
seedhi dhaar aur kampas nirmaan

jyaamiti mein, seedhi dhaar aur parisar nirmaan-jise shaasak-aur-parisar nirmaan, euclidean nirmaan, yaa shaastriya nirmaan key roop mein bhee jaanaa jaataa haye-kewal ack aadarsh shaasak aur kampas kii ack jodee kaa upayog karke lambai, konn aur anya jyaamitiya aakritiyon kaa nirmaan haye.
aadarsh shaasak, jise ack seedhi dhaar key roop mein jaanaa jaataa haye, koo lambai mein anant maanaa jaataa haye, kewal ack kinara hotaa haye, aur usa parr koee nishan naheen hotaa haye. kampas koo koee adhikatam yaa nyuntam trijya naheen maanaa jaataa haye, aur prishth seey uthaaye jaane parr "dhwast" maanaa jaataa haye, isaliye iska upayog seedhe doori sthanantarit karane key liye naheen kiya jaa sakata haye. (yeh ack mahatvaheen pratibandh haye kyonki, ack bahu-charaniya prakriya kaa upayog karke, ack doori koo ack dhahane vaale kampas key saath bhee sthanantarit kiya jaa sakata haye; kampas samatulya pramey dekhein. haalaanki dhyaan dein kii ack seedhi dhaar key khilaf aayojit ack gair-takaraane vaalaa compass ise chihnit karane key barabar lugg sakata haye, nyusis nirmaan abhee bhee asweekarya haye aur iska vaastav mein kyaa arth haih neeche chihnit shasakon koo dekhein.) adhik aupchaarik roop seey, kewal anumeya nirmaan way hain joe euclid key "tatvon" key pahale teen abhidhaaranaon dwara diye gaye hain.
yeh mamla yeh nikla haye kii seedhi dhaar aur kampas kaa upayog karke harr nirmaan yogya bindu kaa nirmaan akele kampas kaa upayog karke kiya jaa sakata haye, yaa akele seedhi dhaar dwara yadi ack ekal vritt aur iska kendra diyaa jaae.
pracheen yunani ganitvidon ney pahali baar seedhi-aur-vyaapak nirmaan kii kalpana kii, aur samtal jyaamiti mein kaee pracheen samasyaaen iss pratibandh koo lagu karati hain. pracheen yoonaniyon ney kaee nirmaan vikasit kiye, lekin kuchh mamalon mein aisaa karane mein asamarth they. goss ney dikhaaya kii kuchh bahubhuj nirmaan yogya hain lekin adhikansh naheen hain. 1837 mein kshetra siddhant kaa upayog karke chhed kii kamee dwara kuchh sabse prasiddh seedhi-aur-vyaapak samasyaaen asambhav saabit huin, arthaat ack manmaane konn koo trigunit karna aur ack ghana kii matra koo doguna karna (§ asambhav nirmaan dekhein). inmein seey kaee samasyaaen aasaani seey hull kii jaa sakane vaali hain basharte kii anya jyaamitiya parivartanon kii anumati hoe; udaaharan key liye, poorv dou samasyaon koo hull karane key liye nyusis nirmaan kaa upayog kiya jaa sakata haye.
beejganit key sandarbh mein, ack lambai nirmaan yogya haye yadi aur kewal tabhi jaba yeh ack nirmaan yogya sankhya kaa pratinidhitva karati haye, aur ack konn nirmaan yogya haye yadi aur kewal tabhi jaba isaki cosine ack nirmaan yogya sankhya hoe. ack sankhya nirmaan yogya haye yadi aur kewal tabhi jaba ise chaar buniyaadi ankganitiya sanchaalan aur vargamool key nishkarshan kaa upayog karke likhaa jaa sakata haye lekin koee uchch crum mool naheen haye.
seedhi dhaar aur kampas upkaran.
seedhi dhaar aur vyaapak nirmaan key "seedhi dhaar" aur "kampas" vaastavik duniya key shasakon aur compass key aadarsh sanskaran hain.
