Ngā Mahi i runga i ngā Tau Matatini
He ariā pāngarau ngā tau matatini e whakakotahi ana i ngā tau tūturu me ngā tau pohewa. He mea nui ēnei ki ngā peka maha o te pūtaiao, tae atu ki te ahupūngao, te hangarau, me te pāngarau tonu. I roto i tēnei tuhinga, ka tūhuratia e tātou ngā mahi rerekē ka taea te mahi i runga i ngā tau matatini, tae atu ki te tāpiri, te tango, te whakarea, te wehewehe, me ētahi atu.
Te Mārama ki ngā Tau Uaua
Ka taea te tuhi i ia tau matatini ki te āhua \(a+bi\), ina ko \(a\) me \(b\) he tau tūturu, ā, ko \(i\) he waeine whakaaro noa e tutuki ana i te \(i^2 = -1\). Ka kiia te kupu \(a\) ko te wāhanga tūturu, ko \(b\) ia ko te wāhanga whakaaro noa o te tau matatini. Hei tauira, ko \(3 + 4i\) he tau matatini me te wāhanga tūturu 3 me te wāhanga whakaaro noa 4.
Hei whārite taketake, kei a tātou:
\[ i^2 = -1 \]
Ko te tikanga ko te \(i\) te pūtake tapawhā o te -1.
Tāpiri me te Tango
Ka taea te tāpiri me te tango i ngā tau matatini mā te tāpiri me te tango i ngā wāhanga tūturu me ngā wāhanga pohewa. Mehemea he rua ā tātou tau matatini \( z_1 = a + bi \) me \( z_2 = c + di \), kātahi:
\[ z_1 + z_2 = (a + bi) + (c + di) = (a + c) + (b + d)i \]
\[ z_1 – z_2 = (a + bi) – (c + di) = (a – c) + (b – d)i \]
Tauira:
Me kī ko \( z_1 = 3 + 4i \) me \( z_2 = 1 + 2i \), kātahi:
\[ z_1 + z_2 = (3+1) + (4+2)i = 4 + 6i \]
\[ z_1 – z_2 = (3-1) + (4-2)i = 2 + 2i \]
Te Whakarea
Ka whakamahia ngā tauwehewehenga hei whakarea i ngā tau matatini pērā i te arapū engari ka whakaarohia \( i^2 = -1 \). Me kī ko \( z_1 = a + bi \) me \( z_2 = c + di \), kātahi:
\[ z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
\[ = ac + adi + bci + bd(-1) \]
\[ = ac + adi + bci – bd \]
\[ = (ac – bd) + (ad + bc)i \]
Tauira:
Me kī ko \( z_1 = 3 + 4i \) me \( z_2 = 1 + 2i \), kātahi:
\[ z_1 \cdot z_2 = (3 + 4i)(1 + 2i) \]
\[ = 3 \cdot 1 + 3 \cdot 2i + 4i \cdot 1 + 4i \cdot 2i \]
\[ = 3 + 6i + 4i + 8i^2 \]
\[ = 3 + 10i + 8(-1) \]
\[ = 3 + 10i – 8 \]
\[ = -5 + 10i \]
Tohatoha
Mā te whakarea i te taupū me te tauwehenga ki te hononga o te tauwehenga ka taea te wehewehe i ngā tau matatini. Ko te hononga o te tau matatini \( z = a + bi \) ko \( \overline{z} = a – bi \).
Me kī ko \( z_1 = a + bi \) me \( z_2 = c + di \), kātahi:
\[ \frac{z_1}{z_2} = \frac{a + bi}{c + di} \]
Whakareatia ki te hononga o te taupū:
\[ = \frac{(a + bi)(c – di)}{(c + di)(c – di)} \]
\[ = \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} \]
Tauira:
Me kī ko \( z_1 = 3 + 4i \) me \( z_2 = 1 + 2i \), kātahi:
Hononga \( z_2 = 1 – 2i \).
\[ \frac{z_1}{z_2} = \frac{3 + 4i}{1 + 2i} \cdot \frac{1 – 2i}{1 – 2i} \]
\[ = \frac{(3 + 4i)(1 – 2i)}{(1 + 2i)(1 – 2i)} \]
\[ = \frac{3 – 6i + 4i – 8i^2}{1 – 4i^2} \]
E mōhiotia ana ko \( i^2 = -1 \):
\[ = \frac{3 – 6i + 4i + 8}{1 + 4} \]
\[ = \frac{11 – 2i}{5} \]
\[ = \frac{11}{5} – \frac{2i}{5} \]
\[ = 2.2 – 0.4i \]
Ngā Tauira me ngā Tautohe
Ko te modulus o tētahi tau matatini ko te tawhiti mai i te pūtake i roto i te papa matatini ki te pūwāhi e tohuhia ana e te tau matatini. Ko te modulus o tētahi tau matatini \( z = a + bi \) e whakaatuhia ana ko \( |z| \) ā, ka tatauhia mā:
\[ |z| = \sqrt{a^2 + b^2} \]
Tauira:
Mena ko \( z = 3 + 4i \), kāti:
\[ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Ko te tautohe o tētahi tau matatini ko te koki e hangaia ana e te tau matatini ki te tuaka tūturu i roto i te papa matatini, ā, e whakaatuhia ana i roto i ngā rātiana, i ngā nekehanga rānei.
Puka Pōro
Ka taea hoki te whakaatu i ngā tau matatini i roto i te āhua porowhita. He maha ngā wā ka whakangawarihia e tēnei āhua ngā tataunga e whakamahi ana i ngā mana me ngā pūtake o ngā tau matatini. Ka taea te whakaatu i tētahi tau matatini penei:
\[ z = r(\cos \theta + i \sin \theta) \]
ko \( r \) te modulus, ā, ko \( \theta \) te tautohe o te tau matatini.
Ngā Kaiwhakahaere Matatini Kē: Te Taupūnga me te Tauira
Ka taea te huri i ngā tau matatini ki te āhua taupū mā te whakamahi i te tātai a Euler:
\[ z = re^{i\theta} \]
ko \( e \) te pūtake o te logarithm maori, ā, ko \( \theta \) te tautohe o \( z \).
He tino whai hua te taupūtanga o ngā tau matatini i roto i ngā mahi maha, inā koa i roto i te tātari Fourier me ngā panonitanga Laplace.
Whakamutunga
He taputapu taketake ngā tau matatini e tino whai hua ana mō te whakaoti rapanga maha i roto i te pāngarau me te pūtaiao. Ko te matatau ki ngā mahi taketake pēnei i te tāpiri, te tango, te whakarea, me te wehewehe he taahiraa tuatahi nui. Hei tāpiri, ko te mārama ki ngā ariā o te modulus, te tautohe, me te hurihanga ki ngā āhua porowhita me te taupūnga ka whakarei ake i tō tātou kaha ki te tūhura i ngā tono o ngā tau matatini i roto i te maha o ngā mara.
Mā te mārama me te whakamahi i ngā tau matatini, ka taea e tātou te whakaoti rapanga e uaua ana, e kore rānei e taea te whakaoti mā te whakamahi i ngā tau tūturu anake. Hei taputapu tātari kaha, kei te noho tonu ngā tau matatini hei wāhanga nui o te pāngarau me ngā tono pūtaiao tae noa mai ki tēnei rā.
