अंकगणित

aankaganit (aangreji mean arithametik) ganit ki vah shakha hai jisamean keval aankoan aur sankhyaoan se ganana ki jati hai. isamean n sanketaksharoan ka prayog hota hai aur n rrin sankhyaoan ka hi, kiantu aankaganit ke niyamoan ki vyakhya mean sanketaksharoan ka prayog hone laga hai. bahudha aisa mana gaya hai ki aankaganit ka vishay vistar abhiganana (kampyuteshan) tak simit hai aur vishay ke pratipadan mean tark ki vishesh mahatta nahian hoti. aankaganit ka tarkayukt vivechan ek alag vishay hai jise sankhya siddhaant (thyori aauv nanbars) kahate haian. kuchh ganitajn ab aankaganit aur sankhya siddhaant ko samanarthak manane lage haian.

do samoohoan mean vastuoan ki sankhya tab saman kahi jati hai jab ek samooh ki pratyek vastu ke lie doosare samooh mean ek jo didar vastu mil sake. is prakar yadi anukram 1, 2, 3, . . ., m ki pratyek sankhya ki jo di kisi samooh ki ek-ek vastu se banaee ja sake to us samooh mean vastuoan ki sankhya m hai. is sankhya ka jnan prapt karana vastuoan ki ganana karana, arthath‌ ginana, kaha jata hai. ginane ki vidhi se jo sankhyaean milati haian unhean prakritik sankhyaean athava poorn sankhyaean kahate haian.

ghan poorn sankhya sanbandhi mool niyam: yadi ek samooh mean k vastuean aur doosare samooh mean kh vastuean haian to donoan samoohoan mean milakar k + kh vastuean haian. k + kh ko k aur kh ka yogaphal, athava yog, banate haian. yogaphal jnat karane ko jo dana kahate haian. chihn + kh ko dhan kahate haian. ginane ki prakriya se spasht hai ki yog ke lie nimnalikhit mool niyam thik hai:

yog ka kram vinimey (kanpyootetiv) niyam: k + kh = kh + k.

yog ka sahachary (aisoshietiv) niyam: k + kh = (kh + g) = (k + kh) + g.

yadi ch koee aisi ghan poorn sankhya hai ki k = kh+ ch, to kaha jata hai ki k, kh se b di hai (aur is k > kh likhate haian); sath hi kh, k se kam hai (aur ise kh < k likhate haian. is prakar yadi k aur kh koee do dhan poorn sankhyaean haian to ya to k = kh, ya k > kh ya k < kh.

dhan poorn sankhyaoan mean yah gun haian ki kinhian do ya do se adhik aisi sankhyaoan ka yog dhan poorn sankhya hi hota hai, arthath‌ yadi k aur kh ho dhan poorn sankhyaean haian to ek aisi dhan poorn sankhya g avashy hai ki k + kh = g. spasht hai ki g > k.

yadi k+ kh = g, aur sankhyaean k aur g di huee haian to kh ka man g se k ko ghatakar jnat kiya jata hai. is kriya ko vyavakalan kahate haian aur likhate haian kh = g__k. chihn- ko rrin padha jata hai.

poorvokt niyamoan se spasht hai ki ek se adhik sankhyaean chahe jis kram se jo di jaean, unake yogaphal mean koee aantar nahian p data. atev 4 + 4 + 4 ke saman punaragat yog ko 4´ 3 likh sakate haian, jahaan sankhya 3 yah batalati hai ki 4 kitani bar liya gaya hai. ise 4 gunit 3 kahate haian aur is kriya ko gunan, arthath‌ guna karana, kahate haian. 4´ 3 ke parinam ko gunanaphal kahate haian. isamean sankhya 4, jo bar-bar jo di gee sankhya haian, guny haian; aur sankhya 3, arthath‌ jitani bar 4 jo da gaya hai, gunak haian.

yadi ham sankhyaoan ko sanketaksharoan se prakat karean to gunanaphal k´ kh ko pray: k, kh ya keval kakh likha jata hai. yog ki bhaanti hi gunan kriya ke lie nimnalikhit niyam thik haian:

1. gunan ka kram vinimey niyam: k´ kh =kh´ k;

2. gunan ka sahachary niyam: k (kh´ g) = (k´ kh) g.