vaastavik kampas dhwast naheen hotey hain aur aadhunik jyaamitiya nirmaan aksar iss suvidha kaa upayog karate hain. ack 'dhahane vaalaa kampas' ack cumm shaktishali upkaran prateet hoga. haalaanki, euclid key tatvon key pustak 1 key prastaav 2 mein kampas samatulyata pramey dwara, ack dhahane vaale kampas kaa upayog karke koee shakti naheen khoti haye. yadyapi prastaav sahee haye, isake pramanon kaa ack lamba aur jaanchit itihas haye. kissy bhee mamle mein, samaantaa yeh haye kii yeh visheshta aadarsh kampas kii paribhasha mein kyon nirdhaarit naheen haye.
pratyek nirmaan ganitiya roop seey "sateek" honah chaahiye. "netragolak" doori (nirmaan koo dekhate hue aur isaki sateekta kaa anuman lagaate hue) yaa ack shaasak parr nishan kaa upayog karane kii anumati naheen haye. pratyek nirmaan koo bhee "samaapt" honah chaahiye. arthaat, isamen seemit sankhya mein kadam honey chaahiye, aur kabhi bhee nikat anumaanon kii seema naheen honey chaahiye. (yadi aseemit sankhya mein charanon kii anumati dee jaati haye, tou kuchh anyatha-asambhav nirmaan ack seema mein parivartit honey vaale anant anukramon key madhyam seey sambhav hoe jaate hain.)
iss tarah seey kahaa gayaa haye kii seedhi dhaar aur vyaapak nirmaan ack gambhir vyaavhaarik samasya key bajaay ack parlor game prateet hotaa haye, lekin pratibandh kaa uddeshya yeh sunishchit karna haye kii nirmaan koo "bilkul" sahee saabit kiya jaa sakey.
itihas.
pracheen yunani ganitvidon ney pahali baar seedhi-aur-vyaapak nirmaan kaa prayaas kiya, aur unhone paaya kii dee gaee lambai key yog, antar, utpaad, anupaat aur vargamool kaa nirmaan kaise kiya jaataa haye. way ack diye gaye konn key aadhe hisse kaa nirmaan bhee curr sakate hain, ack varg jiska kshetrafal dusare varg seey doguna haye, ack varg jiska kshetrafal diye gaye bahubhuj key samaan haye, aur 3,4, yaa 5 bhujaaon key niyamit bahubhuj (yaa ack diye gaye bahubhuj key bhujaaon kii sankhya key dogune key saath). lekin way kissy diye gaye konn key ack tihai hisse kaa nirmaan naheen curr sakate they, sivaay vishesh mamalon key, yaa ack diye gaye vritt key samaan kshetra key saath ack varg, yaa anya pakshon kii sankhya key saath niyamit bahubhuj. naa hii way ack ghana key usa bhaag kaa nirmaan curr sakate they jiska aayatan ack diye gaye bhaag key saath ack ghana key aayatan seey doguna hoe.
hippocrates aur menecomus ney dikhaaya kii ghana kii matra koo hyperbolas aur parabolas key pratichedanon koo dhundhkar doguna kiya jaa sakata haye, lekin inca nirmaan seedhi dhaar aur kampas dwara naheen kiya jaa sakata haye. eesaa poorv paanchvin shatabdi mein, hippia ney ack vakr kaa upayog kiya jise unhone samanya konn aur vritt koo varg karane key liye ack chaturbhuj kahaa, aur doosari shatabdi eesaa poorv mein nicomedes ney dikhaaya kii ack manmaane konn koo tribhuj karane key liye ack shanku kaa upayog kaise kiya jaae; lekin inn tarikon kaa paalan kewal seedhi dhaar aur kampas key saath bhee naheen kiya jaa sakata haye.
dou sahasraabdiyon taka ansuljhi samasyaon parr koee pragati naheen hui, jaba taka kii 1796 mein goss ney dikhaaya kii 17 pakshon key saath ack niyamit bahubhuj kaa nirmaan kiya jaa sakata haye; paanch saal baad unhone "aine" pakshon key niyamit bahubhuj key liye paryaapt mandand dikhaaya joe nirmaan yogya thaa.
