Ngā Tau Uaua
He ariā pāngarau ngā tau matatini e whai wāhi nui ana ki ngā momo peka pūtaiao, pērā i te ahupūngao, te miihini, te ōhanga, me te pāngarau tonu. Hei toronga atu i ngā tau tūturu e mōhiotia ana e tātou i ia rā, ka whakauruhia mai e ngā tau matatini he taha hou ki te āhua o tā tātou mārama me te whakatauira i ngā āhuatanga rerekē.
Hītori o ngā Tau Matatini
I puta tuatahi mai ngā tau matatini i te hiahia ki te kimi otinga mō ngā whārite tapawhā kāore he otinga i roto i ngā tau tūturu. Mai anō i ngā wā onamata, kua pāngia ngā tohunga pāngarau e ngā raruraru pēnei i te whārite tapawhā \(x^2 + 1 = 0\), kāore he pūtake tūturu. Nā te mea mō ia tau tūturu \(x\), kāore a \(x^2\) i te kino, nō reira kāore e taea e \(x^2 + 1\) te kore.
I tīmata te whanaketanga o te māramatanga hohonu ake mō ngā tau matatini i te rautau 16 nā te mahi a ngā tohunga pāngarau Pākehā pērā i a Girolamo Cardano, nāna i whakamahi ngā pūtake whakaaro i roto i te otinga o ētahi whārite. I ngā rautau 18 me te 19, i whakawhanakehia e ngā tohunga pāngarau pērā i a Leonhard Euler rāua ko Carl Friedrich Gauss ngā turanga o te ariā tau matatini, me te whakarato i tētahi whakamārama pūnaha ake, me te whakauru i te nuinga o ngā tohu e whakamahia tonutia ana i ēnei rā.
Ngā Whakamāramatanga me ngā Tuhituhinga
E rua ngā wāhanga o te tau matatini: he wāhanga tūturu me he wāhanga pohewa. I te nuinga o te wā, ka taea te tuhi i te tau matatini ki te āhua \(a + bi\), arā:
– Ko te wāhanga tūturu ko \(a\).
– Ko te wāhanga pohewa ko \(b\).
– Ko te \(i\) te waeine pohewa, e tautuhia ana ko \(\sqrt{-1}\).
Hei tauira, i roto i te tau matatini \(4 + 3i\):
– Ko te wāhanga tūturu ko \(4\).
– Ko te wāhanga pohewa ko \(3i\).
Ko te taahiraa tuatahi ki te mārama ki ngā tau matatini ko te whakaae he āhuatanga tino whakamere tō \(i\): \(i^2 = -1\).
Ngā Mahi Taketake mō ngā Tau Matatini
Pērā i ngā tau tūturu, ka taea e tātou te mahi i ngā mahi taketake mō ngā tau matatini, pērā i te tāpiri, te tango, te whakarea, me te wehewehe.
Tāpiri me te Tango
Hei tāpiri i ngā tau matatini e rua, ka tāpirihia noa ō rāua wāhanga tūturu me ō rāua wāhanga pohewa. Hei tauira, mō ngā tau matatini e rua \(z_1 = a + bi\) me \(z_2 = c + di\):
\[ z_1 + z_2 = (a + c) + (b + d)i \]
Ka rite te mahi tango, arā, te tango i te wāhanga tūturu me te wāhanga pohewa:
\[ z_1 – z_2 = (a – c) – (b – d)i \]
Te Whakarea
He uaua ake te whakarea i ngā tau matatini, nā te mea me whakarea ngā wāhanga tūturu me ngā wāhanga pohewa, me te whakaaro hoki ki ngā āhuatanga o \(i\). Mō ngā tau matatini e rua \(z_1 = a + bi\) me \(z_2 = c + di\):
\[ z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
Kia maumahara ko \(i^2 = -1\), kia taea ai e tātou te whakangawari ake ki:
\[ z_1 \cdot z_2 = (ac – bd) + (ad + bc)i \]
Tohatoha
Hei wehewehe i ngā tau matatini e rua, ka whakamahia e mātou te ariā o ngā hononga. Ko te hononga o tētahi tau matatini \(a + bi\) ko \(a – bi\). Me kī, me wehewehe e mātou \(z_1 = a + bi\) mā \(z_2 = c + di\):
\[ \frac{z_1}{z_2} = \frac{a + bi}{c + di} \]
Hei whakangawari, ka whakareatia te taupū me te tauwehe ki te hononga o te tauwehe:
\[ \frac{z_1}{z_2} = \frac{(a + bi)(c – di)}{(c + di)(c – di)} = \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} \]
Whakaaturanga Āhuahanga
Ka taea hoki te whakaatu i ngā tau matatini mā te āhua ā-ira i roto i te papa matatini, ko te tuaka whakapae te tohu mō te wāhanga tūturu, ko te tuaka poutū te tohu mō te wāhanga pohewa. He rite tēnei ki te pūnaha taunga Cartesian e whakamahia whānuitia ana i roto i te āhua ā-ira.
