Huakai i loko o ke ao makemakika
Ua kākau au i kēia ʻatikala i ka Pōʻakolu ma hope o ka haʻiʻōlelo a me ka hoʻomaʻamaʻa ʻana ma ke kulanui ʻepekema kamepiula. Ke pale aku nei au iaʻu iho mai ka hoʻohewa ʻana i nā haumāna o kēia kula, ko lākou ʻike, ʻano ʻepekema a ʻo ka mea nui loa: nā mākau aʻo. ʻO kēia ... ʻaʻohe mea e aʻo iā lākou.
No ke aha wau i pale ai? No kahi kumu maʻalahi - aia wau i kahi makahiki, ʻaʻole paha i hoʻomaopopo ʻia ka honua a puni iaʻu. Aʻo paha wau iā lākou pehea e hoʻopaʻa ai a wehe i nā lio, ma mua o ka hoʻokele kaʻa? Aʻo paha wau iā lākou e kākau me ka peni quill? ʻOiai ua ʻoi aku ka maikaʻi o koʻu manaʻo i ke kanaka, ke manaʻoʻiʻo nei au e "hahai", akā...
A hiki i kēia manawa, ma ke kula kiʻekiʻe ua kamaʻilio lākou e pili ana i nā helu paʻakikī. A ma ka Pōʻakolu nei au i hoʻi mai ai i ka hale, haʻalele - aneane ʻaʻohe o nā haumāna i aʻo i ke ʻano a me ka hoʻohana ʻana i kēia mau helu. Ke nānā nei kekahi poʻe i nā makemakika a pau e like me ka kuihi ma ka puka pena. Akā, ua pūʻiwa hoʻi au i ko lākou haʻi ʻana mai iaʻu pehea e aʻo ai. ʻO ka maʻalahi, ʻo kēlā me kēia hola o ke aʻo ʻana he ʻelua hola o ke aʻo ʻana ma ka home: heluhelu i kahi puke, aʻo mua i ka hoʻoponopono ʻana i nā pilikia ma kahi kumuhana, etc. Ma hope o ka hoʻomākaukau ʻana i kēia ala, hele mai mākou i nā hoʻomaʻamaʻa, kahi e hoʻomaikaʻi ai i nā mea āpau ... ʻoluʻolu, ua manaʻo nā haumāna e noho ana i kahi haʻiʻōlelo - ʻo ka nānā pinepine ʻana i ka puka makani - ua hōʻoiaʻiʻo e komo ka ʻike i ke poʻo.
Kāohi! Ua lawa kēlā. E wehewehe au i kaʻu pane i kekahi nīnau i loaʻa iaʻu ma ka papa me nā hoa mai ka National Children's Fund, kahi kula e kākoʻo ana i nā keiki akamai mai nā ʻāina a pau. ʻO ka nīnau (a i ʻole ka manaʻo) ʻo ia:
— Hiki iā ʻoe ke haʻi iā mākou i kekahi mea e pili ana i nā helu maoli ʻole?
"ʻOiaʻiʻo," i pane aku ai au.
ʻO ka ʻoiaʻiʻo o nā helu
"ʻO wau kekahi hoaaloha, ʻo ka pilina ka ratio o nā helu 220 a me 284," wahi a Pythagoras. ʻO ka helu ma ʻaneʻi, ʻo ka huina o nā mea puʻunaue o ka helu 220 ua like ia me 284, a ʻo ka hui pū ʻana o nā mea puʻunaue o ka helu 284 ua like ia me 220.
1 + 2 + 4 + 71 + 142 = 220
1 + 2 + 4 + 5 + 10 = 11 + 20 + 22 + 44 + 55 + 110 = 284. E hoʻomaopopo ma ke ala i hāʻawi ai ʻo Iakoba i ka Baibala iā Esau i nā hipa 220 a me nā hipa kāne i hōʻailona o ka hoaloha (Genesis 32:14).
