hoʻohihi hoʻohuli
Nui nā kamaʻilio e pili ana i ka "charm of opposites", ʻaʻole wale ma ka makemakika. E hoʻomanaʻo, ʻo nā helu kūʻē, ʻo ia ka ʻokoʻa ma ka hōʻailona wale nō: hoʻohui i ka 7 a me ka hōʻemi ʻana i ka 7. ʻO ka huina o nā helu kūʻē he ʻole. Akā, no mākou (ʻo ia hoʻi. nā mea makemakika) ʻoi aku ka hoihoi o nā pānaʻi. Ina ua like ka huina hoonui o na huina me 1, alaila, ua hoololi keia mau huina i kekahi i kekahi. He kū'ē ko kēlā me kēia helu, ʻo kēlā me kēia helu helu ʻole he ʻano hoʻohuli. ʻO ka pānaʻi o ka pānaʻi ʻo ia ka hua.
Hana ʻia ka hoʻohuli ʻana ma nā wahi ʻelua e pili ana kekahi i kekahi, no laila inā e hoʻonui kekahi, e emi ana kekahi ma kahi ʻano like. "Pelevant" 'o ia ho'i, 'a'ole e loli ka huahana o kēia mau nui. Hoʻomanaʻo mākou mai ke kula: he ʻano hoʻohālikelike kēia. Inā makemake au e hōʻea i koʻu huakaʻi ʻelua ʻoi aku ka wikiwiki (ʻo ia hoʻi, ʻoki i ka manawa i ka hapalua), pono wau e pālua i koʻu wikiwiki. Ina e hoemiia ka leo o ka ipu i sila me ke kinoea i n mau manawa, alaila, e pi'i ke kaomi i na manawa.
Ma ka hoʻonaʻauao kula haʻahaʻa, hoʻokaʻawale mākou i waena o nā hoʻohālikelike like ʻole a pili. "Ehia hou aku"? - "ʻEhia manawa hou aku?"
Eia kekahi mau hana kula.
ʻĀpana 1. ʻO nā waiwai maikaʻi ʻelua, ʻo ka mua he 5 mau manawa nui ma mua o ka lua a ma ka manawa like he 5 mau manawa ʻoi aku ka nui ma mua o ka mua. He aha nā ana?
ʻĀpana 2. Ina he 3 ka nui o kekahi ma mua o ka lua, a he 2 ka lua ma mua o ke kolu, pehea ka nui o ka helu mua ma mua o ke kolu? Inā ʻelua ka lua o ka helu maikaʻi mua, a ʻekolu manawa ka helu mua i ke kolu, ʻehia ka nui o ka helu mua ma mua o ke kolu?
ʻĀpana 3. Ma ka hana 2, ʻae ʻia nā helu kūlohelohe wale nō. Hiki ke hoʻonohonoho ʻia e like me ka mea i wehewehe ʻia ma laila?
ʻĀpana 4. ʻO nā waiwai maikaʻi ʻelua, ʻo ka mua he 5 manawa o ka lua, a ʻo ka lua he 5 manawa ka mua. Hiki paha?
He mea maʻalahi ka manaʻo o ka "average" a i ʻole "average". Inā holo kaʻa au i 55 km i ka Pōʻakahi, 45 km i ka Pōʻalua, a me 80 km i ka Pōʻakolu, ma ka awelika, holo au i 60 km i kēlā me kēia lā. Ua ʻae loa mākou i kēia mau helu ʻana, ʻoiai he mea ʻē aʻe lākou no ka mea ʻaʻole wau i holo i 60 km i ka lā hoʻokahi. E like me ka maʻalahi o kā mākou ʻae ʻana i nā ʻāpana o ke kanaka: inā ʻelua haneli poʻe e kipa i kahi hale ʻaina i loko o ʻeono lā, a laila ʻo ka awelika o kēlā me kēia lā he 33 a me ke kolu o nā kānaka. HM!