pahale niyam ki satyata ki jaanch ke lie k panktiyoan mean se pratyek mean kh goliyaan is prakar rakhean ki sab panktiyoan ki pahali goliyaan ek sidh mean rahean, doosari goliyaan ek sidh mean, ityadi. is prakar kh stanbh mileange, jinamean se pratyek mean k goliyaan haian. stanbhoan ke hisab se kul goliyoan ki sankhya m: k ´ kh hai aur panktiyoan ke hisab se kh´ k; kiantu goliyaan kul milakar donoan bar utani hi haian; isilie k´ kh =kh´ k.

doosare niyam ki satyata ki jaanch ke lie kh samoohoan mean se pratyek mean g stanbh rahean aur pratyek stanbh mean k goliyaan. ye samooh ek ke niche ek rakhe jaean. is prakar g stanbh baneange aur pratyek mean k´ kh goliyaan raheangi. isase pratyaksh hai ki kul goliyoan ki sankhya (k´ kh) ´ g hai. ab ye samooh is prakar rakhe jaean ki inaki pahali panktiyaan sab ek sidh mean rahaian, unake niche sab samoohoan ki doosari panktiyaan ek sidh mean rahean, ityadi. is prakar pratyek pankti mean sab samoohoan ko milakar kh´ g goliyaan rahaiangi aur un goliyoan ki aisi panktiyaan k hoangi. isalie ab goliyoan ki sankhya = k ´ (kh´ g). goliyoan ki sankhya vahi rahati hai; isalie k ´ (kh x g) = (k´ kh) ´ g.

in do niyamoan ke atirikt gunan kriya ke lie nimnaankit niyam bhi haian:

3. vitaran niyam: (k+ kh) g = kag + khag;

isaki satyata ki jaanch goliyoan se poorvavath‌ ki ja sakati hai. any niyam ghat sanbandhi haian. jis prakar ch bar punaragat yog k+ k+ . . . + k ko chak likha jata hai, usi prakar ch bar punaragat gunanaphal k´ k ´ . . . ´ k ko k likha jata hai. ch ko ghataank ya keval ghat aur k ko adhar kahate haian. paribhasha se ghat sanbandhi nimnalikhit niyamoan ki satyata spasht hai: 4. k^ch ´ k^chh = k^ch + chh;

5. (k^ch)^chh = k^ch+ ^chh;

6. k^ch kh^chh = (kakh)^chh.

yadi k aur kh koee do dhan poorn sankhyaean haian to k x kh bhi koee dhan poorn sankhya g hogi. yadi g aisi sankhya di huee hai jo do sankhyaoan ke gunanaphal ke barabar hai aur unamean se ek sankhya k aisi jnat hai jo shoony se bhinn hai, to doosari sankhya kh ka man g ko k se vibhajit karane par prapt hota hai. ham likhate haian:

kh = g ¸ k athava g/ k, athava g/ k.

chihn bhag ka chihn kahate haian aur bhajit padhate haian. chihn / ko bata ya bate padhate haian. udaharanarth:, 8 bhajit 4 (arthath‌ 8¸ 4) 2; athava 8/4 arthath‌ = 2.

vibhajan ke lie ghat sanbandhi niyam yah hai:

7. k ¸k = k, jahaan m > s.

paribhasha se isaki satyata ki jaanch karana saral hai.

bhajak siddhaant__yadi tin dhan poorn sankhyaoan kakh = g mean sanbandh kakh g hai, to k aur kh ko g ke bhajak athava gunanakhand kahate haian. kabhi-kabhi itana kahana paryapt samajha jata hai ki k, g ko vibhajit karata hai. g, k ka apavarty athava gunaj kahalata hai, aur k, g ka apavartak. sankhya 1 ekak kahalati hai aur spasht hai ki yah pratyek poorn sankhya ka bhajak hai tatha pratyek sankhya svayan apana bhajak hai. yadi g = kakh, aur k tatha kh mean se pratyek 1 se b di hai, to g ko sanyukt sankhya kahate haian, anyatha abhajy sankhya. udaharanat:, 2, 3, 5, 7, 11, 13, abhajy sankhyaean haian. yooklid ne elimeants, khand 9, sadhy 20, mean siddh kar diya hai ki abhajy sankhyaean ginati mean anant hai. usane yah bhi siddh kiya tha ki pratyek sanyukt sankhya ko abhajy sankhyaoan ke gunanaphal ke roop mean pradarshit karane ki, unake kram mean herapher ko chho dana keval ek hi vidhi hai.

