ganit

Pune mein aaryabhat ki moorti 476-550

shabd 'hisaab' yahaaain anupreshit hota hai. 'hisaab' ke saamaanya arth ke liye dekhein bahi-khaata

ganit aisi vidyaaon ka samooh hai jo sankhyaaon, maatraaon, parimaanon, roopon aur unke aapasi rishton, gun, svabhaav ityaadi ka adhyayan karti hain. ganit ek amoort ya niraakaar (abstract) aur nigmanaatmak pranaali hai. ganit ki kai shaakhaaeain hain : ankaganit, rekhaaganit, trikonamiti, saankhyiki, beejaganit, kalan, ityaadi. ganit mein abhyast vyakti ya khoj karne vaale vaigyaanik ko ganitjnya kehte hain.

beesaveen shataabdi ke prakhyaat british ganitjnya aur daarshanik barteind Russel ke anusaar ''ganit ko ek aise vishay ke roop mein paribhaashit kiya ja sakta hai jismein ham jaante hi naheen ki ham kya kah rahe hain, na hi hamein yeh pata hota hai ki jo ham kah rahe hain vah satya bhi hai ya naheen.''

ganit kuchh amoort dhaaranaaon evam niyamon ka sankalan maatr hi naheen hai, balki dainandin jeevan ka moolaadhaar hai.

anukram

ganit ka mahatva

chitr:Brahmagupta089.jpg

brahmagupt

brahmagupt ka prameya, iske anusaar AF = FD.

puraatan kaal se hi sabhi prakaar ke gyaan-vigyaan mein ganit ka sthaan sarvopari raha hai-

yatha shikha mayooraanaan naagaanaan manayo yatha.

tatha vedaaङagashaastraanaan ganitan moordhni sthitm. (vedaang jyotish)

( jis prakaar moron mein shikha aur naagon mein mani ka sthaan sabse upar hai, usi prakaar sabhi vedaang aur shaastron me ganit ka sthaan sabse oopar hai.)

mahaan ganitjnya gaaus ne kaha tha ki ganit sabhi vijnyaaanon ki raani hai. ganit, vigyaan aur praudyogiki ka ek mahatvapoorn upakaran (tool) hai. bhautiki, rasaayan vigyaan, khagol vigyaan aadi ganit ke bina naheen samajhe ja sakte. aitihaasik roop se dekha jaaya to vaastav mein ganit ki anek shaakhaaon ka vikaas hi isaliye kiya gaya ki praakrutik vigyaan mein iski aavashyakta aa padi thi.

kuchh had tak ham sab ke sab ganitjnya hain. apne dainik jeevan mein rojaana hi ham ganit ka istemaal karte hain - us vakt jab samay jaanane ke liye ham ghadi dekhte hain, apne khareede gaye saamaan ya khareedaari ke baad bachane waali rejagaari ka hisaab jodte hain ya fir football tennis ya cricket khelte samay banane vaale score ka lekha-jokha rakhate hain.

vyavasaaya aur udyogon se judi lekha sambandhi sankriyaaen ganit par hi aadhaarit hain. beema (inshyoreins) sambandhi gananaaen to adhikaanshataya byaaj ki chakravruddhi dar par hi nirbhar hai. jalayaan ya vimaan ka chaalak maarg ke disha-nirdhaaran ke liye jyaamiti ka prayog karta hai. bhaugolik sarvekshan ka to adhikaansh kaarya hi trikonamiti par aadhaarit hota hai. yahaan tak ki kisi chitrakaar ke aarekhan kaarya mein bhi ganit madadgaar hota hai, jaise ki sandarbh (parsapektiv) mein jismein ki chitrakaar ko trivimeeya duniya mein jis tarah se insaan aur vastuen asal mein dikhaai padte hain, unheen ka tadanuroop chitran vah samatal dharaatal par karta hai.

sangeet mein svaragraam tatha sannaadi (haarmoni) aur pratibindu (kaauntarapaaint) ke siddhaant ganit par hi aashrit hote hain. ganit ka vigyaan mein itna mahatva hai tatha vigyaan ki itni shaakhaaon mein iski upayogita hai ki ganitjnya erik tempal bail ne ise 'vigyaan ki saamraajnyai aur sevika' ki sanjnyaa di hai. kisi bhautikvijnyaaani ke liye anumaapan tatha ganit ka vibhinn tareekon ka bada mahatva hota hai. rasaayanavijnyaaani kisi vastu ki amleeyata ko soochit karne vaale pi H (pH) maan ke aakalan ke liye laghuganak ka istemaal karte hain. konon aur kshetrafalon ke anumaapan dvaara hi khagolavijnyaaani soorya, taaron, chandra aur grahon aadi ki gati ki ganana karte hain. praani-vigyaan mein kuchh jeev-jantuon ke vruddhi-paitarnon ke vishleshan ke liye vimeeya vishleshan ki madad li jaati hai.