ʻO kekahi mea like ʻole ma waena o nā helu 220 a me 284 ʻo ia kēia: ʻo nā helu ʻumikūmāhiku kiʻekiʻe loa he 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, a me 59.
ʻO ko lākou huina he 2x220, a ʻo ka huina o nā huinahā he 59x284.
Ka mua. ʻAʻohe manaʻo o "helu maoli". Me he mea lā ma hope o ka heluhelu ʻana i kahi ʻatikala e pili ana i nā ʻelepani e nīnau ʻoe, "I kēia manawa e noi mākou i nā ʻelepani ʻole." Aia nā mea holoʻokoʻa a piha ʻole, noʻonoʻo a noʻonoʻo ʻole, akā ʻaʻohe mea maoli ʻole. kikoʻī: ʻAʻole i kapa ʻia nā helu ʻaʻole pololei. Nui nā ʻano "helu" ma ka makemakika, a he ʻokoʻa lākou mai kekahi i kekahi e like me ka hoʻohālikelike zoological-he elepani a me kahi elo honua.
ʻO ka lua, e hana mākou i nā hana āu i ʻike mua ai ua pāpā ʻia: lawe i nā aʻa huinahā o nā helu maikaʻi ʻole. ʻAe, e lanakila ka makemakika i kēlā mau pale. He manaʻo paha kēia? Ma ka makemakika, e like me ka ʻepekema ʻē aʻe: inā paha e komo mau ke kumumanaʻo i ka waihona o ka ʻike e pili ana ... ma kāna noi. Inā he mea ʻole ia, a laila pau i ka ʻōpala, a laila i kekahi ʻōpala i ka moʻolelo o ka ʻike. Me ka ʻole o nā helu aʻu e kamaʻilio nei ma ka hopena o kēia ʻatikala, hiki ʻole ke hoʻomohala i ka makemakika. Akā, e hoʻomaka kākou me kekahi mau mea liʻiliʻi. ʻIke ʻoe i nā helu maoli. Hoʻopiha piha lākou i ka laina helu me ka ʻole o nā hakahaka. ʻIke ʻoe i nā helu kūlohelohe: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …….. - ʻaʻole kūpono lākou a pau ka hoomanao nui loa. He inoa nani ko lākou: kūlohelohe. Nui ko lakou mau waiwai hoihoi. Pehea kou makemake i kēia:
1 + 15 + 42 + 98 + 123 + 179 + 206 + 220 = 3 + 11 + 46 + 92 + 129 + 175 + 210 + 218
12 + 152 + 422 + 982 + 1232 + 1792 + 2062 + 2202 = 32 + 112 + 462 + 922 + 1292 + 1752 + 2102 + 2182
13 + 153 + 423 + 983 + 1233 + 1793 + 2063 + 2203 = 33 + 113 + 463 + 923 + 1293 + 1753 + 2103 + 2183
14 + 154 + 424 + 984 + 1234 + 1794 + 2064 + 2204 = 34 + 114 + 464 + 924 + 1294 + 1754 + 2104 + 2184
15 + 155 + 425 + 985 + 1235 + 1795 + 2065 + 2205 = 35 + 115 + 465 + 925 + 1295 + 1755 + 2105 + 2185
16 + 156 + 426 + 983 + 1236 + 1796 + 2066 + 2206 = 36 + 116 + 466 + 926 + 1296 + 1756 + 2106 + 2186
17 + 157 + 427 + 983 + 1237 + 1797 + 2067 + 2207 = 37 + 117 + 467 + 927 + 1297 + 1757 + 2107 + 2187
“He mea maʻamau ka hoihoi i nā helu kūlohelohe,” wahi a Carl Lindenholm, a ʻo Leopold Kronecker (1823–1891) i ʻōlelo maʻalahi: “Na ke Akua i hana i nā helu kūlohelohe—ʻo nā mea ʻē aʻe he hana a ke kanaka!” ʻO nā hakina (i kapa ʻia nā helu rational e ka poʻe makemakika) he mau waiwai kupanaha kekahi:
a ma ke kaulike.