1837 mein piyare visale ney ack manmaane konn koo trivibhajan karane yaa ack ghana kii matra koo doguna karane kii asambhavata kaa pramaan prakaashit kiya, joe lambai key ghana jadon key nirmaan kii asambhavata parr adharit thaa. unhone yeh bhee dikhaaya kii niyamit bahubhuj key liye goss kii paryaapt nirmaan kshamata kii sthiti bhee aavashyak haye.
phir 1882 mein lindeman ney dikhaaya kii sootra _ 1 ack divya sankhya haye, aur iss prakaar ack diye gaye vritt key samaan kshetra key saath ack varg kaa nirmaan karna seedhi dhaar aur compass dwara asambhav haye.
buniyaadi nirmaan.
sabhi seedhi dhaar aur vyaapak nirmaanon mein pahale seey hii banaae gaye binduon, rekhaaon aur vritton kaa upayog karke paanch buniyaadi nirmaanon kaa baar-baar upayog kiya jaataa haye. yeye hainh
udaaharan key liye, kewal dou alag-alag binduon seey shuru karate hue, humm ack rekha yaa dou vritton mein seey kissy ack koo banaa sakate hain (badale mein, pratyek bindu koo kendra key roop mein upayog karke aur dusare bindu seey gujarate hue). yadi humm donon vritton koo kheenchate hain, tou unake pratichhedan parr dou nae bindu banaae jaate hain. dou mool binduon aur inn nae binduon mein seey ack key beach rekhaaen kheenchane seey ack sambahu trikon kaa nirmaan puuraa hotaa haye.
isaliye, kissy bhee jyaamitiya samasya mein hamaare paas prateekon (binduon aur rekhaaon) kaa ack praarambhik sett, ack algorithm aur kuchh parinam hotey hain. iss drishtikon seey, jyaamiti ack swayamsiddh beejganit key barabar haye, joe isake tatvon koo prateekon dwara pratisthaapit karati haye. shayed gaus ney pahale ise mahsoos kiya, aur iska upayog kuchh nirmaanon kii asambhavata koo saabit karane key liye kiya; bahut baad mein hilbert ney jyaamiti key liye swayamsiddh kaa ack puuraa sett paaya.
samanya seedhi dhaar aur vyaapak nirmaan.
sabse adhik upayog kiye jaane vaale seedhe kinare aur vyaapak nirmaanon mein shamil hainh
nirmaan yogya bindu.
dou rekhaaon seey bunny cartasian samanvaya pranali kaa upayog karke ack beejganit koo hamari jyaamiti seey jodaa jaa sakata haye, aur vactor dwara hamaare tala key binduon kaa pratinidhitva curr sakata haye. anta mein humm inn vactors koo jatil sankhyaon key roop mein likh sakate hain.
rekhaaon aur vritton key liye samikaranon kaa upayog karke, koee bhee yeh dikhaa sakata haye kii jinn binduon parr way pratichhed karate hain, way sabse chhote kshetra "f" key ack vargaatmak vistaar mein sthit hain, jisamen rekha parr dou bindu, vritt kaa kendra aur vritt kii trijya hoti haye. arthaat, way roop key hain, kahaan, aur hain.
choonki nirmaan yogya binduon kaa kshetra "vargamool" key tahat bund hotaa haye, isaliye isamen way sabhi bindu hotey hain jinhen parimeya gunank key saath jatil sankhyaon key kshetra key dwighaat vistaar key ack seemit anukram dwara praapt kiya jaa sakata haye. uparokt paragraph seey, koee bhee yeh dikhaa sakata haye kii kissy bhee rachanaatmak bindu koo vistaar key iss tarah key anukram seey praapt kiya jaa sakata haye. isake ack upsamhaar key roop mein, koee bhee paataa haye kii ack nirmaan yogya bindu (aur isaliye kissy bhee nirmaan yogya lambai) key liye nyuntam bahupad kii degree 2 kii shakti haye. vishesh roop seey, koee bhee nirmaan yogya bindu (yaa lambai) ack beejganitiya sankhya haye, halanki pratyek beejganitiya sankhya nirmaan yogya naheen haye; udaaharan key liye, beejganitiya haye lekin nirmaan yogya naheen haye.
nirmaan yogya konn.
kissy bhee nirmaan yogya vritt parr nirmaan yogya konon aur nirmaan yogya binduon key beach ack dwibhajan hotaa haye. joe konn nirmaan yogya hain, way jodd modulo 2π key tahat ack abelian samooh banaate hain (joe jatil sankhyaon key roop mein dekhe jaane vaale ikai vritt parr binduon key gunan seey mell khaata haye). joe konn nirmaan yogya hain way thheek vahee hain jinaki sparshrekha (yaa samkaksh, saain yaa cosine) ack sankhya key roop mein nirmaan yogya haye. udaaharan key liye, niyamit heptadecagan (satrah-pakshiya niyamit bahubhuj) nirmaan yogya haye kyonki
jaisa kii gaus dwara khoja gayaa thaa.