uchch gativaale sanganakon dvaara gananaaon ko doosari vidhiyon dvaara ki gayi gananaaon ki apeksha ek ansh maatr samay ke andar hi sampann kiya ja sakta hai. is tarah kamyootaron ke aavishkaar ne un sabhi prakaar ki gananaaon mein kraanti la di hai jahaan ganit upayogi ho sakta hai. jaise-jaise khagoleeya tatha kaal maapan sambandhi gananaaon ki praamaanikta mein vruddhi hoti gayi, vaise-vaise nausanchaalan bhi aasaan hota gaya tatha Christopher kolambas aur uske paravarti kaal se maanav sudooragaami nae pradeshon ki khoj mein ghar se nikal pada. saath hi, aage ke maarg ka naksha bhi vah banaata gaya. ganit ka upayog behtar kism ke samudri jahaaj, rel ke engine, motor kaaron se lekar havaai jahaajon ke nirmaan tak mein hua hai. raadaar pranaaliyon ki abhiklpana tatha chaand aur grahon aadi tak raaket yaan bhejane mein bhi ganit se kaam liya gaya hai.

bhautiki mein ganit ka mahatva

vidyutachumbakeeya siddhaant samajhne evam usaka upayog karne ke liye ke liye sadish vishleshan bahut mahatvapoorn hai.

group siddhaant, spektroskopi, kvaantam yaantriki, thos avastha bhautiki evam naabhikeeya bhautiki ke liye bahut upayogi hai.

bhautiki mein sabhi tarah ke rekheeya sankaayon ke vishleshan ke liye furiar ki yuktiyaaain upayogi hai.

kvaantam yaantriki ko samajhne ke liye maitriks vishleshan jaroori hai.

vidyutachumbakeeya tarangon ka varnan karne evam kvaantam yaantriki ke liye samishr sankhyaaon ka upayog hota hai.

ganit ka itihaas

mukhya lekh : ganit ka itihaas

maanav gyaan ki kuchh praathamik vidhaaon mein sanbhavataya ganit bhi aata hai aur yeh maanav sabhyata jitna hi puraana hai. maanav jeevan ke vistaar aur ismein jatiltaaon mein vruddhi ke saath ganit ka bhi vistaar hua hai aur usaki jatiltaaen bhi badhi hain. sabhyata ke itihaas ke poore daur mein gufa mein rahane vaale maanav ke saral jeevan se lekar aadhunik kaal ke ghor jatil evam bahuaayaami manushya tak aate-aate maanav jeevan mein dheere-dheere parivartan aaya hai. iske saath hi maanav gyaan-vigyaan ki ek vyaapak evam samruddh shaakha ke roop mein ganit ka vikaas bhi hua hai. haalaanki ek aam aadmi ko ek hajaar saal se bahut adhik peechhe ke ganit ke itihaas se utana sarokaar naheen hona chaahiye, parantu vaigyaanik, ganitjnya, praudyogikeevid, arthashaastri evam kai anya visheshagya rojamarra ke jeevan mein ganit ki samunnat pranaaliyon ka kisi na kisi roop mein ek vishaal, akalpaneeya paimaane par istemaal karte hain. aajkal ganit dainik jeevan ke saath sarvavyaapi roop mein samaaya hua dikhta hai.