Hiki iā ʻoe, e hoʻomaka ana mai ka ʻaoʻao hema, e ʻoki i nā mea hoʻohui a hoʻololi iā lākou me nā hōʻailona hoʻonui - a e mau ka kūlike:
A pela aku.
E like me ka mea i ʻike ʻia, no nā hakina a/b, kahi a a me b nā helu helu a me b ≠ 0, ʻōlelo lākou. helu kuʻuna. Akā kapa wale lākou iā lākou iho ma ka Polani. 'Ōlelo Pelekania, Palani, Kelemānia a me Lūkini. helu kuʻuna. Ma ka ʻōlelo Pelekania: rational numbers. Helu kuhi hewa He noʻonoʻo ʻole, noʻonoʻo. Ke kamaʻilio pū nei mākou ma ka Polani e pili ana i nā manaʻo noʻonoʻo, nā manaʻo a me nā hana - he pupule kēia, noʻonoʻo, ʻaʻole hiki ke wehewehe. 'Ōlelo lākou he makaʻu nā wāhine i nāʻiole - pehea ka noʻonoʻoʻole?
I ka wā kahiko, he ʻuhane ko nā helu. ʻO kēlā me kēia mea i kekahi mea, kēlā me kēia mea i hōʻailona i kekahi mea, ua hōʻike kēlā me kēia i kahi ʻāpana o kēlā lokahi o ke Universe, ʻo ia hoʻi, ma ka ʻōlelo Helene, ka Cosmos. ʻO ka huaʻōlelo "cosmos" ponoʻī ʻo ia ka "kauoha, kauoha." ʻO ka mea nui loa he ʻeono (ka helu kūpono) a me ka ʻumi, ʻo ka huina o nā helu hope 1+2+3+4, i hana ʻia me nā helu ʻē aʻe, ʻo ia ka hōʻailona i ola a hiki i kēia lā. No laila ua aʻo ʻo Pythagoras ʻo nā helu ka hoʻomaka a me ke kumu o nā mea āpau, a ʻo ka ʻike wale nō helu kuhi hewa ua hoʻohuli i ka neʻe Pythagorean i ke ʻano geometry. ʻIke mākou i ke kumu o ke kula
√2 - helu kuhi hewa
No ka mea, aia no: a aole hiki ke hoemiia keia hakina. ʻO ka mea nui, he ʻokoʻa nā p a me q. E huinahalike kakou: 2q2=p2. ʻAʻole hiki ke ʻokoʻa ka helu p, mai ia manawa p2 ʻo ia hoʻi, a ʻo ka ʻaoʻao hema o ka like ʻana he paʻi ia o 2. No laila, p like like, ʻo ia hoʻi, p = 2r, no laila p2= 4r2. Hoemi makou i ka hoohalike 2q2= 4r2 e 2. Loaʻa iā mākou q2= 2r2 a ke ike nei makou he pono pu no q, a ua manao makou aole pela. ʻO ka hopena kū'ē e hoʻopau i ka hōʻoia - hiki ke ʻike pinepine ʻia kēia ʻano kumu ma kēlā me kēia puke makemakika. ʻO kēia hōʻike kūʻokoʻa kahi ʻenehana punahele a ka poʻe sophists.
ʻAʻole hiki ke hoʻomaopopo ʻia kēia nui e ka Pythagoreans. Pono e wehewehe ʻia nā mea a pau e nā helu, a ʻo ka diagonal o kahi huinahā, hiki i kekahi ke kaha kiʻi me ka lāʻau i ke one, ʻaʻohe, ʻo ia hoʻi, hiki ke ana, lōʻihi. "Ua makehewa ko mākou manaʻoʻiʻo," me he mea lā ka poʻe Pythagoreans. Pehea? He ʻano... noʻonoʻo ʻole. Ua hoʻāʻo ka Union e hoʻopakele iā ia iho ma nā ʻano sectarian. ʻO ka mea aʻa e hōʻike i ko lākou ola helu kuhi hewa, e hoʻopaʻi ʻia e ka make, a me he mea lā, ua hoʻokō ʻia ka hoʻopaʻi mua e ka haku ponoʻī.
