Nā nīnau hoʻohālike e kūkākūkā ana i nā Conjugates, Moduli, a me nā hoʻopaʻapaʻa o nā helu paʻakikī a me ko lākou mau waiwai
He ʻāpana koʻikoʻi nā helu paʻakikī o ka makemakika, ʻoi aku hoʻi ma ke kahua o ka loiloi paʻakikī. Aia nā helu paʻakikī i kahi ʻāpana maoli a me kahi ʻāpana hoʻokalakupua, i hōʻike pinepine ʻia ma ke ʻano \( z = a + bi \), kahi \( a \) a me \( b \) he mau helu maoli a ʻo \( i \) ka ʻāpana hoʻokalakupua e hoʻokō ana i \( i^2 = -1 \). No ka hoʻomaopopo hohonu ʻana i nā helu paʻakikī, pono mākou e ʻike i nā manaʻo o ka conjugate, modulus, a me ka hoʻopaʻapaʻa o nā helu paʻakikī a me ko lākou mau waiwai.
Hoʻohuihui o nā Helu Paʻakikī
ʻO ka hui pū ʻana o ka helu paʻakikī \( z = a + bi \) ʻo \( \overline{z} = a – bi \). Hoʻololi ka hui pū ʻana o kahi helu paʻakikī i ka hōʻailona o ka ʻāpana hoʻokalakupua me ka ʻole o ka hoʻololi ʻana i ka hōʻailona o ka ʻāpana maoli.
Nā Waiwai o nā Conjugates
1. \( \overline{\overline{z}} = z \)
– ʻO ka hui o ka hui o kahi helu paʻakikī ʻo ia ka helu paʻakikī ponoʻī.
2. \( \overline{z + w} = \overline{z} + \overline{w} \)
– ʻO ka hui pū ʻana o ka huina o nā helu paʻakikī ʻelua, ʻo ia ka huina o nā hui pū ʻana o kēlā me kēia helu paʻakikī.
3. \( \overline{z \cdot w} = \overline{z} \cdot \overline{w} \)
– ʻO ka hui pū ʻana o ka huahana o nā helu paʻakikī ʻelua, ʻo ia ka huahana o nā hui pū ʻana o kēlā me kēia helu paʻakikī.
4. \( \overline{\left( \dfrac{z}{w} \right)} = \dfrac{\overline{z}}{\overline{w}} \)
– ʻO ka hui pū ʻana o ka māhele ʻana o nā helu paʻakikī ʻelua, ʻo ia ka māhele ʻana o kā lākou mau hui pū ʻana.
Modulus Helu Paʻakikī
ʻO ka modulus o kahi helu paʻakikī \( z = a + bi \) ka lōʻihi a i ʻole ka nui o ka vector e hōʻike ana iā \( z \) ma ka mokulele paʻakikī. Hōʻike ʻia ka modulus e \( |z| \) a ua helu ʻia e ke ʻano hana
\[ |z| = \sqrt{a^2 + b^2} \]
Nā Waiwai o Modulus
1. \( |z| \geq 0 \)
– ʻAʻole maikaʻi ʻole mau ka modulus o kahi helu paʻakikī.
2. \( |z| = 0 \iff z = 0 \)
– ʻO ka modulus o kahi helu paʻakikī he zero inā a inā wale nō he zero ka helu paʻakikī.
3. \( |z \cdot w| = |z| \cdot |w| \)
– ʻO ka modulus o ka huahana o nā helu paʻakikī ʻelua ka huahana o nā moduli o kēlā me kēia helu paʻakikī.
4. \( \hema| \dfrac{z}{w} \akau| = \dfrac{|z|}{|w|} \), \( w \neq 0 \)
– ʻO ka modulus o ka mahele ʻana o nā helu paʻakikī ʻelua ka mahele ʻana o kā lākou moduli.
5. \( |z + w| \leq |z| + |w| \)
– ʻO ke kaulike ʻole o ka huinakolu no ka modulus o kahi helu paʻakikī.