Aia nā pilikia me ka nui waena. Makemake au i ka paikikala. No laila ua hoʻohana au i ka hāʻawi ʻana o ka hui huakaʻi "E hele pū me mākou" - hāʻawi lākou i nā ukana i ka hōkele kahi e hele ai ka mea kūʻai aku ma ke kaʻa no nā hana leʻaleʻa. Ma ka Pōʻalima ua holo au no ʻehā mau hola: ʻo ka mua ʻelua ma ka wikiwiki o 24 km i kēlā me kēia hola. A laila ua luhi au no nā ʻelua aʻe e hana wale ana au i 16 i hoʻokahi hola. He aha koʻu māmā māmā? ʻOiaʻiʻo (24+16)/2=20km=20km/h.
I ka Poaono nae, ua waihoia na ukana ma ka hotele, a ua hele au e nana i na wahi i koe o ka hale kakela, he 24 km ka mamao, a ike aku au, hoi aku la au. Hoʻokahi hola au i kaʻa ai ma kahi ʻaoʻao, hoʻi mālie i hope, ma ka wikiwiki o 16 km i kēlā me kēia hola. He aha koʻu māmā māmā ma ke ala hōkele-kakela-hotel? 20 km i ka hola? ʻAʻole naʻe. Ma hope o nā mea a pau, ua holo au i ka huina o 48 km a ua lawe iaʻu i hoʻokahi hola ("ma laila") a hoʻokahi hola a me ka hapa i hope. 48 km i loko o ʻelua hola a me ka hapa, ʻo ia hoʻi. hola 48/2,5=192/10=19,2 km! Ma kēia kūlana, ʻaʻole ʻo ka awelika wikiwiki ka helu helu, akā ʻo ka harmonic o nā waiwai i hāʻawi ʻia:
a hiki ke heluhelu ʻia kēia ʻano papa ʻelua penei: ʻo ka mean harmonic o nā helu maikaʻi ʻo ia ka pānaʻi o ka mean helu o ko lākou pānaʻi. Ka reciprocal o ka huina o ka inverses ikeia ma na choruses he nui kula haʻawina: ina kekahi limahana e eli hola, ka 'ē aʻe - b hola, a laila, hana pu, eli lakou i ka manawa. loko wai (hoʻokahi i kēlā me kēia hola, ʻo kekahi i nā hola b). Inā loaʻa iā R1 kekahi pale a ʻo R2 kekahi, a laila loaʻa iā lākou ke kūʻē like.
Inā hiki i hoʻokahi kamepiula ke hoʻoponopono i kahi pilikia i kekona, ʻo kekahi kamepiula i b kekona, a laila ke hana pū lākou...
Kū! ʻO kēia kahi e pau ai ka hoʻohālikelike, no ka mea, pili nā mea a pau i ka wikiwiki o ka pūnaewele: ka pono o nā pilina. Hiki i nā limahana ke keʻakeʻa a kōkua kekahi i kekahi. Inā hiki i hoʻokahi kanaka ke ʻeli i ka luawai i loko o ʻewalu hola, hiki i nā limahana he kanawalu ke hana i loko o ka 1/10 o ka hola (a i ʻole 6 mau minuke)? Inā lawe ʻeono mau mea lawe i ka piano i ka papahele mua i loko o 6 mau minuke, pehea ka lōʻihi o ka lawe ʻana o kekahi o lākou i ka piano i ka papahele kanaono? ʻO ka lapuwale o ia mau pilikia e hoʻomanaʻo i ka palena palena o nā makemakika āpau i nā pilikia "mai ke ola".
E pili ana i kahi mea kūʻai ikaika
ʻAʻole hoʻohana hou ʻia nā unahi. E hoʻomanaʻo kākou, ua kau ʻia ke kaumaha ma luna o kekahi pola o ia mau unahi, ua kau ʻia ka ukana ma luna o kekahi, a i ke kaupaona ʻana, ua like ke kaumaha me ke kaumaha. ʻOiaʻiʻo, pono ka lōʻihi o nā lima ʻelua o ke kaumaha, i ʻole e hewa ke kaupaona ʻana.