ganit ki utpatti kaise hui, yeh aaj itihaas ke pannon mein hi vismrut hai. magar hamein maaloom hai ki aaj ke 4000 varsh pehle bebeelon tatha misr sabhyataaen ganit ka istemaal panchaang (calendar) banaane ke liye kiya karti theen jisse unhein poorv jaankaari rahati thi ki kab fasal ki buaai ki jaani chaahiye ya kab neel nadi mein baadh aaegi, ya fir iska prayog ve varg sameekaranon ko hal karne ke liye kiya karti theen. unhein to us prameya (thyoram) tak ke baare mein jaankaari thi jiska ki galat shreya paaithaagoras ko diya jaata hai. unki sanskrutiyaaain krushi par aadhaarit theen aur unhein sitaaron aur grahon ke pathon ke shuddh aalekhan aur sarvekshan ke liye sahi tareekon ke gyaan ki jaroorat thi. ankaganit ka prayog vyaapaar mein rupayon-paison aur vastuon ke vinimay ya hisaab-kitaab rakhane ke liye kiya jaata tha. jyaamiti ka istemaal kheton ke chaaron taraf ki seemaaon ke nirdhaaran tatha piraamid jaise smaarakon ke nirmaan mein hota tha.

myaan sankhya

miletas nivaasi thels (645-546 isa poorv) ko hi sabse pehla saiddhaantik ganitjnya maana jaata hai. usane bataaya ki kisi bhi vastu ki oonchaai ko maapan chhadi dvaara nikshepit parachhaai se tulana karke maapa ja sakta hai. aisa maanate hain ki usane ek soorya grahan ke hone ke baare mein bhi bhavishyavaani ki thi. uske shishya paaithaagoras ne jyaamiti ko yoonaaniyon ke beech ek maanya vigyaan ka svaroop dilaakar yooklid aur aarkimideej ke liye aage ka maarg prashast kiya.

bebeelon nivaasiyon ke viraasat mein mile gyaan mein yoonaaniyon ne kaafi vruddhi ki. iske alaava ganit ko ek tarkasangat paddhati ke roop mein unhonne sthaapit bhi kiya-ek aisi paddhati jismein kuchh mool tathyon ya dhaaranaaon ko satya maankar (jinhein prameya kehte hain) nishkarshon (jinhein upapatti ya pramaan kehte hain) tak pahuncha jaata hai.

Bhaarat ke liye yeh gaurav ki baat hai ki baarahaveen sadi tak ganit ki sampoorn vikaas-yaatra mein uske unnayan ke liye kiye gaye saare mahatvapoorn prayaas adhikaanshataya bhaarateeya ganitjnyaon ki khojon par hi aadhaarit the.

ise bhi dekhein : bhaarateeya ganit, bhaarateeya ganitjnya soochi

ganit kaaryapaddhati

ganit maanav mastishk ki upaj hai. maanav ki gatividhiyon evam prakruti ke nireekshan dvaara hi ganit ka udbhav hua. maanav mastishk ki chintan prakriyaaon ke mool mein paith kar hi ganit mukhar roop se unki abhivyakti karta hai aur vaastavik sansaar avadhaaranaaon ki duniya mein badal jaata hai. ganit vaastavik jagat ko niyamit karne waali moort dhaaranaaon ke peechhe kaam karne vaale niyamon ka adhyayan karta hai. jyaadaatar dainik jeevan ka ganit in mool dhaaranaaon ka hi saar hai aur isliye ise aasaani se samjha-boojha ja sakta hai. haalaanki adhikaansh dhaaranaaen ant:prajnyaa ke dvaara hi ham par prakat hoti hai, fir bhi shuddh evam sankshep roop mein un dhaaranaaon ko vyakt karne ke liye uchit shabdaavali evam kuchh niyamon aur prateekon ki aavashyakta padti hai.

at: ganit ki apni alag hi bhaasha evam lipi hoti hai jise pehle jaanana-samajhna jaroori hota hai. shaayad yahi kaaran hai ki dainik jeevan se asanbaddhit maankar ise samajhne ki drushti se kathin maana jaata hai, jabki hakeekat mein yeh vaastavik jeevan ke saath abhinn roop se juda hi naheen hai, balki usi se iski utpatti bhi hui hai. yeh vidanbana hi hai ki jyaadaatar log ganit ke prati vimukhata dikha kar usase door bhaagte hain, jabki vastusthiti yeh hai ki jeevan tatha gyaan ke har kshetr mein iski upayogita hai. yeh keval sanyog naheen hai ki aarkimideej, nyootan, gaus aur laigaraanj jaise mahaan vaigyaaniko ke vigyaan ke saath-saath ganit mein bhi apna mahaan yogadaan diya hai.

ganit ka vargeekaran

vartamaan mein ganit ko mote taur par do bhaagon mein baanta jaata hai:

anuprayukt ganit ya niyojya ganit (Applied Mathematics) aur
shuddh ganit (Pure Mathematics).

