Amanani antsonkothileyo ngala amanani aqulathe icandelo lokwenyani kunye nenxalenye yentelekelelo.

Bamele iseti yazo zonke izibini ezi-odolweyo (x, y), izinto zazo ezizezeseti yamanani okwenyani (R).

Iseti yamanani antsonkothileyo iboniswa ngu C kwaye ichazwe ngokusebenza:

• Ukulingana: (a, b) = (c, d) = a = c kunye b = d
• Kwakhona: (a, b) + (c, d) = (a + b + c + d) (
• Ukuphindaphinda: (a, b). (c, d) = (ac - bd, intengiso + bc)

## Icandelo lokucinga (i)

Ibonakalisiwe ngonobumba yo, iyunithi yokucinga yokucinga ngababini aba-odoliweyo (0, 1). Ilogo:

mna. i = –1 ↔ i2 = -1

Ngaloo ndlela, yo yingcambu ka - 1.

## Uhlobo lweAlgebra lweZ

Ifom ye-algebraic ye-Z isetyenziselwa ukumela inani elintsokothileyo kusetyenziswa ifomula:

Z = x + yi

Kuphi:

• x linani lokwenyani eliboniswe x = Re (Z), ebizwa Inxalenye yokwenene yeZ.
• y linani lokwenyani eliboniswe ngu y = Im (Z), elibizwa njalo Inxalenye yentelekelelo kaZ.

## Ukudibanisa inani elinzima

Ukudityaniswa kwenani elinzima kuboniswe ngu z, ichazwe ngu z = a-bi. Ke ngoko, uphawu lwenxalenye yokucinga luyatshintshana.

Ke ukuba z = a + bi, emva koko z = a - bi

Xa siphinda-phinda inani elintsonkothileyo ngokudibanisa kwalo, iziphumo ziya kuba linani lokwenyani.

## Ukulingana phakathi kwamanani antsonkothileyo

Amanani amabini antsonkothileyo ngu-Z1 = (a, b) kunye no-Z2 = (c, d), bayalingana xa a = c kunye b = d. Kungenxa yokuba banamalungu afanayo okwenyani kunye nawentelekelelo. Ke:

a + bi = c + di xa i = cyb = d

## Umsebenzi weenombolo ezinzima

Amanani entsonkothileyo kunokwenzeka ukwenza imisebenzi yokudibanisa, ukuthabatha, ukuphindaphinda kunye nokwahlulahlula. Jonga iinkcazo kunye nemizekelo engezantsi:

### Kwakhona

Z1 + Z2 = (a + c, b + d) =

Kwifom yealgebra, sine:

(a + bi) + (c + di) = (a + c) + mna (b + d)

Umzekelo:

(2 + 3i) + (–4 + 5i)
(2-4) + mna (3 + 5)
–2 + 8i

### Ukuthabatha

Z1 -Z2 = (a - c, b-d)

Kwifom yealgebra, sine:

(a + bi) - (c + di) = (a - c) + i (b-d)

Umzekelo:

(4 - 5i) - (2 + i)
(4 - 2) + i (–5 –1)
2 - 6i

### Ukuphindaphinda

(a, b) (c, d) = (ac - bd, ad + bc)

Kwifom yealgebra, sisebenzisa i ukuhambisa ipropathi:

(a + bi) usetyenziso lweWindows kwi- Ivenkile yeWindows (c + di) = ac + adi + bci + bdi2 (i2 = -1)
(a + bi) usetyenziso lweWindows kwi- Ivenkile yeWindows (c + di) = ac + adi + bci - bd
(a + bi) usetyenziso lweWindows kwi- Ivenkile yeWindows (c + di) = (ac - bd) + i (ad + bc)

Umzekelo:

(4 + 3i). (2 - 5i)
8 - 20i + 6i - 15i2
8 - 14i + 15
23 - 14i

### ICandelo

Z1/ Z2 = Z3
Z1 = Z2 . Z3

Kulingano olungentla, ukuba uZ3 = x + yi, sine:

Z1 = Z2 . Z3

a + bi = (c + di). (x + yi)
a + bi = (cx - dy) + i (cy + dx)

Ngokwenkqubo yokungaziwa x kwaye y sine:

cx - idy = a
dx + cy = b

Emva koko

x = ac + bd / c2 + d2
y = bc - intengiso / c2 + d2

Umzekelo:

2 - 5i / i
2 - 5i /. (- i) / (- i)
–2i + 5i2/ -I2
5 - 2i

Ukwazi ngakumbi, jonga kwakhona

## Ukuzivocavoca umzimba ngeempendulo

1. (UF-TO) Cinga yo Iyunithi yokucinga yeenombolo ezinzima. Ixabiso lentetho (i + 1)8 :

a) 32i
b) 32
c) Ishumi elinambini
d) 16i

2. (UEL-PR) Inani eliyinkimbinkimbi z eliqinisekisa i-equation iz - 2w (1 + i) = 0 (w ibonisa ukuba ikhonkrithi ye-z) yile:

a) z = 1 + i
b) z = (1/3) - i
c) z = (1 - i) / 3
d) z = 1 + (i / 3)
e) z = 1 - i

3. (Vunesp-SP) Cinga ngenani elintsonkothileyo z = cos π / 6 + i sin π / 6. Ixabiso lika Z3 + Z6 6 + Z12 :

a) - i
b) ½ + √3 / 2i
c) i - 2
wanika
e) 2i

## Ividiyo yaseklasini

Ukwandisa ukuqonda kwakho kwamanani antsonkothileyo, bukela ividiyo «Intshayelelo kumanani antsonkothileyo»

## Imbali yamanani antsonkothileyo.

Ukufunyanwa kwamanani antsonkothileyo kwenziwa ngenkulungwane ye-1501 ngenxa yegalelo lesazi semathematics uGirolamo Cardano (1576-XNUMX).

Nangona kunjalo, bekungekho kwinkulungwane ye-1777 apho ezi zifundo zazisenziwa ngokusemthethweni sisazi sezibalo uCarl Friedrich Gauss (1855-XNUMX).

Ibiyinkqubela phambili ebalulekileyo kwimathematika, kuba inani elingelilo linengcambu ezikwisikwere, nalapho ukufunyanwa kwamanani antsonkothileyo kwakuthathwa njengokungenakwenzeka.