nirmaan yogya konon kaa samooh usa operation key tahat bund hotaa haye joe konon koo aadhaa kartaa haye (joe jatil sankhyaon mein vargamool lene key anuroop hotaa haye). dou binduon seey shuru honey vaale seemit crum key ekmatra konn way hain jinka crum yaa tou dou kii shakti haye, yaa dou kii shakti kaa utpaad haye aur alag-alag firmate prime kaa ack sett haye. isake alawa anant crum key nirmaan yogya konon kaa ack ghanaa samooh haye.
jatil ankganit seey sambandh.
euclidean tala mein binduon kaa ack samooh diye jaane parr, unamen seey kissy ack koo 0 aur dusare koo 1 kahane key liye chunana, abhivinyas key manmaane vikalp key saath hamein binduon koo jatil sankhyaon key ack samooh key roop mein vichaar karane kii anumati detaa haye.
jatil sankhyaon key roop mein binduon key ack samooh kii aisee kissy bhee vyakhya koo dekhate hue, kewal vaidh seedhi dhaar aur samagr nirmaan kaa upayog karke nirmaan yogya bindu sateek roop seey sabse chhote kshetra key tatva hain jinamen binduon kaa mool sett hotaa haye aur jatil sanyugma aur vargamool sanchaalan key tahat bund hotey hain (aspashtata seey bachane key liye, humm π seey cumm jatil tark key saath vargamool nirdisht curr sakate hain). iss kshetra key tatva thheek way hain jinhen mool binduon mein kewal jodd, ghataav, gunan, vibhajan, jatil sanyugma aur vargamool key sanchaalan kaa upayog karke ack sootra key roop mein vyakt kiya jaa sakata haye, jise aasaani seey samtal kaa ack ganana yogya ghanaa upasamuchchay maanaa jaataa haye. inn chhah karyon mein seey pratyek ack saral seedhi-aur-vyaapak nirmaan key anuroop haye. iss tarah key sootra seey pratyek ankganitiya sanchaalan key liye nirmaanon koo jodkar sambandhit bindu kaa nirmaan karna seedha haye. iss tarah kii gananaon mein binduon key ack vishesh samooh kaa adhik kushal nirmaan shortcut key anuroop hotaa haye.
samatulya roop seey (aur manmaane dhang seey dou ankon koo chunane kii aavashyakta key binaa) humm kaha sakate hain kii, abhivinyas key manmaane vikalp koo dekhate hue, ankon kaa ack sett kissy bhee dou jode ankon key beach antar key anupaat dwara diye gaye jatil anupaaton kaa ack sett nirdhaarit kartaa haye. iss tarah key anupaaton key samooh seey seedhi dhaar aur kampas kaa upayog karke nirmaan yogya anupaaton kaa samooh sateek roop seey sabse chhota kshetra haye jisamen mool anupaat hotey hain aur jatil sanyugma aur vargamool lene key tahat bund hotey hain.
udaaharan key liye, ack bindu yaa anupaat "z" kaa vaastavik bhaag, kaalpanik bhaag aur mapank (oopar diye gaye dou drishtikon mein seey ack koo lete hue) rachanaatmak hain kyonki inhen iss prakaar vyakt kiya jaa sakata haye -
"ghana koo doguna karna" aur "ack konn kaa trivibhajan" (kissy bhee "φ" jaise vishesh konon koo chhodkar jaise kii "φ"/(2) ack parimeya sankhya haye jisamen bhaajak 3 seey vibhaajya naheen haye) key liye anupaat kii aavashyakta hoti haye joe ghana samikaranon kaa samadhan hain, jabaki "vritt koo varg karane" key liye ack divya anupaat kii aavashyakta hoti haye. inmein seey koee bhee varnit kshetron mein naheen haye, isaliye inke liye koee seedhi-aur-vyaapak nirmaan maujood naheen haye.
asambhav nirmaan.