anuprayukt ganit

vigyaan, arthashaastr aur anya kai kshetron mein prayog kiya jaane wala ganit praayogik ganit hai aur ismein adhyayan ki jaane waali ganiteeya samasyaaon ka srot kisi aur kshetr mein hota hai. iske antargat yantrashaastr, bhoomaapan, bhoopadaarth vigyaan, jyotish aadi vishay hai.


ganiteeya bhautiki	taral gatiki	ishtatamakaran	praayikta	saankhyiki	ganiteeya vitt	khel siddhaant

shuddh ganit

shuddh ganit swayam ganit mein upaji un samasyaaon ka hal dhoondhata hai jinka anya kshetron se seedha sambandh naheen hai. kai baar samay ke saath-saath shuddh ganit ke anuprayog milte jaate hain aur is prakaar usaka kuchh hissa praayogik ganit mein aata jaata hai. shuddh ganit ke antargat, beejaganit, jyaamiti aur sankhya siddhaant aadi aate hain. frama ka suprasiddh prameya, sankhya siddhaant ka hi ek ang hai. shuddh ganit ka vikaas beesaveen shataabdi mein bahut adhik hua aur iske vikaas mein 1900 mein David hilbart ke dvaara peris mein diye gaye vyaakhyaan ka bahut yogadaan raha.

sankhyaaeain

$1,2,3,\ldots \!$ $1,2,3,\ldots \!$	$\ldots ,-2,-1,0,1,2\,\ldots \!$ $\ldots ,-2,-1,0,1,2\,\ldots \!$	$-2,{\frac {2}{3}},1.21\,\!$ $-2,{\frac {2}{3}},1.21\,\!$	$-e,{\sqrt {2}},3,\pi \,\!$ $-e,{\sqrt {2}},3,\pi \,\!$	$2,i,-2+3i,2e^{i{\frac {4\pi }{3}}}\,\!$ $2,i,-2+3i,2e^{i{\frac {4\pi }{3}}}\,\!$
praakrutik sankhyaaeain	poornaank	parimeya sankhyaaeain	vaastavik sankhyaaeain	samishr sankhyaaeain

sanrachanaaeain (structures)

${\begin{matrix}(1,2,3)&(1,3,2)\\(2,1,3)&(2,3,1)\\(3,1,2)&(3,2,1)\end{matrix}}$ ${\begin{matrix}(1,2,3)&(1,3,2)\\(2,1,3)&(2,3,1)\\(3,1,2)&(3,2,1)\end{matrix}}$
saanyogiki	sankhya siddhaant	samooh siddhaant	graaf siddhaant	kram siddhaant (Order theory)	beejaganit

aakaash (space)


jyaamiti	trikonamiti	avakal jyaamiti	topoloji	fraiktal jyaamiti	maapan siddhaant

roopaantaran (transformation)


kailakulas	sadish kailakulas	avakal sameekaran	gateeya nikaaya	akram siddhaant (chaos theory)	samishr vishleshan

vivikt ganit (Discrete mathematics)

saiddhaantik sanganak vigyaan mein kaam aane vaale ganit ka saamaanya naam vivikt ganit hai. ismein sanganan siddhaant (Theory of Computation), sangananaatmak jatilta siddhaant, tatha saidhaantik kampyootar vigyaan shaamil hain.

${\begin{matrix}\left[1,2,3\right]&\left[1,3,2\right]\\\left[2,1,3\right]&\left[2,3,1\right]\\\left[3,1,2\right]&\left[3,2,1\right]\end{matrix}}$ ${\begin{matrix}\left[1,2,3\right]&\left[1,3,2\right]\\\left[2,1,3\right]&\left[2,3,1\right]\\\left[3,1,2\right]&\left[3,2,1\right]\end{matrix}}$
saanyojiki	sahaj kulak siddhaant	sanganan siddhaant	kootan (Encryption)	graaf siddhaant

sanganan ke upakaran

neeche kuchh muktasrot computer sauftaveyaron ka naam diya gaya hai jo ganit ke vibhinn kaarya karne ke liye bahut upayogi hain.

	maiksima (Maxima (software))	http://maxima.sourceforge.net/
110px	saailaib (Scilab)	http://scilabsoft.inria.fr/
	aar (softaveyar) (R (software))	http://www.r-project.org/
	GNU aaktev	http://www.octave.org