ʻAe pololei. E noʻonoʻo i kahi mea kūʻai aku i ke kaumaha me ka leverage like ʻole. Eia naʻe, makemakeʻo ia eʻoiaʻiʻo me nā mea kūʻai a kaumaha i nā waiwai iʻelua mau pahu. ʻO ka mua, kau ʻo ia i ke kaumaha ma luna o kekahi pā, a ma kekahi ʻē aʻe i ka nui o nā waiwai - i kaulike nā unahi. A laila kaupaona ʻo ia i ka lua o ka "hapalua" o nā waiwai ma ke ʻano hoʻohuli, ʻo ia hoʻi, kau ʻo ia i ke kaumaha ma ka lua o ke kīʻaha, a me ka waiwai ma ka mua. No ka like 'ole o na lima, 'a'ole i like ka "hapa". A maʻemaʻe ka lunamanaʻo o ka mea kūʻai aku, a hoʻomaikaʻi ka poʻe kūʻai aku i kāna ʻoiaʻiʻo: "ʻO ka mea aʻu i wehe ai ma ʻaneʻi, ua hoʻohui wau."
Eia naʻe, e nānā pono kākou i ke ʻano o ka mea kūʻai aku e makemake ana e ʻoiaʻiʻo me ke kaumaha ʻole. E loaʻa i nā lima o ke kaulike ka loa a me ka b. Ina ua hooiliia kekahi o na kiaha i ka paona kilo, a ua hooiliia kekahi i ka x waiwai, alaila ua kaulike na unahi ina ax = b i ka manawa mua a me bx = a i ka lua o ka manawa. No laila, ua like ka hapa mua o ka huahana me ka b/a kilokani, ua like ka hapa lua me ka a/b. ʻO ke kaumaha maikaʻi he a = b, ʻo ia hoʻi e loaʻa i ka mea kūʻai aku 2 kg o nā waiwai. E ʻike kākou i ka hopena inā he ≠ b. A laila, a - b ≠ 0 a mai ka hoʻonui hoʻonui i hoʻopau ʻia iā mākou
Ua hiki mai mākou i kahi hopena i manaʻo ʻole ʻia: ʻo ke ʻano kūpono o ka "averaging" i ke ana i kēia hihia e hana i ka pono o ka mea kūʻai aku, ka mea i loaʻa i nā waiwai hou aʻe.
5 kalepa. (He mea nui, ʻaʻole loa ma ka makemakika!). ʻO ke kaumaha o ka makika he 2,5 milligrams, a ʻo ka ʻelepani ʻelima mau tona (he ʻikepili pololei kēia). E helu i ka mean helu, mean geometric, a me mean harmonic o ka nui o ka makika a me ka Elepani (kaumaha). E nānā i ka helu ʻana a ʻike inā he manaʻo ko lākou ʻokoʻa o nā haʻawina helu. E nānā kākou i nā hiʻohiʻona ʻē aʻe o nā helu makemakika i kūpono ʻole i ka "ola maoli". Manaʻo kōkua: Ua nānā mua mākou i kekahi laʻana ma kēia ʻatikala. 'O ia ho'i, ua pololei kekahi haumāna inoa 'ole nona ka mana'o a'u i 'ike ma ka Pūnaewele: "Ho'opunipuni ka makemakika i ka po'e me nā helu"?
ʻAe, ʻae wau i ka nui o ka makemakika hiki iā ʻoe ke "hoʻopunipuni" i ka poʻe - ʻōlelo kēlā me kēia hoʻolaha shampoo lua e hoʻonui ai i ka frizz ma kekahi pakeneka. E ʻimi anei mākou i nā hiʻohiʻona hou aku o nā mea hana pono o kēlā me kēia lā i hiki ke hoʻohana ʻia no ka hana hewa?
Grams!