Akā "ua hala ka manaʻo me ka pōʻino ʻole." Ua hiki mai ka makahiki gula. Ua lanakila nā Helene i nā Peresia (Marathon 490, Plache 479). Ua hoʻoikaika ʻia ka Democracy, ua puka mai nā kikowaena noʻonoʻo noʻonoʻo a me nā kula hou. ʻO ka poʻe hahai o Pythagoreanism e hakakā mau ana me nā helu helu ʻole. Ua hai mai kekahi poe, aole makou e hoomaopopo i keia mea pohihihi; hiki iā mākou ke noʻonoʻo a mahalo iā Uncharted. ʻOi aku ka pragmatic o ka hope a ʻaʻole mahalo i ka mea huna. I kēlā manawa, ua puka mai ʻelua mau manaʻo noʻonoʻo i hiki ke hoʻomaopopo i nā helu irrational. ʻO ka ʻoiaʻiʻo o kā mākou hoʻomaopopo maikaʻi ʻana iā lākou i kēia lā no Eudoxus (XNUMXth century BC), a ma ka hopena o ke kenekulia XNUMX wale nō i hāʻawi ai ka mea makemakika Kelemania ʻo Richard Dedekind i ka manaʻo o Eudoxus i ka hoʻomohala ʻana e like me nā koi o ka loiloi makemakika koʻikoʻi.
Nui nā helu a hoʻomāinoino paha
Hiki iā ʻoe ke ola me ka helu ʻole? ʻOiai inā, he aha ke ʻano o ke ola ... Pono mākou e hele i ka hale kūʻai e kūʻai i nā kāmaʻa me kahi lāʻau, a mākou i ana mua ai i ka lōʻihi o ka wāwae. “Makemake au i ka ʻāpala, e, eia nō!” - e hōʻike mākou i nā mea kūʻai aku ma ka mākeke. "Pehea ka mamao mai Modlin a i Nowy Dwór Mazowiecki"? “Kokoke loa!”
Hoʻohana ʻia nā helu e ana. Hoʻohana pū mākou iā lākou e hōʻike i nā manaʻo ʻē aʻe. No ka laʻana, hōʻike ka pālākiō o ka palapala ʻāina i ka emi ʻana o ka ʻāpana o ka ʻāina. ʻO ka pālākiō ʻelua a hoʻokahi, a i ʻole 2 wale nō, e hōʻike ana i ka ʻoiaʻiʻo ua pālua ʻia kekahi mea. E ʻōlelo kākou me ka makemakika: pili kēlā me kēia homogeneity i kahi helu - kona unahi.
ʻĀina. Hana mākou i kope xerographic, hoʻonui i ke kiʻi i nā manawa he nui. A laila ua hoʻonui hou ʻia ka ʻāpana i hoʻonui ʻia i nā manawa b. He aha ka unahi hoʻonui nui? Pane: a × b i hoonuiia me ka b. Pono e hoonui i keia mau unahi. ʻO ka helu hoʻemi i hoʻokahi, -1, pili i hoʻokahi kikoʻī i waena, ʻo ia hoʻi, ka hoʻololi ʻana o 180 degere. He aha ka helu e pili ana i ka hoʻololi 90 degere? ʻAʻohe helu like. ʻO ia, ʻo ia ... a i ʻole, e hiki koke mai ana. Mākaukau ʻoe no ka hoʻomāinoino noʻonoʻo? E koa a lawe i ke kumu huinahalike o ka hapa hookahi. Ke hoʻolohe nei au iā? He aha kāu hiki ʻole ke hana? Ma hope o nā mea a pau, ua ʻōlelo wau iā ʻoe e koa. Huki i waho! ʻEā, e huki, huki... E kōkua au... Eia: −1 I kēia manawa ua loaʻa iā mākou, e hoʻāʻo kākou e hoʻohana ia... ʻOiaʻiʻo, hiki iā mākou ke lawe i nā aʻa o nā helu maikaʻi ʻole, no ka mea laʻana.:
√-4 = 2√-1, √-16 = 4√-1
- "me ka nānā ʻole i ka ʻeha noʻonoʻo e pili ana i kēia." ʻO kēia ka mea a Girolamo Cardano i kākau ai i ka makahiki 1539, e hoʻāʻo ana e lanakila i nā pilikia noʻonoʻo e pili ana - ʻoiai ua kapa ʻia ʻo ia - nui noonoo. Ua like kona manao...
