Amḍan asemlal

Deg tusnakt, **amḍan asemlal** d amḍan yezmer ad yettwasumer s talɣa a + bi, anda (a) d (b) id sin n yimḍanen ilawen, ma d i d yiwet n tigget tasnumkant (d anamek-is ur nezmir ara ad tt-naf deg tilawt) yettwammlen akken i² = −1. Amḍan ( a ) yettwasemma d **amur ilaw** n umḍan asemlal, ma d ( b ) d **amur asugnan**-ines.

Tagrumma n yimḍanen isemlalen tettwasemma s uzemz C. Tesnulfa-d akken ad tesnerni tagrumma n yimḍanen ilaw R s umḍan i yesɛan tazmert ad yefk tifrat i kra n tenkarin ur nesɛi ara tifrat deg R, am x² = −1. Yal amḍan ilaw d amḍan asemlal anda amur asugnan-ines d ilem. Ihi, tagrumma n yimḍanen ilaw d tadugrumma n tegrumma n yimḍanen isemlalen.

Yella waṭas n yisnasen n yimḍanen isemlalen deg yiḥricen yemgaraden n tusnakt d tussna. Deg tsengama, ttusemrasen deg uselmed n wahilen n trisiti akked umussu ahuzzan. Deg tsengama tatrart, ttusemrasen deg tenfalit n yimrigen imaẓlayen. Deg tesnileswalt tanmettit, ttusemrasen deg uselmed n yiferdisen n tutlayt. Ttusemrasen daɣen deg uẓawan d uselmed n yiẓuran n tenkarin.

## '''Amḍan i'''
Amḍan i d tigget tasugnant. Yettwammel s i² = -1. Asekcam n umḍan-agi yefka tazmert i tusnakt ad taf tifrat i waṭas n tenkarin ur nesɛi ara tifrat deg umaɣrad n yimḍanen ilaw.

## '''Iḥricen n umḍan asemlal  x+iy'''
Amḍan asemlal z = a + bi yegber sin n yiḥricen:

# Amur ilaw (a): D amḍan ilaw yettwazemren ad yili d amagnu neɣ d ameɣẓan.
# Amur asugnan (bi): Yegber amḍan ilaw b yeddan d tigget tasugnant i.

## '''Tanekda'''

### <big>Talɣa tajebrant</big>
Z = a + bi Anda:

* a d amur ilaw
* b d amur asugnan
* i d tigget tasugnant

### <big>Talɣa tasfaylut</big>
Z = r(cos(ϕ) + i sin(ϕ)) Anda:

* r  neɣ |z|       d azal amagdez (module)
* ϕ neɣ arg(z) d tiɣiret (argument)
Tikwal amḍan asemlal yettwaru deg yiwet n talɣiwin igi: 

* Z = r * \exp(\Theta*i)  **talɣa** **tasusmirt** s usumres n tanfalit n Euler
* z = (r, θ) **Talɣa tasfaylut**
* z = r (cosθ + i sinθ) = r cis (θ)  (s tuggza cis)



azal amagdez n umḍan asemlal z d aẓar amkuẓ  n timernit n yimkuẓen n yiḥricen ilawen akked isugnanen : 

|Z| = \sqrt{a^2+b^2}  

akken ad nsiḍen θ sef talɣa taljebrit a + bi, nezmer ad nessemres tiwuriwin arccos, arcsin neɣ arctan : 
\theta=\begin{cases}
\arccos\left(\tfrac ar\right)&\text{ma } b\ge0\\
-\arccos\left(\tfrac ar\right)&\text{ma } b0\\
\arctan\left(\tfrac ba\right)+\pi&\text{ma} a

## '''Timhal d wassaɣen'''

### <big>Timernit</big>
Ma yella Z{\scriptstyle\text{1}} = a + bi  d Z{\scriptstyle\text{2}} = c + di  sin n yimḍanen isemlalen:


Z1 + Z2 = (a + bi) + ( c+di)  =  (a+c) + (b+d)i 

Timernit n sin yimḍanen isemlalen d timernit n yiḥricen ilawanen d yiḥricen isugnanen yal yiwen ɣer wayeḍ.

### <big>Afaris</big>
Ma yella Z{\scriptstyle\text{1}} = a + bi  d Z{\scriptstyle\text{2}} = c + di  sin n yimḍanen isemlalen:


Z1 * Z2 = (a + bi) * ( c+di)  =  (ac-bd) + (ad+bc)i 


Afaris n sin yimḍanen isemlalen yettwaxdem s usemres n tsuddest (a+b)(c+d) = ac + ad + bc + bd  s usmekti bell  ii^2 = -1 
### <big>Taẓunt <chem>z1 / z2</chem></big>
Ma yella   Z{\scriptstyle\text{1}} = a + bi  d Z{\scriptstyle\text{2}} = c + di  sin n yimḍanen isemlalen (Z{\scriptstyle\text{2}} \neq 0 ):


Z1 / Z2 = (a + bi) / ( c+di)  = [(a+bi)(c-di))]/[(c+di)(c-di)] = [(ac + bd) + (bc-ad)i]/(c^2 + d^2) 

Taẓunt n sin yimḍanen isemlalen tettwaxdem s usemres n uneftay n umḍan asemlal deg umetṭerf d wadda, sakin s ubeṭṭu n ufaris ɣef tigget.

### <big>Aneftay  <chem>\bar{Z}</chem></big>

Aneftay n umḍan asemlal Z = a + bi          id        \bar{Z} = a - bi. 

Aneftay yesɛa amur ilaw yecrek d umḍan amenzu, ma d amur asugnan yettuɣal d amgal-is.

## '''Amezruy'''
Tikti n umḍan asemlal d tin yettwaxedmen s kra n yimekdiyen n tusnakt. Deg tazwara, imussnawen xeddmen fell-as akken ad afen tifrat i tenkarin ur nesɛi ara tifrat deg umagrad n yimḍanen ilaw. Amedya amezwaru n usemres n yiwet n tigget tasugnant i uselkin n tenkarit n uswir wis sin yettwaf-d deg yixeddimen n Gerolamo Cardano deg 1545. Deg useggas n 1572, Rafael Bombelli yesnulfa-d ilugan n tmahelt ɣef yimḍanen isemlalen deg udlis-ines L'Algebra. Deg lqern wis 18, imussnawen am Abraham de Moivre d Leonhard Euler ssnernin aselmed n yimḍanen isemlalen, ladɣa s usemres-nsen deg tmahilin tisnumkanin. Dɣa d Euler i d-yefkan azamul i i tigget tasugnant. 

* Deg lqern wis 16, imussnawen n tusnakt bdan ttnadint ɣef tifrat n tenkarin n uswir wis kraḍ d wis kuẓ.
* Girolamo Cardano (1501-1576) d win i d-yewwin tikti tamezwarut n yimḍanen isemlalen.
* Rafael Bombelli (1526-1572) yerra-d lwelha ɣer wamek ara nesemres imḍanen isemlalen deg tenkarin.
* Leonhard Euler (1707-1783) d netta i d-yefkan isem i d tigget tasugnant.
* Carl Friedrich Gauss (1777-1855) yessebded tazrawt tamatant n yimḍanen isemlalen.