ʻO ke poʻo inoa o kēia paukū he huaʻōlelo (kanaka mua plural) ʻaʻole he noun (nominative plural o hoʻokahi tausani o ka kilokani). Hōʻike ka Harmony i ka hoʻonohonoho a me ke mele. No ka poʻe Helene kahiko, he lālā ʻepekema ke mele - pono e ʻae ʻia inā e ʻōlelo mākou pēlā, hoʻololi mākou i ke ʻano o kēia manawa o ka huaʻōlelo "ʻepekema" i ka manawa ma mua o ko mākou au. Noho ʻo Pythagoras i ka makahiki XNUMX BC ʻAʻole wale ʻo ia i ʻike i kahi kamepiula, kelepona kelepona a me ka leka uila, akā ʻaʻole ʻo ia i ʻike ʻo wai ʻo Robert Lewandowski, Mieszko I, Charlemagne a me Cicero. ʻAʻole ʻo ia i ʻike i nā helu ʻAlapia a i ʻole Roma (ua hoʻohana ʻia lākou ma kahi o ke kenekulia XNUMX BC), ʻaʻole ʻo ia i ʻike i ke ʻano o ka Punic Wars ... Akā ʻike ʻo ia i ke mele ...
Ua ʻike ʻo ia ma nā mea kani i hoʻohālikelike ʻia nā coefficients o ka haʻalulu i ka lōʻihi o nā ʻāpana haʻalulu o nā kaula. ʻIke ʻo ia, ʻike ʻo ia, ʻaʻole hiki iā ia ke hōʻike i ke ʻano o kā mākou hana ʻana i kēia lā.
ʻO nā alapine o nā haʻalulu kaula ʻelua i hana i ka octave aia ma ka ratio 1:2, ʻo ia hoʻi, ʻo ke alapine o ka memo kiʻekiʻe he pālua ia i ka alapine o ka memo haʻahaʻa. ʻO ka lākiō haʻalulu kūpono no ka ʻelima he 2:3, ʻehā ʻo 3:4, ʻekolu kolu nui maʻemaʻe ʻo 4:5, ʻekolu kolu liʻiliʻi he 5:6. He mau manawa consonant maikaʻi kēia. A laila aia ʻelua mau mea kūʻokoʻa, me ka vibration ratios o 6:7 a me 7:8, a laila dissonant - he leo nui (8:9), kahi leo liʻiliʻi (9:10). Ua like kēia mau hakina (ratio) me nā lakio o nā huaʻōlelo ʻē aʻe o kahi kaʻina, i kapa ʻia e ka poʻe makemakika (no kēia kumu) he pūʻulu harmonic:
- ma ka manaʻo he nui palena ʻole. Hiki ke kākau ʻia ka ratio o nā haʻalulu octave ma ke ʻano he 2:4 a kau i ka hapalima ma waena o lākou: 2:3:4, ʻo ia hoʻi, puʻunaue mākou i ka octave i ka hapalima a me ka hā. Ua kapa ʻia kēia mahele ʻāpana harmonic ma ka makemakika:
Laiki. 1. No ka mea hookani pila: e puunaue ana i ka octave AB i ka elima AC.No ka Mathematician: Harmonic Segmentation
He aha kaʻu e ʻōlelo ai ke kamaʻilio wau (ma luna) o kahi huina ʻike ʻole, e like me ka hui harmonic? ʻIke ʻia e hiki i kēlā ʻano helu ke helu nui, ʻo ka mea nui e hoʻohui mākou no ka manawa lōʻihi. He liʻiliʻi a liʻiliʻi nā meaʻai, akā ʻoi aku ka nui o lākou. He aha ka lanakila? Ma ʻaneʻi mākou e komo ai i ke ʻano o ka nānā ʻana i ka makemakika. ʻIke ʻia ua pau nā mea ʻai, akā ʻaʻole wikiwiki loa. E hōʻike wau ma ka lawe ʻana i nā meaʻai lawa, hiki iaʻu ke hōʻuluʻulu:
nui wale. E lawe kākou i "no ka laʻana" n = 1024. E hui kākou i nā huaʻōlelo e like me ka mea i hōʻike ʻia ma ke kiʻi:
Ma kēlā me kēia pale, ʻoi aku ka nui o kēlā me kēia huaʻōlelo ma mua o ka mea ma mua, koe naʻe, ʻoiaʻiʻo, ʻo ka hope, ʻo ia ka mea like me ia iho. I loko o nā pūʻulu ma lalo nei, loaʻa iā mākou nā ʻāpana 2, 4, 8, 16, 32, 64, 128 a me 512; ʻoi aku ka nui o ka waiwai o ka huina ma kēlā me kēia pani ma mua o ka ½. ʻOi aku kēia ma mua o 5½. E hōʻike ana nā helu pololei ʻoi aku ka nui o kēia huina ma kahi o 7,50918. ʻAʻole nui, akā i nā manawa a pau, a hiki iā ʻoe ke ʻike i ka lawe ʻana i nā mea nui, hiki iaʻu ke hoʻokō i kekahi helu. He lohi loa kēia (no ka laʻana, ʻo mākou ma luna o ʻumi me nā meaʻai wale nō), akā ʻo ka ulu palena ʻole ka mea hoihoi i ka poʻe makemakika.