...ʻĀina. E puunaue i 10 i elua apana, ua like ka hua me ka 40. Ke hoomanao nei au mai ka moolelo mua aku, ua kakau oia i kekahi mea penei: Obviously impossible. Eia naʻe, e hana kākou i kēia: e puʻunaue i ka 10 i ʻelua mau ʻāpana like, ua like kēlā me kēia me ka 5. E hoʻonui iā lākou - loaʻa iā mākou he 25. Mai ka hopena 25, e unuhi i kēia manawa i 40, inā makemake ʻoe, a loaʻa iā mākou -15. E nānā: √-15 i hoʻohui ʻia a unuhi ʻia mai ka 5 e hāʻawi iā ʻoe i ka hua o 40. ʻO ia mau helu he 5-√-15 a me 5 + √-15. Ua hōʻoia ʻia ka hopena e Cardano penei:
“Me ka ʻole o ka ʻeha o ka noʻonoʻo, e hoʻonui i ka 5 + √-15 me ka 5-√-15. Loaʻa iā mākou he 25 – (-15), ua like ia me 25 + 15. No laila, ʻo ka hua he 40…. He paʻakikī loa."
ʻAe, ehia ia: (1 + √-1) (1-√-1)? E hoonui kakou. E hoʻomanaʻo ʻo √-1 × √-1 = -1. Nui. I kēia manawa he pilikia paʻakikī: mai a + b√-1 a i ab√-1. He aha i hana ai? ʻOiaʻiʻo, penei: (a + b√-1) (ab√-1) = a2+b2
He aha ka mea hoihoi loa i kēia? No ka laʻana, hiki iā mākou ke helu i nā ʻōlelo a mākou i "ʻike ʻole ma mua." Hōʻuluʻulu hoʻonui pōkole no2-b2 hoʻomanaʻo paha ʻoe i ke kumu hoʻohālike no2+b2 ʻaʻole hiki ke hiki mai. Ma ke kahua o nā helu maoli, ka polynomial2+b2 ʻaʻole hiki ke pale ʻia kēia. E hōʻike kākou i ke kumu huinahalike o ka huaʻōlelo i.2= -1. ʻO kēia ka helu kumu "ʻaʻole maoli". A ʻo ia ka mea e wehewehe ana i kahi mokulele e huli ana i 90 degere. No ke aha mai? Ma hope o nā mea āpau,2= -1, a me ka hui ʻana i hoʻokahi 90-degere kahuli ʻana me kekahi ʻano hoʻololi ʻē aʻe e hoʻopuka i kahi 180-degere kahuli. He aha ke ʻano o ka hoʻololi ʻana i wehewehe ʻia? Ua maopopo - he 45 degere huli. He aha ke ʻano o ka helu -i? ʻOi aku ka paʻakikī:
(-I)2 = -i × (-i) = + i2 = -1
No laila -i kekahi wehewehe ʻana i ka 90 degere ka hoʻololi ʻana, ma ka ʻaoʻao ʻē aʻe i ka huli ʻana o i. ʻO wai ka hema a ʻo wai ka ʻākau? Pono ʻoe e hoʻonohonoho. Manaʻo mākou ʻo ka helu i kuhikuhi i ka huli ʻana ma ke ala i manaʻo ʻia e ka poʻe makemakika maikaʻi: counterclockwise. Hōʻike ka helu -i i ka huli ʻana ma ke ala e neʻe ai nā kuhikuhi.