dhan poorn sankhyaoan k₁ k₂ . . . kam ke saman pratyek parimit sangh ke lie ek aisi sabase b di poorn sankhya m rahati hai jinase sab ki pratyek sankhya poora-poori vibhajit ho sakati hai. is sankhya ko mahattam samapavartak (m.s.) kahate haian. yadi m = 1, to sankhyaean ek-doosare ke sapeksh abhajy kahalati haian. pratyek sankhya sangh ke lie sabase chhoti ek aisi sankhya bhi hoti hai jo sangh ki pratyek sankhya se vibhajy hoti hai. is sankhya ko laghuttam samapavarty (l.s.) kahate haian. m.s. aur l.s. jnat karane ki ek vidhi mean sankhyaoan ko abhajy sankhyaoan ke gunanaphaloan ke roop mean prakat karana hota hai (vidhi ka varnan aankaganit ki pray: sabhi pustakoan mean mil jaega). udaharan ke lie yadi sankhyaean @52, 420, 1176 hoan, to 252 = 2², 3², 7, 420 = 2², 3, 5, 7, 1176 = 2³, 3, 72 isalie isaka m.s. = 2², 3, 7= 84 hai aur l.s. = 2³, 3², 5, 7² = 17, 640. do sankhyaoan ka, bina unake gunanakhand kie, m.s. jnat karane ki ek vidhi vibhajan ki hai. isamean pahale chhoti sankhya se b di sankhya ko bhag diya jata hai, phir shesh se chhoti ko, arthath‌ poorvagami bhajak ko; yahi kram tab tak chalata rahata hai jab tak shesh shoony n a jae. aantim bhajak abhisht m.s. hai. is vidhi ka avishkar bhi yooklid ne kiya tha. udaharanarth, 252, 420 ke lie kriya yah hogi:

is prakar abhisht m.s. 84 hai. sankshipt roop mean ise is prakar likh sakate haian: thumb|center|300px

is prakar abhisht mean m.s. 84 hai. sankshipt roop mean ise is prakar likh sakate haian: thumb|center|300px

aantim aur pratham stanbhoan mean kramanusar bhagaphal aur bhajak haian.

do sankhyaoan ka gunanaphal unake m.s. aur l.s. ke gunanaphal ke barabar hota hai. m.s. jnat hone par, is niyam se, un sankhyaoan ka bina gunanakhand kie l.s. jnat kiya ja sakata hai.

sadharan bhinn__bhinn 1/k ka arth hai vah sankhya jisako k se guna karane par 1 prapt hota hai. yahaan k koee dhan poorn sankhya hai. g 1/k ko g/k athava g/k likhate haian. g/k ko sadharan bhinn kahate haian. ise vah bhagaphal mana ja sakata hai jo g ko k se bhag dene par milata hai. g aur k bhinn ke do avayav haian. g ko aansh (nyoomaretar) aur k ko har (dinaminetar) kahate haian. jab g k, to g/k ko uchit bhinn kahate haian, anyatha anuchit bhinn. jab g aur k paraspar abhajy hoan, arthath‌ aisi koee sankhya n ho jo donoan ko vibhajit kar sake, to bhinn g/k ka roop laghuttam padoan vala kaha jata hai. bhinnoan ke yog, vyavakalan, gunan, bhajan, adi ke lie bhinn shirshak lekh dekhean.

aparimey sankhyaean__poorn sankhyaoan aur sadharan bhinnoan ko parivesh sankhya kahate haian. jo sankhya poorn n ho aur sadharanat: bhinn ke roop mean prakat n ki ja sake vah aparimey sankhya kahalati hai, jaise Ö2, p. inaka vivechan sankhya namak lekh mean milega.

dashamalav paddhati__prachalit sankhya paddhati ko, jisamean ek sau teees ko 123likha jata hai, dashamalav paddhati kahate haian. CXXIII dashamalav paddhati mean nahian hai, roman paddhati mean hai. dashamalav paddhati apanane par hi aankaganit ki charoan kriyaoan ki saral vidhiyaan prayog mean ane lagian. (is paddhati ka, tatha any paddhatiyoan ka, vivaran sankhyaank paddhatiyaan shirshak lekh mean milega). dashamalav paddhati mean sankhya ko vastut: 10 ke ghatoan ki sahayata se vyanjit kiya jata hai. udaharanat:

3467 = 3.10³ + 4.10² + 6.¹⁰ + 7.