pracheen yoonaniyon ney socha kii way jinn nirmaan samasyaon koo hull naheen curr sakate they, way kewal jiddi theen, naa kii ansuljhi. aadhunik tarikon key saath, halanki, inn seedhi dhaar aur vyaapak nirmaanon koo taarkik roop seey asambhav dikhaaya gayaa haye. (haalaanki, samasyaaen swayam hull karane yogya hain, aur yunani jaanate they kii kewal seedhi dhaar aur kampas key saath kaam karane kii baadha key binaa unhein kaise hull kiya jaae.)
vritt koo chaukor karna.
inn samasyaon mein seey sabse prasiddh, vritt koo varg karna, jise anyatha vritt key chaturbhuj key roop mein jaanaa jaataa haye, mein kewal seedhi dhaar aur kampas kaa upayog karke diye gaye vritt key samaan kshetra key saath ack varg kaa nirmaan karna shamil haye.
vritt koo varg karna asambhav saabit huaa haye, kyonki isamen ack divya sankhya, yaani, utpann karna shamil haye. kewal kuchh beejganitiya sankhyaon kaa nirmaan kewal shaasak aur kampas key saath kiya jaa sakata haye, arthaat joe yog, ghataav, gunan, vibhajan aur vargamool lene key sanchaalan key ack seemit anukram key saath purnankon seey nirmit hain. vakyansh "vritt koo varg karna" kaa arth aksar iss kaaran seey "asambhav koo karna" key liye kiya jaataa haye.
kewal shaasak aur kampas dwara samadhan kii aavashyakta kii baadha key binaa, samasya koo jyaamitiya aur beejganitiya sadhanon kii ack vistrit vividhata dwara aasaani seey hull kiya jaa sakata haye, aur pracheen kaal mein kaee baar hull kiya gayaa thaa.
ack vidhi joe "vritt key chaturbhuj" key lagbhag aaney key bahut karib aatee haye, ack capler trikon kaa upayog karke praapt kii jaa sakati haye.
ghana koo doguna karein.
ghana koo doguna karna ack ghana key kinare kaa nirmaan haye, jisamen kewal ack seedhi dhaar aur kampas kaa upayog kiya jaataa haye, jisamen ack diye gaye kinare key saath ack ghana kii matra doguni hoti haye. yeh asambhav haye kyonki 2 key ghana mool kii ganana purnank seey jodd, ghataav, gunan, vibhajan aur vargamool laker naheen kii jaa sakati haye. iska kaaran yeh haye kii tarksangat parr isaki nyuntam bahupad mein degree 3 haye. yeh nirmaan dou nishan aur ack kampas key saath ack seedhi dhaar kaa upayog karke sambhav haye.
konn trichhedan.
konn trichhedan ack aise konn kaa nirmaan haye, jisamen kewal ack seedhi dhaar aur ack kampas kaa upayog kiya jaataa haye, joe ack diye gaye manmaane konn kaa ack tihai haye. samanya mamle mein yeh asambhav haye. udaaharan key liye, konn 2/5 radian (72° = 360°/5) koo trikoniya kiya jaa sakata haye, lekin konn/3 radian (60°) koo trikoniya naheen kiya jaa sakata haye. samanya trisection samasya koo bhee aasaani seey hull kiya jaa sakata haye jaba usa parr dou nishan vaale ack seedhi dhaar kii anumati dee jaati haye (ack nyusis nirmaan).
ack deerghavritt taka kii doori.
samtal mein kissy bhee bindu seey vritt parr nikatatam bindu taka rekha khand kaa nirmaan kiya jaa sakata haye, lekin samtal mein kissy bhee bindu seey sakaratmak vikendrikaran key deerghavritt parr nikatatam bindu taka khand kaa nirmaan samanya roop seey naheen kiya jaa sakata haye.
alhazen kii samasya.
1997 mein, oxford key ganitshastri peater aim. neuman ney iss pramey koo saabit curr diyaa kii pracheen alhazen kii samasya (billiard samasya yaa golakar darpan seey pratibimb) key samanya samadhan key liye koee shaasak-aur-samagr nirmaan naheen haye.
niyamit bahubhuj kaa nirmaan.
kuchh niyamit bahubhuj (udaa.jee. ack panchbhuj) koo seedhi dhaar aur compass key saath banaana aasaan haye; anya naheen hain. isase sawal yeh huaah kyaa sabhi niyamit bahubhuj kaa nirmaan seedhi dhaar aur kampas key saath karna sambhav haye?