He whakamārama anō hoki ngā koki me ngā roa o tēnei whakaaturanga. Ko te roa, te modulus rānei o tētahi tau matatini \(z = a + bi\) ko te tawhiti mai i taua pūwāhi ki te pūtake (0,0), ā, ka taea te tatau mā te whakamahi i te ariā Pythagorean:
\[ |z| = \sqrt{a^2 + b^2} \]
I taua wā, ko te koki, ko te tautohe rānei o tētahi tau matatini ko te koki i hangaia e te rārangi e hono ana i te pūwāhi ki te pūtake me te tuaka tūturu pai, e whakaatuhia ana i roto i ngā rātiana.
Ngā Taupānga Tau Matatini
He whānuitia ngā tono mahi a ngā tau matatini, mai i te hangarau ki te ahupūngao matū. Ko ētahi tauira o ngā tono o ngā tau matatini ko:
Te Hangarau Hiko me te Hikohiko
I roto i te tātaritanga ara iahiko AC (Au Hurihuri), ka whakamahia ngā tau matatini hei tohu i te aukati, te ngaohiko, me te au. Ko te aukati i tēnei horopaki he mehua matatini o te ātete e uru ana ki te ātete parakore me te tauhohenga hoki.
Ahupūngao Kuantum
I roto i te ahupūngao matūriki, ko te mahi ngaru e whakaahua ana i te āhua o tētahi matūriki iti-atomika ka whakaatuhia pinepinetia hei tau matatini. He mea nui tēnei mahi ngaru ki te whakatau i te tūponotanga o te taunga o tētahi matūriki me te pūngao kei roto i tētahi pūnaha.
Te Tukatuka Tohu
I roto i te tukatuka tohu, he taputapu nui te Whakawhitinga Fourier e whakamahi ana i ngā tau matatini. Ka wehea e te Whakawhitinga Fourier te tohu wā ki ngā wāhanga auau ka taea te tātari me te whakarerekē motuhake.
Te Āhuatanga Wai me te Āhuatanga Hauātea
I roto i te miihini wai, ka whakamahia ngā tau matatini hei whakaoti rapanga maha e pā ana ki te rere rua-ahu. Ka āwhina te tikanga pūmanawa matatini ki te whakatau i ngā tauira rere me te whakamahi i ngā ariā aerodynamic.
Whakamutunga
He ariā kaha, he ariā whānui hoki ngā tau matatini i roto i te pāngarau. Ahakoa te āhua uaua, he tawhiti rawa atu i te ao o ia rā, ko ō rātou whakamahinga i roto i ngā momo mara pūtaiao e whakaatu ana i te hiranga o te mārama me te matatau ki tēnei ariā.
Nā te nui o te hītori me te whānui o ngā whakamahinga, kāore i te whakawhānui noa i te whānuitanga o te pāngarau ngā tau matatini, engari i whakawātea hoki i te ara mō ngā mahi auaha me ngā kitenga maha i roto i te pūtaiao me te hangarau. Hei toronga atu i te pūnaha tau tūturu, he wāhanga tino nui ngā tau matatini mō te tātari me te whakaoti rapanga uaua ake o te ao tūturu.