pratyek ghat ka gunaank 0 se 9 tak (in das sankhyaoan) mean se koee bhi ho sakata hai. b di sankhyaoan ka ekak sthan ke aank se aranbh kar tin-tin aankoan ke avartakoan mean baantane ki pratha pashchaty hai. bharatiy pratha mean ekak aank ke aranbh kar pahale tin aankoan ka ek avartak aur bad mean do-do aankoan ke avartak banae jate haian. udaharanat: 2306472 ko pashchaty pratha ke anusar 2,306,472 likhate haian; bharatiy pratha mean 23,06,472. aisa karane ka karan spasht hai. bharatiy ganana mean sau hajar ka ek lakh, sau lakh ka 1 karo d ityadi hota hai. pashchaty pratha mean 10 lakh ko ek miliyan kahate haian.

amarika aur phraans mean hajar miliyan (ek arab) ko biliyan kahate haian, parantu ianglaiand mean miliyan miliyan (= das kharab) ko biliyan kahate haian.

in dashamalav paddhati ke prayog dvara ve bhinnean bhi likhi ja sakati haian jinaka har 10 ka koee ghat ho; yatha: thumb|center|250px = 35 + 7 ´ 10.1+ 0 ´ 10-2+6 ´ 10-3+4 ´ 10-4

arthath‌ dashamalav biandu ke daeean or ke pahale aank ko 10^-1 se guna karake dashamalav ke baeean or ki poorn sankhya mean jo dana hota hai. doosare ko 10^-2 se guna kar pahale ke yog mean jo date haian aur isi prakar any aankoan ko bhi guna karake jo dana p data hai.

dashamalav mean yog aur vyavakalan-dashamalav paddhati mean yog jnat karane ki nimnaankit paddhati ab pray: sarvamany hai. sankhyaoan ko ek ke niche ek is prakar likhana chahie ki dashamalav biandu sab ek stanbh mean arthath‌ ek ke niche ek rahean. is prakar ekak ke sabhi aank ek stanbh mean p deange, dahaee ke sthan vale aank ek any stanbh mean, ityadi; udaharanat: 53.76, 236081, 408346 ka yog yoan nikalega: thumb|center|250px

spasht hai ki dashamalavoan ka yog sadharan jo d ke saman hi hai. oopar ki kriya vastut: nimnalikhit ka sankshipt roop hai:

5 ´ 10 + 3+ 7 ´ 10^-1+ 6 ´ 10^-2

2 ´ 10²+ 3 ´ 10+ 6 + 0 ´ 10^-1+ 8 ´ 10^-2+ 1 ´ 10^-3

4 ´ 10²+ 0 ´ 10 + 8 + 3 x 10^-1+ 4 ´ 10-2+ 6 ´ 10^-3

---

= 6 ´ 10²+ 8 ´ 10+ 17 + 10 ´ 10^-1+ 18 ´ 10^-2+ 7 ´ 10^-3 = 6 ´ 102+ 9 ´ 10+ 8 + 1 ´ 10-1+ 8 ´ 10-2+ 7 ´ 10-3

vyavakalan ke lie poorvokt kriya ko ulatana hota hai. b di sankhya ko oopar aur chhoti ko niche is prakar likhana chahie jisamean dashamalav biandu ek-doosare ke thumb|center|250px

niche rahean; phir sadharan riti se ghatana chahie. shesh mean dashamalav biandu ko oopar likhi sankhyaoan ke dashamalav bianduoan ke thik niche rakhana chahie, jaisa bagal mean dikhaya gaya hai. guna karane ki vidhi vitaran niyam par adharit hai aur aankaganit ki adhikaansh pustakoan mean isaka varnan mil jaega.

yadi do dashamalav sankhyaoan ka sannikat gunanaphal, man lean; dashamalav sthanoan tak shuddh, jnat karana hai, to sugamata isamean hai ki isamean se ek sankhya ka (jise gunak kaheange) dashamalav baeean or ya dahini or hatakar us sankhya ko 1 aur 10 ke bich mean laya jae, phir utane hi sthan viparit disha mean doosari sankhya ka (jise guny kaheange) dashamalav bhi hataya jae tab guny ke tisare dashamalav sthan se gunak ke ekak vale aank ko guna aranbh karana chahie. gunak ke dashamalav vale aank se guny ke dashamalav ke doosare sthan se guna aranbh karana chahie, ityadi. jin aank se guna karana aranbh kiya jae usake dahini or vale aank ke guna karake hath lagane vali sankhya le leni chahie. yah kriya nimnalikhit udaharan spasht ho jaegi:

424.33643 ´ 12.732 = 4243.3643 ´ 1.2732

thumb|center|200px

dashamalav biandu ke bad ane vale sthan mean 1 ho to vah vastut: 1/10 barabar hai, usake bad vale sthan mean 1 ho to vah vastut: 1/100 ke barabar hai, ityadi. isase spasht hai ki dashamalav aank ke bad bahut se aankoan ke rakhane ki avashyakata vyavahar mean nahian p dati, kyoanki aankoan ko man uttarottar shighrata se ghatata jata hai. isilie bahudha dashamalav ke pashchath‌ isake tisare ya chauthe sthan ke bad ke sab aank ko chho d die jate haian; parantu yadi chho de hue aankoan mean se pahala aank 5 ya 5 se b da ho to rakhe ge aankoan mean se aranbh aank mean 1 jo d diya jata hai, kyoanki tab uttar adhik shuddh ho jata hai.

ek pankti mean gunan__jo vyakti maukhik yog mean pravin ho, yah ek pankti mean do sankhyaoan ka gunanaphal nikal sakata hai. man lean dashamalav par dhyan n dete hue guny mean ekak ke sthan mean aank k, hai, dahaee (dasham) ke sthan mean k2, ityadi, aur gunak mean in sthanoan ke aank kramanusar kh1, kh2, ityadi hai. man lean:

k₁kh₁= 10 h₁+ g₁,

k₁kh₂ + k₂kh₁, + h₁= 10 h₂+ g₂,

k₁kh₃ + k₂kh₂, k₃kh₁+ h₁= 10 h₃+ g₃,

ityadi, jahaan g₁, g₂, . . . pratyek 10 se kam hai; to gunanaphal ke ekak ke sthan mean g1, dahaee ke sthan mean g2, saik de ke sthan mean g3 .. . hoange. vastavik prakriya mean sugamata isamean hoti hai ki gunak ko ulatakar lie liya jae. tab samaantar rekhaoan mean sthit aankoan ke maukhik gunanaphal ka yog jnat karana hota hai:

udaharanat: 34608 ko 5387 se guna karane mean kriya itani likhi jaegi: thumb|center|250px

yahaan gunanaphal ka aank 2 yog 7 ´ 6 + 8 ´ 0 + 3 ´ 8 + hasil ke 6 ka ekak vala aank hai. aant mean gunanaphal mean dashamalav is prakar lagaya jata hai ki usake dahini or utane hi aank rahean jitane gunak aur guny mean milakar hoan.

is dashamalav sankhya mean doosari sankhya ka bhag dene mean suvidha isamean hoti hai ki bhajak se dashamalav hata diya jae aur bhajy mean dashamalav ko bhi utane hi sthan tak daeean or hata diya jae. isake bad sadharan riti se bhag ki kriya ki jati hai. bhagaphal mean dashamalav us aank ke bad lagega jo bhajy mean ekak vale sthan ke aank ko utar kar bhag dene par milata hai. kriya nimnalikhit udaharan se spasht ho jaegi:

63802¸ 73.1 = 6380.2¸731 thumb|center|200px

spasht he ki shesh mean dashamalav biandu ko ekak sthan se utane hi sthan baeean or hatakar lagana chahie jitane dashamalav sthan par aantim utara hua aank mool bhajy mean tha. yahaan aantim utara hua aank 2 mool bhajy mean doosare dashamalav sthan par tha. atev shesh 2.05 hai.

uparyukt kriya mean bhajy mean 2 ke ago ichchhanusar shoony badhakar bhagaphal ichchhanusar dashamalavoan tak jnat kiya ja sakata hai.

vargamool__vargamool jnat karane ki kriya nimnalikhit sootr par adharit haian:

(k+ kh)² = (k+2kh) k¹+kh²

di huee sankhya ke dashamalav sthan se aranbh kar baeean or aur dahini or do-do aankoan ke jo de bana lean. ab sankhya ke baean sire par pratham khand ya to ek poora jo da hoga ya keval ek aank. 1 se 9 tak ke vargoan ki sarani se dekhean ki yah khand kin sankhyaoan ke vargoan ke bich mean hai. chhoti sankhya ko vargamool mean likhean. isake varg ko khand se ghataean aur shesh ke age doosara khand utarean; yah doosara bhajy hai. bhajak ke likhe ab tak prapt vargamool ka doona likhean aur dekhean ki usake age dirghatam kaun-sa aank b badhaya jae ki badhane par prapt bhajy ka b guna doosare bhajy se kam rahe. is prakar vargamool ka doosara aank b hua. isi prakar any aank jnat karean. yah kriya oopar bagal mean dikhae ge udaharan se spasht ho jaegi jisamean 325.649 ka vargamool jnat kiya gaya hai. thumb|center|200px

isake bad ham 207400 ki 3604 se bhag de sakate haian.

