Kor u qaadista tiro kakan oo ah awood dabiici ah

Daabacaaddan, waxaan ku tixgalin doonaa sida tirada adag kor loogu qaadi karo awoodda (oo ay ku jirto isticmaalka qaaciidada De Moivre). Maaddada aragtida waxa la socda tusaalayaal si loo fahmo wanaagsan.

Kor u qaadista nambar adag oo koronto ah

Marka hore, xasuusnoow in nambar kakan uu leeyahay qaabka guud: z = a + bi (qaab aljabra ah).

Hadda waxaan si toos ah ugu socon karnaa xallinta dhibaatada.

Tirada labajibbaaran

Waxaan u matali karnaa shahaadada wax soo saar isku mid ah, ka dibna hel badeecadooda (innagoo taas xasuusanayna i² = -1).

z² = (a + bi)² = (a + bi) (a + bi)

Tusaale 1:

z=3+5i

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

Waxa kale oo aad isticmaali kartaa, kuwaas oo ah afargeeska wadarta:

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

Fiiro gaar ah: Si la mid ah, haddii loo baahdo, waxaa la heli karaa qaabab loogu talagalay afargeeska faraqa, cube ee wadarta / farqiga, iwm.

Darajada Nth

Kor u qaad tiro kakan z nooc ah n aad u sahlan haddii lagu matalo qaab trigonometric ah.

Xusuusnow in, guud ahaan, calaamadaynta lambarku ay u egtahay sidan: z = |z| ⋅ (cos φ + i ⋅ sin φ).

Si aad u sharaxdo, waxaad isticmaali kartaa qaacidada De Moivre (oo loogu magac daray xisaabyahan Ingiriis Abraham de Moivre):

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

Qaaciddada waxaa lagu helaa iyadoo loo qorayo qaab trigonometric ah (qaybaha waa la badiyay, doodahana waa lagu daray).

Tusaale 2

Kor u qaad tiro kakan z = 2 ⋅ (cos 35° + i ⋅ dembi 35°) ilaa heerka sideedaad.

Solution

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