1796 mein carle friedrich goss ney dikhaaya kii ack niyamit 17-pakshiya bahubhuj kaa nirmaan kiya jaa sakata haye, aur paanch saal baad dikhaaya kii ack niyamit "aine"-pakshiya bahubhuj kaa nirmaan seedhi dhaar aur compass key saath kiya jaa sakata haye yadi "aine" key visham pramukh kaarak alag-alag firmate prime hain. goss ney anuman lagaaya kii yeh sthiti bhee aavashyak thee; 1837 mein pier visale dwara anuman saabit kiya gayaa thaa.
pahale kuchh rachanaatmak niyamit bahubhuj mein nimnalikhit bhujaaen hoti hainh
ack samaan sankhya mein bhujaaon key saath rachanaatmak niyamit bahubhuj kaa ack anantank gyaat haye (kyonki yadi ack niyamit "aine"-gonn rachanaatmak haye, tou ack niyamit 2 "aine"-gonn haye aur isaliye ack niyamit 4 "aine"-gonn, 8 "aine"-gonn, aadi haye.). haalaanki, kewal 31 gyaat rachanaatmak niyamit "aine"-guns hain jinake paksh visham sankhya mein hain.
teen vishisht binduon yaa lambai seey ack trikon kaa nirmaan.
ack trikon key solah pramukh bindu isake sheersh, isake kinaaron key madhya bindu, isaki oonchaai key pair, isake aantarik konn dwibhaajak key pair aur iska paridhi, kendrak, orthocenter aur protsaahan hain. teen binduon seey ack trikon key nirmaan kii 139 alag-alag gair-tuchchh samasyaon koo utpann karane key liye inhen ack samay mein teen liya jaa sakata haye. inn samasyaon mein seey teen mein ack bindu shamil haye jise anya dou binduon seey advitiya roop seey banaaya jaa sakata haye; 23 kaa gair-advitiya roop seey nirmaan kiya jaa sakata haye (vaastav mein anant roop seey kaee samadhanon key liye) lekin kewal tabhi jaba binduon key sthaan kuchh baadhaon kaa paalan karate hain; 74 mein samasya samanya mamle mein nirmaan yogya haye; aur 39 mein aavashyak trikon maujood haye lekin nirmaan yogya naheen haye.
ack tribhuj kii baarah pramukh lambai teen taraf kii lambai, teen oonchaai, teen madhya aur teen konn dwibhaajak hain. teen konon key saath, yeye 95 alag-alag sanyojan dete hain, jinamen seey 63 ack nirmaan yogya trikon koo janm dete hain, jinamen seey 30 naheen, aur jinamen seey dou cumm paribhaashit hain.
pratibandhit nirmaan.
vibhinn niyamon key tahat nirmaan key liye sweekarya upkaranon koo pratibandhit karane key liye vibhinn prayaas kiye gaye hain, taaki yeh nirdhaarit kiya jaa sakey kii abhee bhee kyaa nirmaan yogya haye aur iska nirmaan kaise kiya jaa sakata haye, saath hii saath nyuntam mandand nirdhaarit kiya jaa sakey joe abhee bhee compass aur seedhi dhaar vaale sabhi nirmaan mein saksham honey key liye aavashyak haye.
kewal shaasak yaa kewal kampas key saath nirmaan.
yeh sambhav haye (mohar-masteuroni pramey key anusaar) kii kewal ack kampas key saath kuchh bhee banaaya jaae yadi ise ack shaasak aur kampas key saath banaaya jaa sakata haye, basharte kii diye gaye detta aur paae jaane vaale detta mein asatat bindu (rekhaaen yaa vritt naheen) houn. iss pramey kaa satya archimedies key swayamsiddh key satya parr nirbhar kartaa haye, joe prakriti mein pratham crum naheen haye. kampas-maatr nirmaan key udaharanon mein napolean kii samasya shamil haye.
kewal ack shaasak key saath ack vargamool lena asambhav haye, isaliye kuchh cheejen jinka nirmaan ack shaasak key saath naheen kiya jaa sakata haye, unakaa nirmaan ack compass key saath kiya jaa sakata haye; lekin (ponsalate-steiner pramey dwara) ack ekal vritt aur isake kendra koo dekhate hue, unakaa nirmaan kiya jaa sakata haye.
vistaarit nirmaan.
