Kukweza nambala yovuta kukhala mphamvu yachilengedwe

M'buku lino, tiwona momwe nambala yovuta ingakwezeredwe kukhala mphamvu (kuphatikiza kugwiritsa ntchito fomula ya De Moivre). Zinthu zangongole zimatsagana ndi zitsanzo zomvetsetsa bwino.

Kukweza nambala yovuta kukhala mphamvu

Choyamba, kumbukirani kuti nambala yovuta imakhala ndi mawonekedwe onse: z = ndi + bi (mawonekedwe a algebraic).

Tsopano titha kupitilira njira yothetsera vutolo.

Nambala ya square

Titha kuyimira digiriyo ngati chinthu chazinthu zomwezo, kenako ndikupeza zomwe zili (pokumbukira izi i² =-1).

z² = (a + bi)² = (a + bi) (a + bi)

Chitsanzo 1:

z=3+5i

z² = (3 + 5i)² = (3 + 5i) (3 + 5i) = 9 + 15i + 15i + 25i² = -16 + 30i

Mukhozanso kugwiritsa ntchito, kutanthauza square of the sum:

z² = (a + bi)² = a² + 2 ⋅ a ⋅ bi + (bi)² = a² + 2abi – b²

Zindikirani: Momwemonso, ngati kuli kofunikira, mafomu a lalikulu la kusiyana, cube ya kuchuluka / kusiyana, ndi zina zotero.

Nth digiri

Kwezani nambala yovuta z mwanjira n zosavuta kwambiri ngati zikuimiridwa mu trigonometric mawonekedwe.

Kumbukirani kuti, kawirikawiri, kulemba kwa nambala kumawoneka motere: z = |z| ⋅ (cos φ + i ⋅ tchimo φ).

Kwa exponentiation, mungagwiritse ntchito Fomula ya De Moivre (wotchedwa dzina la katswiri wa masamu wachingelezi Abraham de Moivre):

zⁿ = | z |ⁿ ⋅ (cos(nφ) + i ⋅ sin(nφ))

Njirayi imapezeka polemba mu mawonekedwe a trigonometric (ma modules amachulukitsidwa, ndipo zotsutsana zimawonjezeredwa).

Mwachitsanzo 2

Kwezani nambala yovuta z = 2 ⋅ (cos 35° + i ⋅ tchimo 35°) ku digiri yachisanu ndi chitatu.

Anakonza

z⁸ = 2⁸ ⋅ (cos(8 ⋅ 35°) + i ⋅ tchimo(8 ⋅ 35°)) = 256 ⋅ (cos 280° + i sin 280°).