He aha mea he maikaʻi helu? History, laulā, ano

Makemakika hookaawale, mai ka mau akeakamai e pili ana i ke ono o ka kenekulia BC. e ka., a mai ia manawa, hoomaka ia kona lanakila hele ana a puni ka honua. Kēlā me kēia kahua o ka? Acaeoey lawe mai i kekahi mea hou - he kula haʻahaʻaa mooolelo o Hawaiʻi, lilo iho i loko o ka palena pau a me ka integral calculus, ano 'ē aʻeo ke kenekulia, ka haʻilula lilo hou i hookahuli ai, a hiki mai i ka manawa o "ka hoʻomaka o ka loa pilikia makemakika -. Ia nalowale mai a pau na helu" Akā, mea e waiho ana ma ke kua?

Ka wahi ho omaka

Nā helu maoli ua ma luna o ka par me ka mea mua makemakika hana. Once hoʻi,ʻelua hoʻi,ʻekolu spine ... Ua ikea mai ke aloha i ka Indian nä känaka 'epekema, ka poe i lawe mua i ka positional helu nenoaiu. Ka hua'ōlelo "positional" 'o ia hoʻi i ka wahi o kēlā me kēia huahelu i loko o ka helu o ka pololei i ho'ākāka' a me ka hoʻopili like me kona waeʻano. No ka laʻana, na helu 784 a me 487 - nā helu i ka ia, akā, na helu e i ka ia me ka mea mua mai nā '7 haneri, akā, i ka lua o - 4. wale HOU Indians lalau aku la ia i ka Arabia, ka poe i lawe mai i ka helu o ka laha' ana, ua ike Now.

I ka wā kahiko, i ka helu i pili mystical manaʻo nui, i ka mea nui mathematician Pythagoras manaoio i ka helu mea ma ka naau o ka honua ma luna o ka par me ke kumu o hehee - ke ahi, ka wai, ka honua, ea. Inā mākou e noonoo a pau wale me kaʻaoʻao makemakika, a laila, i ka mea he maikaʻi helu? Ke kahua o maoli huahelu ua denoted like N a mea he mana loa moʻo o ka helu i ka mea maikaʻi a me ka integers 1, 2, 3, ... + ∞. 'Aʻohe ua hoʻokoe'. ʻano nui hoʻohana no ka helu i nā ikamu a me ka hōʻike 'i ke kauoha.

He aha mea he helu maoli i ka makemakika? axioms o Peano

Field N o ka waihona ipu ma iʻanuʻu kula haʻahaʻaa makemakika. Over manawa, ke kaʻawale kahua integers, ka naauao nui, luna 'helu.

I ka hana o ka Italia mathematician Dzhuzeppe Peano i hiki i ka hou structuring o ka makemakika, ua hoʻolilo ihola iā ia i nā formalities, a hoomakaukau iho la i ka lepo no ka hou hopena a e hele ma kela aoao o ke kahua māhele N. He aha ka mea he helu maoli, ka mea, ua loaʻa mua ma alodio 'ōlelo, ka eia e e noʻonoʻo i ke kumu o ka makemakika ka wehewehena o ka Peano axioms.

• Unit Ua manaʻo like me ka helu maoli.
• Ka helu i hahai i ka helu maoli, o ke ano maoli.
• Ma mua o ka mokuna mea i helu maoli.
• Inā ka helu b. Pono e nā ka helu pela aku, a me ka helu o ka n, alaila, e = d.
• Ke axiom o ka hoʻokomo, a ma ka huli paipai i kekahi maoli helu, ina he olelo e hilinaʻi nui ma luna o ka aiao He oiaio no i ka helu 1, laila, ua kuhi ia mea, hana no N helu o nā mahinaʻai o nā maoli helu N. A laila, i ka assertion He oiaio no N = 1, mai ke kahua mai o maoli helu N.

XIX. Ana no ke kahua o ka maoli helu

Mai ka mahinaai N i ka mua i ka makemakika ho, ia mea me ke kikowaena o ka ho'ākāka 'ana, a me ka wahi ma lalo o ka helu o nā hana Hawaiʻi i ka e hana. Ka mea i paa, aʻaʻohe. I ka papa kuhikuhiE koena unuhi o ia ua hoʻohiki ka hana ua i haʻalele i kekahi paʻa result i loko o ka huapalapala N, nānā 'ole o ka mea nui loa o ka pae'āina. He nui iho la ia i ka mea, e kūlohelohe. I ka hope o ka i koe laulā AaOIeIOUIeXAOAaIN mea,ʻaʻole like straightforward a me ka hilinaʻi nui ma luna o ka mea e ai no ka poe i komo i loko o ka olelo, e like me ka mea e e kue ana i ke kumu o ka wehewehena. Ke'ōlelo mai, no ka hoopaa ana:

• Hou - m + y me = Z, kahi m, y me, Z mea, mai ke kahua N;
• multiplication - m * y me = Z, kahi m, y me, Z mea, mai ke kahua N;
• exponentiation - m ^{y me,} kahi m, y me ka mai N. Field

Ke koe ana, i ka hopena o ka i hiki ole nei i loko o ka manao paa o ka pōʻaiapili "i mea he maoli helu" like penei:

• Subtraction - m - y me = Z. Field maoli helu apono 'ia ina ka hou m y me wale;
• mahele - m / y me = Z. Field maoli helu e leie aku ia wale nō inā ua puunaueia e y me i koena, a laila Z kōā kaulike.

Waiwai o ka nui, no a hiki i ke kula N

All hou makemakika a kuhi e e ma muli o kēia mau waiwai, i ka loa AICIeOEOI, akā,ʻaʻohe emi nui.

• Commutative waiwai o ka hou - m + y me = y me + m, ma ka helu o ka m, y me komo i loko o ka pahu N. A i ka luawai-ike "mai ua ole hoʻololi ka relocation o ka huina."
• Commutative waiwai o ka multiplication - m * y me = y me * m, ma ka m helu, y me ka mai N. Field
• Associative waiwai o ka hou - (m + y me) + Z = m + (y me + Z), kahi m, y me, Z mea mai N. Field
• Associative waiwai o ka multiplication - (m * y me) * Z = m * (y me * Z), kahi o ka m helu, y me, Z mea mai N. Field
• distributive waiwai - m (y me + Z) = m * y me + m * Z, kahi o ka m helu, y me, Z mea mai N. Field

Table o Pythagoras

Kekahi o na anuu, mua i loko o ka ike o na haumana ma ke kula haʻahaʻa makemakika hale a ma hope ka mea, ike no lakou iho ka nui i kapaia ma kona ano maoli, o ka papaʻaina o Pythagoras. Ua hiki ke noonoo ole wale mai i ka wahi o ka Hawaii o ka 'epekema, akā, i like me ka waiwai' epekema kia.

Kēia multiplication papaʻaina i undergone i ka helu o nā loli ma manawa: ka mea, i wehe 'ia mai' Aʻohe, a me ka nui, mai 1 a hiki i 10 ku no lakou iho, Ma waho aʻeo ka kauoha o ka nui (haneri, tausani ...). He He papaʻaina i loko o a me ke kuleana o ka lalani a me kolamu - i ka helu a me Contents o na keena o ke kuʻina, ua like me ka huina hoonui o lākou iho.

Ma ka hana o nooaai? I ka hope mau makahiki ua i ka pono no ka mea Pythagorean papa aʻo ' "ma ka mea",ʻo ia hoʻi, mua hele ma memorization. Multiplication 1 Ua hōʻike pū 'ana, mai ka hopena mea like i 1 a nui ololi. Ia manawa, i loko o ka papaʻaina hiki ke ike i ka mea kapa ole maka kumu hoʻohālike: i ka huina hoonui o ka helu mahuahua ma kekahi anu u, i mea like inoa kui. Ke'ī mai, i ka lua o ka ololi hōʻike mai ana i na manawa e pono e lawe i ka mua, i ka mea e loaa i ka makemake huahana. Kēia nenoaiu Ua like ka oi pono kekahi mea i hana i loko o ka Middle Ages: i ka ike ole i ka mea he maikaʻi helu, a pehea ia mea AICIeOEOI, kanaka e hoʻokele i complicate iho maʻamau ma ka hoʻohana 'ana i ka'ōnaehana i i ka nānā' ana ma luna o nā degere o elua.

A subset like me ka cradle o ka makemakika

Ma ka manawa, i ke kahua o maoli helu N Ua manaoia wale me kekahi o na subsets o ka luna 'helu, akā, ka mea, aole e hana mai ia lakou e emi waiwai ma ka' epekema. Natural helu - ka mua mea i kekahi keiki e ao hou ma këia ia makou iho a me ke ao nei a puni mākou. Once ka manamana lima, elua manamana lima ... Mahalo ia ia, he kanaka i hanaʻia ma ka Logical manaʻo, e like me ka hiki, e hooholo i ke kumu a me ka no o ka auaiaea, ia n oaa i ke ala no ka nui discoveries.