Na-ebuli ọnụ ọgụgụ dị mgbagwoju anya na ike eke

N'akwụkwọ a, anyị ga-atụle otú e nwere ike isi bulie nọmba mgbagwoju anya na ike (gụnyere iji usoro De Moivre). Ihe omuma a na-esonyere ya na ihe atụ maka nghọta ka mma.

Na-ebuli nọmba mgbagwoju anya na ike

Nke mbụ, cheta na nọmba mgbagwoju anya nwere ụdị izugbe: z = a + bi (ụdị algebra).

Ugbu a, anyị nwere ike ịga n'ihu ozugbo na ngwọta nke nsogbu ahụ.

Nọmba square

Anyị nwere ike na-anọchite anya ogo dị ka ngwaahịa nke otu ihe ahụ, wee chọta ngwaahịa ha (mgbe ị na-echeta nke ahụ i² = -1).

z² = (a + bi)² = (a + bi) (a + bi)

Ihe Nlereanya 1:

z=3+5i

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

Ị nwekwara ike iji, ya bụ square nke nchikota:

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

Cheta na: N'otu ụzọ ahụ, ọ bụrụ na ọ dị mkpa, enwere ike nweta usoro maka square nke ọdịiche, cube nke nchikota / iche, wdg.

ogo Nth

Welite nọmba mgbagwoju anya z n'ụdị n ọ dị mfe karị ma ọ bụrụ na egosipụtara ya n'ụdị trigonometric.

Cheta na, n'ozuzu, ndetu nke nọmba dị ka nke a: z = |z| ⋅ (cos φ + i ⋅ sin φ).

Maka nkọwa, ị nwere ike iji Usoro nke De Moivre (nke a na-akpọ aha onye England mathematician Abraham de Moivre):

zⁿ = | z |ⁿ ⋅ (cos (nφ) + i ⋅ mmehie (nφ))

A na-enweta usoro ahụ site na ịde ederede n'ụdị trigonometric (a na-amụba modul, a na-agbakwụnyekwa arụmụka).

atụ 2

Welite nọmba mgbagwoju anya z = 2 ⋅ (cos 35° + i ⋅ mmehie 35°) ruo ogo nke asatọ.

ngwọta

z⁸ = 2⁸ ⋅ (cos (8 ⋅ 35°) + i ⋅ mmehie(8⋅ 35°)) = 256 ⋅ (cos 280° + na mmehie 280°).