa haye.
iss star parr, board aur madhyavarti ganana iss tarah dikhani chaahieh
pratyek pankti mein sankhyaaen "padhee" jaati hain, vargamool huddy seey dusare aur teesare stambhon koo nazarandaj karate hue; yeye darj kiye jaate hain. (udaaharan key liye, chhathi pankti koo iss prakaar padhaa jaataa haih).
jaise pahale dikhaae gaye gunan mein, sankhyaon koo daaen seey baaen padhaa jaataa haye aur oopar-daaen seey baaen-neeche (;;) taka vikarn sankhyaaen jodee jaati hain.
vartamaan shesh seey cumm sabse badee sankhya, 1078 (aathvin pankti seey) paaee jaati haye.
pahale kii tarah, vargamool key agale ank koo praapt karane key liye 8 jodaa jaataa haye aur aathvin pankti kaa moolya, 1024, vartamaan shesh, 1078 seey ghataaya jaataa haye, jisase 54 praapt hotaa haye.
jarh kii huddy, 16, padhee jaati haye aur sankhya board parr nimnanusar sett kii jaati haye.
board parr vartamaan sankhya 12 haye.
16 koo 12 mein jodaa jaataa haye, aur 16 kaa doosara ank parinam key saath jodaa jaataa haye. isaliye board koo iss prakaar nirdhaarit kiya jaanaa chaahieh
board aur madhyavarti gananaaen abb iss tarah dikhati hain.
ack baar phir, vartamaan aanshik shesh, 5453 seey cumm moolya vaali sabse badee pankti paaee jaati haye. iss baar, yeh 4089 key saath teesari pankti haye.
vargamool kaa agala ank 3 haye. pahale kii tarah hii charanon koo dohraya jaataa haye aur 4089 koo vartamaan shesh, 5453 seey ghataakar agale shesh key roop mein 1364 praapt kiya jaataa haye. jaba board koo punarvyavasthit kiya jaataa haye, tou vargamool huddy kaa doosara stambh 6 hotaa haye, ack ank. isaliye board parr 1366 chhodane key liye board parr vartamaan sankhya, 136 key saath 6 jodaa gayaa haye.
prakriya koo phir seey dohraya jaataa haye. abb, board parr sabse badaa moolya vartamaan shesh, 136499 seey chhota haye, joe nauvin pankti seey 123021 haye.
jawab paane key liye aksar harr pankti kaa moolya khojane kii aavashyakta naheen hoti haye. jiss pankti mein uttar haye, uska anuman pahali kuchh haddiyon parr sankhya koo dekhkar aur shesh key pahale kuchh ankon key saath isaki tulanaa karke lagaaya jaa sakata haye. lekin aarekh sabhi panktiyon kaa moolya darshaate hain taaki ise samajhane yogya banaaya jaa sakey.
9 koo parinam key saath jodaa jaataa haye aur 123021 koo vartamaan shesh seey ghataaya jaataa haye.
yadi sabhi ankon kaa upayog kiya gayaa haye, aur shesh bachaa haye, tou purnank bhaag hull hoe jaataa haye, lekin ack aanshik bitt abhee bhee paaya jaanaa chaahiye.
yadi purnank bhaag hull kiya jaataa haye, tou vartamaan parinam varg () 46785899 seey chhota sabse badaa poorna varg honah chaahiye. 
iss vichaar kaa upayog baad mein yeh samajhane key liye kiya jaataa haye kii takneek kaise kaam karati haye, lekin adhik ank utpann kiye jaa sakate hain.
lambe vibhajan mein aanshik bhaag koo khojane key samaan, dou shoonya shesh 1347800 koo praapt karane key liye shesh key saath jode jaate hain. vargamool huddy kii nauvin pankti kaa doosara stambh 18 haye aur board parr vartamaan sankhya 1366 haye. 
board parr 13678 sett karane key liye ganana kii jaati haye.
board aur madhyavarti gananaaen abb iss tarah dikhati hain.
1231101 key saath nauvin pankti sabse badaa moolya haye joe
shesh, isaliye vargamool key aanshik bhaag kaa pahala ank 9 haye.
nauvin pankti kaa moolya shesh seey ghataaya jaataa haye aur kuchh aur shoonya jode jaate hain taaki nae shesh 11669900 koo praapt kiya jaa sakey. nauvin pankti parr doosara stambh 18 haye aur board parr 13678 haye, isaliye
board parr 136798 nirdhaarit karane key liye ganana kii jaati haye.
aavashyak ankon koo khojane key liye aur yadi aavashyak sateekta praapt kii jaati haye tou charanon koo jaari rakhaa jaa sakata haye. yadi shesh shoonya hoe jaataa haye, tou iska matlab haye kii sateek vargamool paaya gayaa thaa.
goal karna.
ankon kii vaanchhit sankhya milane parr, yeh nirdhaarit karna aasaan haye kii ise goal karane kii aavashyakta haye yaa naheen; i.ee.antim ank koo badalana. yeh dekhane key liye ack aur ank khojane kii aavashyakta naheen haye kii kyaa yeh 5.25 key barabar yaa usase adhik haye aur ise mool key saath jodaa gayaa haye aur isaki tulanaa shesh seey kii jaati haye; yadi yeh shesh seey cumm yaa barabar haye, tou agala ank cumm seey cumm paanch hoga aur goal karane kii aavashyakta haye. uparokt udaaharan mein, 6839925 11669900 seey cumm haye, isaliye mool koo 6840.0 taka goal karane kii aavashyakta haye.
ack sankhya key vargamool koo khojane key liye joe ack purnank naheen haye, kahein kii 54782.917, sub kuchh samaan haye, sivaay isake kii dashamlav bindu key baaen aur daaen ankon koo dou mein vargeekrit kiya gayaa haye. 
tou 54782.917 koo iss prakaar samoohbaddh kiya jaayega
taba vargamool koo pahale ullikhit prakriya kaa upayog karke paaya jaa sakata haye.
vikarn sanshodhan.
19veen shatabdi key dauran, napier kii haddiyon koo padhane mein aasaan banaane key liye parivartit kiya gayaa thaa. chhaden lagbhag 65° key konn key saath banaai gaee theen taaki joe trikon jode jaane they unhein sanrekhit kiya jaa sakey. iss mamle mein, chhad key pratyek varg mein ikai daaeen or aur duss (yaa shoonya) baain or hoti haye.
chhaden iss tarah seey banaai gaee theen kii urdhvaadhar aur kshaitij rekhaaen usa rekha kii tulanaa mein adhik dikhaai detee theen jahan chhaden chhui jaati theen, jisase parinam key pratyek ank key dou ghatakon koo padhana aasaan hoe jaataa thaa. iss prakaar, chitra mein yeh turant spasht haye kih
genelle-lucas shasakon.
1891 mein, heinry genail ney napier kii haddiyon key ack prakaar kaa aavishkaar kiya jise genail-lucas shasakon key roop mein jaanaa jaane lagaa. carrie koo graphical roop seey prastut karke, saral gunan samasyaon key parinamon koo seedhe padhaa jaa sakata haye, binaa kissy madhyavarti mansik ganana key.
nimnalikhit udaaharan ganana kartaa haye.