Huakai i ka pau ʻole me ka pūʻulu harmonic
Eia kahi puʻupuʻu i kekahi makemakika koʻikoʻi. Loaʻa iā mākou kahi lako palena ʻole o nā poloka rectangular (he aha kaʻu e ʻōlelo ai, rectangular!) me nā ana, e ʻōlelo, 4 × 2 × 1. E noʻonoʻo i kahi ʻōnaehana i loaʻa i nā mea he nui (ma fig. 2 - ʻehā) mau poloka i ʻike ʻia ka mua i ka ½ o kona lōʻihi, ʻo ka lua mai luna mai ka ¼ a pēlā aku, ʻo ke kolu me ka hapaono. ʻAe, i mea e paʻa maoli ai, e hoʻoemi iki i ka pōhaku mua. No ka helu ʻana, ʻaʻole pili kēia.
Laiki. 2. Hoʻoholo i ke kikowaena o ka ʻumekaumaha
He mea maʻalahi nō hoʻi ke hoʻomaopopo ʻana ʻoiai ʻo ke kiʻi i haku ʻia o nā poloka mua ʻelua (helu ʻia mai luna mai) he kikowaena o ka symmetry ma ke kiko B, a laila ʻo B ke kikowaena o ka umekaumaha. E wehewehe pono kākou i ke kikowaena o ka umekaumaha o ka ʻōnaehana, i haku ʻia me nā poloka luna ʻekolu. Ua lawa ka hoʻopaʻapaʻa maʻalahi maʻaneʻi. E mahele noʻonoʻo kākou i ka haku mele ʻekolu i ʻelua o luna a me ke kolu o lalo. Pono kēia kikowaena ma ka ʻāpana e hoʻopili ana i nā kikowaena o ka ʻumekaumaha o nā ʻāpana ʻelua. I ka manawa hea o kēia ʻāpana?
ʻElua ala e koho ai. Ma ka mua, e hoʻohana mākou i ka nānā ʻana e moe kēia kikowaena i waenakonu o kahi pyramid ʻekolu-pākaʻi, ʻo ia hoʻi, ma kahi laina pololei e ʻoki ana i ka lua, waena waena. Ma ka ʻaoʻao ʻelua, hoʻomaopopo mākou ʻoiai ʻo ka nui o nā poloka luna ʻelua he ʻelua ka nui o ka nui o ka pahu hoʻokahi #3 (luna), ʻo ke kikowaena o ka ʻumekaumaha ma kēia ʻāpana he ʻelua ʻoi aku ka pili ʻana iā B ma mua o ke kikowaena. S o ka poloka ekolu. Pēlā nō, ʻike mākou i ka helu aʻe: hoʻohui mākou i ke kikowaena i ʻike ʻia o nā poloka ʻekolu me ke kikowaena S o ka poloka ʻehā. Aia ke kikowaena o ka ʻōnaehana holoʻokoʻa ma ke kiʻekiʻe 2 a ma ke kiko e puunaue ana i ka ʻāpana me 1 a 3 (ʻo ia hoʻi, ma ka ¾ o kona lōʻihi).
