Misalan Tambayoyi da Tattaunawa kan Ayyuka kan Lambobi Masu Hadari
Lambobi masu rikitarwa wani ƙarin bayani ne na manufar lambobi na gaske don haɗawa da lambobi na tunani. Tsarin gabaɗaya na lamba mai rikitarwa shine + bi, inda a da b lambobi ne na gaske, kuma i raka'a ce ta tunani tare da mallakar i² = -1. Ayyukan akan lambobi masu rikitarwa sun haɗa da ƙari, ragi, ninkawa, da rabawa. Wannan labarin zai samar da misalai da yawa na matsaloli da tattaunawa don ayyuka daban-daban akan lambobi masu rikitarwa.
Ƙarawa da Ragewa na Lambobi Masu Hadaka
Misali Tambaya ta 1
Ƙara waɗannan lambobi masu rikitarwa: (3 + 4i) da (1 + 2i).
Tattaunawa:
Ƙara lambobi masu rikitarwa ana yin su ne ta hanyar ƙara ainihin sassansu da na tunanin daban-daban.
\[ (3 + 4i) + (1 + 2i) = (3 + 1) + (4i + 2i) = 4 + 6i \]
Don haka, sakamakon ƙara (3 + 4i) da (1 + 2i) shine 4 + 6i.
Misali Tambaya ta 2
Cire lambar hadaddun (2 + 5i) daga (6 + 3i).
Tattaunawa:
Ana cire lambobi masu rikitarwa ta hanyar cire ainihin ɓangaren da kuma ɓangaren tunani daban-daban.
\[ (6 + 3i) - (2 + 5i) = (6 – 2) + (3i – 5i) = 4 – 2i \]
Don haka, sakamakon cire (2 + 5i) daga (6 + 3i) shine 4 – 2i.
Yawan Lambobi Masu Hadaka
Misali Tambaya ta 3
A ninka waɗannan lambobi masu rikitarwa: (2 + 3i) da (4 + i).
Tattaunawa:
Ana yin ninka lambobi masu rikitarwa ta amfani da rarrabawa ko tsari na yau da kullun, kamar ninka binomials guda biyu a cikin algebra ta yau da kullun.
\[
(2 + 3i) \cdot (4 + i) = 2 \cdot 4 + 2 \cdot i + 3i \cdot 4 + 3i \cdot i
\]
Sannan mu yi lissafi dalla-dalla:
\[
= 8 + 2i + 12i + 3i^2
\]
Tunda \( i^2 = -1 \):
\[
= 8 + 14i + 3(-1)
\]
\[
= 8 + 14i – 3
\]
\[
= 5 + 14i
\]
Don haka, sakamakon ninkawa (2 + 3i) da (4 + i) shine 5 + 14i.
Rarraba Lambobi Masu Hadaka
Misali Tambaya ta 4
Raba wannan lambar mai rikitarwa: (5 + 6i) ta (2 + i).
Tattaunawa:
Rarraba lambobi masu rikitarwa ta amfani da haɗin ma'aunin ma'auni. Haɗin ma'aunin ma'auni na \(2 + i\) shine \(2 – i\).
Muna ninka lambobi da ma'auni ta hanyar haɗakar ma'auni:
\[
\frac{5 + 6i}{2 + i} \cdot \frac{2 – i}{2 – i}
\]
Yanzu muna lissafin mai ƙidaya da mai ƙidaya daban-daban:
\[
= \frac{(5 + 6i) \cdot (2 – i)}{(2 + i) \cdot (2 – i)}
\]
Yawan ma'auni:
\[
(2 + i) \cdot (2 – i) = 2^2 – i^2 = 4 – (-1) = 4 + 1 = 5
\]
Yawan masu ƙidaya:
\[
(5 + 6i) \cdot (2 – i) = 5 \cdot 2 + 5 \cdot (-i) + 6i \cdot 2 + 6i \cdot (-i)
= 10 – 5i + 12i – 6i^2
= 10 + 7i – 6(-1)
= 10 + 7i + 6
= 16 + 7i
\]
Don haka, rabon shine:
\[
= \frac{16 + 7i}{5} = \frac{16}{5} + \frac{7i}{5} = 3.2 + 1.4i
\]
Don haka sakamakon raba (5 + 6i) da (2 + i) shine 3.2 + 1.4i.
Karin Tattaunawa: Modulus da Haɗaɗɗen Lambobi Masu Hadaka
Misali Tambaya ta 5
Nemo tsarin da haɗin lambar hadaddun \(z = 3 + 4i\).
Tattaunawa:
Modulus na lambar hadaddun \(z = a + bi\) shine:
\[
|z| = \sqrt{a^2 + b^2}
\]
Domin \(z = 3 + 4i\):
\[
|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Haɗin lambar mai rikitarwa \(z = a + bi\) shine \(z^ = a – bi\).
Domin \(z = 3 + 4i\):
\[
z^ = 3 – 4i
\]
Don haka tsarin \(3 + 4i\) shine 5, kuma haɗinsa shine \(3 – 4i\).
Kammalawa
Lambobi masu rikitarwa suna taka muhimmiyar rawa a fannoni daban-daban na lissafi da aikace-aikacen fasaha. Fahimtar ayyukan asali akan lambobi masu rikitarwa, kamar ƙari, ragi, ninkawa, da rabawa, shine mabuɗin amfani da waɗannan ra'ayoyi don magance matsaloli masu rikitarwa. Yin aiki da nau'ikan matsaloli daban-daban, kamar waɗanda aka bayyana a sama, zai taimaka wajen ƙarfafa fahimtarka da ƙwarewarka wajen aiki da lambobi masu rikitarwa.
