Ngā Tauira Pātai e Matapaki ana i ngā Hononga, ngā Moduli, me ngā Tautohe o ngā Tau Matatini me ō rātou Āhuatanga
He wāhanga nui ngā tau matatini o te pāngarau, inā koa i roto i te ao tātari matatini. Kei roto i ngā tau matatini he wāhanga tūturu me he wāhanga pohewa, e whakaatuhia ana i te nuinga o te wā i roto i te āhua \( z = a + bi \), ko \( a \) me \( b \) he tau tūturu, ā, ko \( i \) te wae pohewa e tutuki ana i \( i^2 = -1 \). Hei mārama ake i ngā tau matatini, me mōhio tātou ki ngā ariā o te hononga, te modulus, me te tautohe o ngā tau matatini me ō rātou āhuatanga.
Hononga o ngā Tau Matatini
Ko te hononga o te tau matatini \( z = a + bi \) ko \( \overline{z} = a – bi \). Ka whakarerekē te hononga o te tau matatini i te tohu o te wāhanga pohewa me te kore e whakarerekē i te tohu o te wāhanga tūturu.
Ngā Āhuatanga o ngā Hononga
1. \( \overline{\overline{z}} = z \)
– Ko te hononga o te hononga o tētahi tau matatini ko te tau matatini tonu.
2. \( \overline{z + w} = \overline{z} + \overline{w} \)
– Ko te hononga o te tapeke o ngā tau matatini e rua ko te tapeke o ngā hononga o ia tau matatini.
3. \( \overline{z \cdot w} = \overline{z} \cdot \overline{w} \)
– Ko te hononga o te hua o ngā tau matatini e rua ko te hua o ngā hononga o ia tau matatini.
4. \( \overline{\left( \dfrac{z}{w} \right)} = \dfrac{\overline{z}}{\overline{w}} \)
– Ko te hononga o te wehenga o ngā tau matatini e rua ko te wehenga o ō rāua hononga.
Tauwehenga Tau Matatini
Ko te modulus o tētahi tau matatini \( z = a + bi \) ko te roa, te rahi rānei o te ira e tohu ana i a \( z \) i roto i te papa matatini. Ko te modulus e tohuhia ana e \( |z| \) ā, ka tatauhia mā te tātai.
\[ |z| = \sqrt{a^2 + b^2} \]
Ngā Āhuatanga o te Modulus
1. \( |z| \geq 0 \)
– Kāore te modulus o tētahi tau matatini i te tōraro i ngā wā katoa.
2. \( |z| = 0 \iff z = 0 \)
– Kore te modulus o tētahi tau matatini mēnā he kore te tau matatini, ā, koinā anake.
3. \( |z \cdot w| = |z| \cdot |w| \)
– Ko te modulus o te hua o ngā tau matatini e rua ko te hua o ngā modulus o ia tau matatini.
4. \( \maui| \dfrac{z}{w} \matau| = \dfrac{|z|}{|w|} \), \( w \neq 0 \)
– Ko te modulus o te wehenga o ngā tau matatini e rua ko te wehenga o ō rāua modulus.
5. \( |z + w| \leq |z| + |w| \)
– Te kore taurite o te tapatoru mō te modulus o tētahi tau matatini.
Ngā Tautohe Tau Matatini
Ko te tautohe o tētahi tau matatini \( z = a + bi \) ko te koki e hangaia ana e te wetere e tohu ana i a \( z \) me te tuaka tūturu pai i roto i te papa matatini. Ko te tautohe e tohuhia ana e \( \arg(z) \) ā, ko te tikanga ka whakaaturia i roto i ngā rātiana.
Ngā Āhuatanga o ngā Tautohe
1. \( \arg(z^n) = n \cdot \arg(z) \)
– Ko te tautohe mō tētahi tau matatini kua whakanuia ki te mana ko te hua o te whakarea i te mana ki te tautohe o te tau matatini.
2. \( \arg\left(\dfrac{z}{w}\right) = \arg(z) – \arg(w) \)
– Ko te tautohe mō te wehenga o ngā tau matatini e rua ko te rerekētanga i waenga i ngā tautohe o te taupū me te tauwehe.
Ngā Pātai Tauira me te Kōrero
Raru 1: Ngā Hononga o ngā Tau Matatini
Kimihia te hononga o te tau matatini \( z = 3 + 4i \).
Kōrero:
Ko te hononga o \( z \) ko \( \overline{z} = 3 – 4i \).
Pātai 2: Te Tauwehenga o ngā Tau Matatini
Tātaihia te modulus o te tau matatini \( z = 1 – i \).
Kōrero:
\[ |z| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Pātai 3: Ngā Tautohe Tau Uaua
Whakatauhia te tautohe o te tau matatini \( z = -1 + \sqrt{3}i \).
Kōrero:
Hei kimi i te tautohe, me kimi e tātou te koki i hangaia e \( z \) i roto i te papa matatini.
Kei te hauwhā II te tau matatini \( -1 + \sqrt{3}i \).
\[ \arg(z) = \tan^{-1}\left(\dfrac{\sqrt{3}}{-1}\right) + \pi = \tan^{-1}(-\sqrt{3}) + \pi \]
E mōhio ana tātou ko \( \tan(\dfrac{\pi}{3}) = \sqrt{3}\), nō reira
\[ \arg(z) = \dfrac{2\pi}{3} \]
Pātai 4: Te Whakarea o ngā Tau Uaua
Tātaihia te hua \( z_1 = 2 + 3i \) me \( z_2 = 1 – i \), ka tatau i te modulus o te hua.
Kōrero:
\[ z_1 \cdot z_2 = (2 + 3i)(1 – i) = 2 + 2i – 3i – 3i^2 = 2 – i + 3 = 5 – i \]
Te tauwehenga o \( z_1 \cdot z_2 \):
\[ |5 – i| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \]
Whakamutunga
He ariā taketake ngā tau matatini i roto i te pāngarau me te hangarau. Mā te mārama ki te hononga, te modulus, me te tautohe, ka taea e tātou te mārama ake me te whakahaere i ngā tau matatini kia pai ake te whai hua. Ko ngā āhuatanga o te hononga, te modulus, me te tautohe he taputapu kaha mō te tātari atu me ngā tono whānui i roto i ngā peka pūtaiao. Mā roto i ngā tauira e whakaaturia ana, ko te tumanako ka pai ake te mārama me te matatau o ngā kaipānui ki te whakamahinga o ngā tau matatini i roto i ngā horopaki rerekē.