ʻO ka helu ʻana a mākou e hana hou aku ai e alakaʻi i ka hopena i hōʻike ʻia ma Fig. fiku 3. Wehe ʻia nā kikowaena o ka ʻumekaumaha mai ka ʻaoʻao ʻākau o ka poloka haʻahaʻa e:
No laila, aia mau ke kuhi o ke kikowaena o ka umekaumaha o ka pyramid i loko o ke kumu. ʻAʻole e hiolo ka hale kiaʻi. I kēia manawa e nānā kākou fig. 3 a no kekahi manawa, e hoʻohana kākou i ka ʻāpana ʻelima mai luna mai i kumu (ʻo ka mea i kaha ʻia me ke kala ʻoi aku ka mālamalama). Kūlana kiʻekiʻe:
no laila, ʻoi aku ka 1 o kona ʻaoʻao hema ma mua o ka ʻaoʻao ʻākau o ke kumu. Eia ke kowali a'e:
He aha ka kowali nui loa? Ua ʻike mua mākou! ʻAʻohe mea nui loa! ʻO ka lawe ʻana i nā poloka liʻiliʻi loa, hiki iā ʻoe ke loaʻa i kahi overhang o hoʻokahi kilomita - akā naʻe, ma ka makemakika wale nō: ʻaʻole lawa ka honua holoʻokoʻa e kūkulu i nā poloka he nui!
Laiki. 3. Hoʻohui hou i nā poloka
I kēia manawa nā helu a mākou i waiho ai ma luna. E helu mākou i nā mamao āpau "horizontally" ma ka x-axis, no ka mea, ʻo ia wale nō. ʻO ka helu A (ke kikowaena o ka ʻumekaumaha o ka poloka mua) he 1/2 mai ka ʻaoʻao ʻākau. ʻO ka helu B (ke kikowaena o nā ʻōnaehana poloka ʻelua) he 1/4 ka mamao mai ka ʻaoʻao ʻākau o ka poloka ʻelua. E hoʻomaka i ka hopena o ka ʻāpana ʻelua (e neʻe mākou i ke kolu). No ka laʻana, aia ma hea ke kikowaena o ka umekaumaha o ka poloka #3? ʻO ka hapalua o ka lōʻihi o kēia poloka, no laila, ʻo 1/2 + 1/4 = 3/4 mai ko mākou wahi kuhikuhi. Aia ma hea ka lae C? I loko o ʻelua hapakolu o ka māhele ma waena o 3/4 a me 1/4, ʻo ia hoʻi ma ke kiko ma mua, hoʻololi mākou i ka helu kuhikuhi i ka ʻaoʻao ʻākau o ke kolu o ka poloka. Ua wehe ʻia ke kikowaena o ka ʻumekaumaha o ka ʻōnaehana ʻekolu poloka mai ka helu kuhikuhi hou, a pēlā aku. Ke kikowaena o ka umekaumaha Cn ʻO kahi hale kiaʻi i haku ʻia me nā poloka n he 1/2n ka mamao mai ke kiko kuhikuhi koke, ʻo ia ka ʻaoʻao ʻākau o ka poloka kumu, ʻo ia hoʻi ka poloka nth mai luna mai.
Ma muli o ka ʻokoʻa ʻana o nā ʻano like ʻole, hiki iā mākou ke loaʻa i kekahi ʻano nui. Hiki paha ke hoʻokō ʻia kēia? Ua like ia me ka hale kiaʻi pōhaku pau ʻole - e hāʻule ana ma lalo o kona kaumaha ponoʻī. Ma kā mākou papahana, ʻo ka liʻiliʻi o nā hemahema o ka hoʻokomo ʻana i ka poloka (a me ka piʻi mālie o nā huina hapa o ka moʻo) ʻo ia ka mea ʻaʻole mākou e hele mamao loa.