pracheen yoonaniyon ney nirmaan koo teen pramukh shreniyon mein vargeekrit kiya, joe unake samadhan key liye aavashyak upkaranon kii jatilata parr nirbhar kartaa haye. yadi kissy nirmaan mein kewal ack seedhi dhaar aur kampas kaa upayog kiya jaataa haye, tou ise samtal kahaa jaataa thaa; yadi isake liye ack yaa adhik shanku khandon (vritt key alawa) kii bhee aavashyakta hoti haye, tou ise thhos kahaa jaataa thaa; teesari shreni mein una sabhi nirmaanon koo shamil kiya jaataa thaa joe anya dou shreniyon mein seey kissy mein bhee naheen aatey they. yeh vargeekaran aadhunik beejganitiya drishtikon key saath achchi tarah seey mell khaata haye. ack jatil sankhya jise kewal kshetra sanchaalan aur vargamool (jaisa kii oopar varnit haye) kaa upayog karke vyakt kiya jaa sakata haye, ack samtal nirmaan haye. ack jatil sankhya jisamen ghana jadon kaa nishkarshan bhee shamil haye, ack thhos nirmaan haye.
kshetron kii bhasha mein, ack jatil sankhya joe samtal haye, mein dou kii shakti hoti haye, aur ack kshetra vistaar mein sthit hoti haye jise kshetron key ack meenar mein vibhajit kiya jaa sakata haye jahaan pratyek vistaar mein dou degree hoti haye. ack jatil sankhya jisamen ack thhos nirmaan hotaa haye, kewal dou aur teen key pramukh karakon key saath degree hoti haye, aur ack kshetra vistaar mein sthit hoti haye joe kshetron key ack meenar key sheersh parr hoti haye jahaan pratyek vistaar mein degree 2 yaa 3 hoti haye.
thhos nirmaan.
ack bindu kaa ack thhos nirmaan hotaa haye yadi iska nirmaan ack seedhi dhaar, kampas aur ack (sambhavatah kaalpanik) shanku rekhaachitra upkaran kaa upayog karke kiya jaa sakata haye joe pahale seey nirmit phocus, directrics aur vilakshanta key saath kissy bhee shanku koo aakarshit curr sakata haye. ack hii bindu samooh kaa nirmaan aksar upkaranon key ack chhote samooh kaa upayog karke kiya jaa sakata haye. udaaharan key liye, ack kampas, seedhi dhaar aur kagaj key ack tukade kaa upayog karke, jiss parr hamaare paas paravalay y = x2 haye, saath mein bindu (0,0) aur (1,0) hain, koee bhee aisee jatil sankhya kaa nirmaan curr sakata haye jisamen ack thhos sanrachana hoe. isi tarah, ack upkaran joe pahale seey nirmit kendra aur pramukh aksh (dou pinn aur ack string kaa tukada sochein) key saath kissy bhee deerghavritt koo kheench sakata haye, utanaa hii shaktishali haye.
pracheen yunani jaanate they kii ghana koo doguna karna aur manmaane konn koo trivibhajan karna donon mein thhos nirmaan they. archimedies ney niyamit 7-gonn kaa ack thhos nirmaan diyaa. vritt key chaturbhuj mein thhos nirmaan naheen hotaa haye.
ack niyamit "aine"-gonn kaa ack thhos nirmaan hotaa haye yadi aur kewal tabhi jaba "aine" = 2 "aeye" 3 "bee" aim "jahaan" aeye "aur" bee "kuchh gair-nakaratmak purnank hain aur" aim "shoonya yaa adhik vishisht chhed-akshar abhaajya kaa utpaad haye (roop 2" aar "3" aiss "+ 1 key abhaajya). isaliye, niyamit "aine"-gonn ack thhos koo sweekaar kartaa haye, lekin samtal naheen, nirmaan yadi aur kewal tabhi jaba "aine" anukram mein hoe.
aine" kaa vaha samooh jisake liye ack niyamit "aine"-gonn kaa koee thhos nirmaan naheen hotaa haye, vaha anukram haye.
firmate prime key saath prashn kii tarah, yeh ack khula prashn haye kii kyaa pierpont prime kii ack anant sankhya haye.
konn trichhedan.