Akā, aia nā helu e like me i a me -i? Are! Hāʻawi wale mākou iā lākou i ke ola. Ke hoʻolohe nei au iā? Aia wale lākou i loko o ko mākou mau poʻo? He aha ka mea e kali ai? Aia nā helu ʻē aʻe a pau i loko o ko mākou manaʻo wale nō. Pono mākou e ʻike inā e ola kā mākou helu hānau hou. ʻOi aku ka pololei, kūpono ka hoʻolālā a pono lākou no kekahi mea? E ʻoluʻolu e lawe i kaʻu ʻōlelo no ka mea ua maikaʻi nā mea āpau a he kōkua maoli kēia mau helu hou. ʻO nā helu e like me 3+i, 5-7i, ma ke ʻano laulā: ua kapa ʻia ʻo a+bi i nā helu paʻakikī. Ua hōʻike au iā ʻoe pehea e loaʻa ai iā lākou ma ka hoʻohuli ʻana i ka mokulele. Hiki iā lākou ke hoʻokomo i nā ʻano like ʻole: ma ke ʻano he mau kiko o ka mokulele, e like me kekahi mau polynomials, e like me kekahi mau pūʻulu helu ... a i kēlā me kēia manawa ua like lākou: equation x2 +1=0 ʻaʻohe mea... hocus pocus ua loaʻa!!!! E hauʻoli kākou a hauʻoli!!!
Pau ka huakaʻi
Hoʻopau kēia i kā mākou huakaʻi mua i ka ʻāina o nā helu hoʻopunipuni. ʻO nā helu unearthly ʻē aʻe, e haʻi pū wau i nā huahelu he nui loa i mua a ʻaʻole ma hope (ua kapa ʻia lākou he 10-adic, no mākou he mea nui ka p-adic, kahi p he helu prime), no ka laʻana X = ... ... ... 96109004106619977392256259918212890625
E helu kāua iā X2. No ka mea? He aha inā mākou e helu i ka huinahā o kahi helu i loaʻa ka helu pau ʻole o nā huahelu ma hope o ia? ʻAe, e hana like kāua. E ʻike kākou i kēlā X2 = H.
E ʻimi kākou i kekahi helu like ʻole me ka helu pau ʻole o nā huahelu i mua e hoʻokō i ka hoohalike. 'Ōlelo Aʻo: ʻO ka huinahā o ka helu e pau ana i ʻeono e pau ana i ʻeono. Ua pau ka huinahalike o kekahi huina i 76 i ka 76. Ua pau hoi ka huinahalike o ka huina i 376 i 376. Ua pau hoi ka huinahalike o ka huina i 9376 i 9376. Ua pau hoi ka huinahalike o ka huina i XNUMX... Aia kekahi mau helu liʻiliʻi, ʻoiai he maikaʻi lākou, ʻoi aku ka liʻiliʻi ma mua o nā helu maikaʻi ʻē aʻe. He liʻiliʻi loa lākou i kekahi manawa ua lawa ia i ka huinahā like e loaʻa ai ka zero. Aia nā helu i kūpono ʻole i ke kūlana a × b = b × a. He mau helu pau ole. Ehia mau helu maoli? He nui loa? ʻAe, akā pehea ka nui? He aha ka helu e hiki ke hōʻike i kēia? Pane: ka liʻiliʻi loa o nā helu palena ʻole; ua kaha ʻia me kahi leka maikaʻi: A a hoʻohui ʻia me kahi helu helu A0 , ʻaleph-zero.
Aia kekahi mau helu ʻaʻole mākou i ʻike i ke ola ... a i ʻole hiki iā mākou ke manaʻoʻiʻo a ʻaʻole manaʻoʻiʻo e like me kou makemake. A e kamaʻilio ana: Manaʻo wau makemake ʻoe i nā helu Unreal, Fantasy Species Numbers.