vargamool nikalane ki riti se milati-julati riti dvara ghanamool bhi jnat kiya ja sakata hai, kiantu laghuganakoan (l aaugairithms) ke prayog se sabhi mool saralata se jnat ho jate haian (niche dekhean). laghu gunak sarani upalabdh n hone par harnar ya nyootan ki vidhi se bhi mool jnat kie ja sakate haian (dr. samikaran siddhaant).

laghuganak__yadi k tatha a dhan sankhyaean haian aur a^l = k, to l ko adhar a ke sapeksh k ka laghugunak kahate haian, aur k ke l ka prati laghuganak. likhate haian: l = laghu, k. jab a = 10 tab sadharan laghuganak prapt hote haian, aur yadi a = ee (=2.71828. . .) to nepiriy laghuganak milate haian. sadharan laghuganakoan ki mudrit saraniyaan bikati haian. sootr laghu (k xkh) = laghu k laghu kh ke prayog se gunan kriya yog kriya mean parivartit ho jati haian, kyoanki yadi gunanaphal kakh jnat karana hai to laghu k aur kh ke yog se laghu (kakh) prapt hota hai, aur isaka pratilaghuganak abhisht gunanaphal kakh hai. yahaan sab laghuganakoan ka adhar 10 hai. vishesh janakari ke lie laghuganak shirshak lekh dekhean.

aikik niyam__yadi kisi prakar ki ek vastu ke lie koee rashi taul, mooly, adi) kh ho to usi prakar ki k vastuoan ke lie yah rashi kh ki guna karane par prapt hota hai. vilomata isi niyam se yadi k saman vastuoan ke lie simit rashi s ho isi pratyek ke lie yah rashi s, k hogi. is niyamoan ke adhar par k vastuoan ka mooly aank jnan rahane par ham kh vastuoan ko mooly adi jnat kar sakate haian. is kriya mean lagane vale niyamoan ko ekak niyam kahate haian. yah nam isilie p da ki is riti se pahale ek vastu ke lie upayukt rashi jnat karani hoti hai.

trairashik__yadi k vastuoan ka mooly kh hai ya g vastuoan ka mooly kitana hoga, aise prashnoan ka trairashik ke niyam se bhi hal kiya ja sakata hai. niyam ka nam trairashik isalie p da ki isamean k, kh, g mean tin rashiyaan ati hai. trairashik niyam ka avishkar bharatiyoan ne kiya. brahmagupt tatha bhaskar n ho vastut isaka trairashik nam diya. shatabdiyoan tak vyapariyoan ke lie yah atyant mahatvapoorn niyam raha. aankaganit ke yooropiy lekhe pahale paryapt vistar se is niyam ki vyakhya karate the. yah niyam samanupat ke siddhaant par ashrit hai. ise vistarapoorvak samajhane ke lie yahaan paryapt sthan nahian hai. keval bhaskar ki lilavati se ek udaharan yahaan diya jata hai.

yadi dhaee pal keshar ka mooly 3/7 nishk hai to 9 nishk kitani kesar ka mooly hoga? trairashik niyam se uttar = 9 x 5/2 5/7= 52 ½ pal.

bhaskar ne pancharashik, saptarashik adi niyam bhi batae haian.

anupat__bhinn k/kh ko k aur kh ka anupat, athava k ka kh se anupat bhi kah sakate haian aur anupat ko k:kh roop mean bhi likhate haian. char sankhyaean k, kh, g, gh tatha samanupat mean kahi jati hai jab k:kh = g:gh. samanupat ko k : kh :: g : gh bhi likhate haian. k, gh samanupat ke aantim pad aur kh, g madhy pad haian. spasht hai ki k x gh= kh x g. tin sankhyaean k, kh, g tab gunottar anupat mean kahi jati hai jab k: kh: g:: kh: g, arthath‌ kag = kh².

ganana yantr__aankaganitiy abhiganana ke lie ab bhaanti-bhaanti ke ganana yantr ban ge haian jinase jatil abhigananaean bhi shighr ho jati hai. inaka vistrit vivaran vivaran ganana yantr namak lekh mean milega.

panne ki pragati avastha

adhar

prarambhik

madhyamik

poornata

shodh

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