kyaa hoga yadi, seedhi dhaar aur kampas key saath, hamaare paas ack upkaran thaa joe (kewal) ack manmana konn koo trivibhajan curr sakata thaa? iss tarah key nirmaan thhos nirmaan hain, lekin thhos nirmaan key saath sankhyaaen maujood hain jinka nirmaan iss tarah key upkaran kaa upayog karke naheen kiya jaa sakata haye. udaaharan key liye, humm iss tarah key upkaran key saath ghana koo doguna naheen curr sakate. doosari or, pratyek niyamit aine-gonn jisamen ack thhos nirmaan hotaa haye, iss tarah key upkaran kaa upayog karke banaaya jaa sakata haye.
origami.
origami kaa ganitiya siddhant seedhi dhaar aur vyaapak nirmaan kii tulanaa mein adhik shaktishali haye. hujita-hatori swayamsiddh koo santusht karane vaale taha ack kampas aur shanku rekhaachitra upkaran kaa upayog karke vistaarit nirmaan key samaan hii binduon kaa nirmaan curr sakate hain. isaliye, origami kaa upayog ghana samikaranon (aur isaliye chaturthak samikaranon) koo hull karane key liye bhee kiya jaa sakata haye, aur iss prakaar dou shaastriya samasyaon koo hull kiya jaa sakata haye.
ullekhaniya shaasak.
archimedies, nicomedes aur apollonius ney ack chihnit shaasak key upayog seey judey nirmaan diye. yeh unhein, udaaharan key liye, ack rekha khand, dou rekhaaen (yaa vritt), aur ack bindu lene kii anumati dega; aur phir ack rekha kheenchen joe diye gaye bindu seey gujarati haye aur dou dee gaee rekhaaon koo kaatati haye, jaise kii pratichhedan key binduon key beach kii doori diye gaye khand key barabar haye. ise greek "nyusis" ("jhukaav", "pravritti" yaa "verging") kahate hain, kyonki naee rekha "bindu kii or" jaati haye.
iss vistaarit yojana mein, humm ack manmaane konn koo trivibhajan curr sakate hain (archimids key trivibhajan koo dekhein) yaa ack manmaane ghana mool (nicomedes key kaaran) nikaal sakate hain. isaliye, koee bhee doori jiska maujooda doori kaa anupaat ack ghana yaa ack chaturthansh sameekaran kaa samadhan haye, nirmaan yogya haye. ack chihnit shaasak kaa upayog karate hue, heptagon kii tarah thhos nirmaan key saath niyamit bahubhuj, nirmaan yogya hain; aur johnn aich. conway aur richard key. aadami unamen seey kaee key liye nirmaan detaa haye.
ack shanku rekhaachitra upkaran kii tulanaa mein nyusis nirmaan adhik shaktishali haye, kyonki koee bhee aisee jatil sankhya kaa nirmaan curr sakata haye jisamen thhos nirmaan naheen hain. vaastav mein, iss upkaran kaa upayog karke koee bhee kuchh aise gunon koo hull curr sakata haye joe mool tatvon kaa upayog karke hull karane yogya naheen hain. yeh gyaat haye kii koee bhee nyusis nirmaan kaa upayog karke 7 seey adhik yaa 7 key barabar prime degree key irreducible bahupad koo hull naheen curr sakata haye, isaliye iss upkaran kaa upayog karke niyamit 23-gonn yaa 29-gonn kaa nirmaan karna sambhav naheen haye. benjamin aur snider ney saabit curr diyaa kii niyamit 11-gonn kaa nirmaan karna sambhav haye, lekin nirmaan naheen diyaa. yeh abhee bhee khula haye kii kyaa iss upkaran kaa upayog karke ack niyamit 25-gonn yaa 31-gonn nirmaan yogya haye.
ack seedhe khand koo tribhuj karein.
ack seedhi rekha khand koo dekhate hue jise aeye. bee. kahaa jaataa haye, ise teen nae samaan khandon mein vibhajit kiya jaa sakata haye aur avarodhan pramey key upayog seey aavashyak kaee bhaagon mein.
dwiaadhaari ankon kii ganana.
1998 mein saiman ploff ney ack shaasak-aur-samagr algorithm diyaa jiska upayog kuchh sankhyaon key dwiaadhaari ankon kii ganana karane key liye kiya jaa sakata haye.
algorithm mein ack konn kaa baar-baar doguna honah shamil haye aur lagbhag 20 dwiaadhaari ankon key baad bhautik roop seey avyavahaarik hoe jaataa haye.