Nā Hoʻopaʻapaʻa Helu Paʻakikī
ʻO ka hoʻopaʻapaʻa o kahi helu paʻakikī \( z = a + bi \) ʻo ia ke kihi a ka vector e hōʻike ana iā \( z \) e hana ai me ka axis maoli maikaʻi ma ka mokulele paʻakikī. Hōʻike ʻia ka hoʻopaʻapaʻa e \( \arg(z) \) a hōʻike pinepine ʻia ma nā radians.
Nā Waiwai o nā Hoʻopaʻapaʻa
1. \( \arg(z^n) = n \cdot \arg(z) \)
– ʻO ka hoʻopaʻapaʻa o kahi helu paʻakikī i hāpai ʻia i ka mana ʻo ia ka hopena o ka hoʻonui ʻana i ka mana me ka hoʻopaʻapaʻa o ka helu paʻakikī.
2. \( \arg\left(\dfrac{z}{w}\right) = \arg(z) – \arg(w) \)
– ʻO ka hoʻopaʻapaʻa o ka mahele ʻana o ʻelua mau helu paʻakikī ka ʻokoʻa ma waena o nā hoʻopaʻapaʻa o ka numerator a me ka denominator.
Nā Nīnau Laʻana a me ke Kūkākūkā
Pilikia 1: Nā Hui o nā Helu Paʻakikī
E huli i ka hui pū ʻana o ka helu paʻakikī \( z = 3 + 4i \).
Kūkākūkā:
ʻO ka hui pū ʻana o \( z \) ʻo \( \overline{z} = 3 – 4i \).
Nīnau 2: Modulus o nā Helu Paʻakikī
E helu i ka modulus o ka helu paʻakikī \( z = 1 – i \).
Kūkākūkā:
\[ |z| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Nīnau 3: Nā Hoʻopaʻapaʻa Helu Paʻakikī
E hoʻoholo i ka hoʻopaʻapaʻa o ka helu paʻakikī \( z = -1 + \sqrt{3}i \).
Kūkākūkā:
No ka loaʻa ʻana o ka hoʻopaʻapaʻa, pono mākou e loaʻa i ke kihi i hoʻokumu ʻia e \( z \) ma ka mokulele paʻakikī.
Aia ka helu paʻakikī \( -1 + \sqrt{3}i \) ma ka quadrant II.
\[ \arg(z) = \tan^{-1}\left(\dfrac{\sqrt{3}}{-1}\'ākau) + \pi = \tan^{-1}(-\sqrt{3}) + \pi \]
Ua ʻike mākou \( \tan(\dfrac{\pi}{3}) = \sqrt{3}\), no laila
\[ \arg(z) = \dfrac{2\pi}{3} \]
Nīnau 4: Hoʻonui ʻia o nā Helu Paʻakikī
E hoʻoholo i ka huahana \(z_1 = 2 + 3i \) a me \(z_2 = 1 – i \), a e helu i ka modulus o ka huahana.
Kūkākūkā:
\[ z_1 \cdot z_2 = (2 + 3i)(1 – i) = 2 + 2i – 3i – 3i^2 = 2 – i + 3 = 5 – i \]
Modulus o \( z_1 \cdot z_2 \):
\[ |5 – i| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \]
Ka hopena
He manaʻo nui nā helu paʻakikī i ka makemakika a me ka ʻenekinia. Ma ka hoʻomaopopo ʻana i ka conjugate, modulus, a me ka hoʻopaʻapaʻa, hiki iā mākou ke hoʻomaopopo maikaʻi a hoʻoponopono i nā helu paʻakikī me ka ʻoi aku ka maikaʻi. Hāʻawi nā waiwai o ka conjugate, modulus, a me ka hoʻopaʻapaʻa i nā mea hana ikaika no ka nānā hou aku a me nā noi ākea ma nā lālā like ʻole o ka ʻepekema. Ma o nā laʻana i hōʻike ʻia, manaʻolana ʻia e loaʻa i ka poʻe heluhelu kahi ʻike maikaʻi a me ka mākaukau i ka hoʻohana ʻana i nā helu paʻakikī ma nā ʻano like ʻole.
