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.

centré|vignette|Kra n imḍanen iselmalen deg uɣawas

 Tanekda 

 Talɣa tajebrant 
Z = a + bi Anda:

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

 Talɣa tasfaylut 
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: 

  talɣa tasusmirt s usumres n tanfalit n Euler
 z = (r, θ) Talɣa tasfaylut
 z = r (cosθ + i sinθ) = r cis (θ)  (s tuggza cis Alan Sultan et Alice F. Artzt, The mathematics that every secondary school math teacher needs to know, Studies in Mathematical Thinking and Learning, Taylor & Francis, 2010, p. 326)

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

akken ad nsiḍen θ sef talɣa taljebrit a + bi, nezmer ad nessemres tiwuriwin arccos, arcsin neɣ arctan : 

 Timhal d wassaɣen 

 Timernit 
Ma yella  d  sin n yimḍanen isemlalen:

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

 Afaris 
Ma yella  d  sin n yimḍanen isemlalen:

Afaris n sin yimḍanen isemlalen yettwaxdem s usemres n tsuddest  s usmekti bell  i
 Taẓunt z1 / z2 
Ma yella    d  sin n yimḍanen isemlalen ():

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

 Aneftay  \bar{Z} 
droite|vignette|Tagensest tudlift n uneftay n umḍan asemlal

Aneftay n umḍan asemlal          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.

+TuggzaAlnamekAmeskarAzemz℞. m. 15amḍan ur yezmir ad yili acku amkuẓ-is (tazmert tis 2 ) tegda-d Cardan1545"tasugnant"Yal tasmekta yesɛan aẓar amkuẓ n umḍan azedranDescartes1637Euler1777azal amagdez razal amagdez n umḍan asemlal a +bi id Argand1806Azal amagdez n ZKarl Weierstrassanaftay \bar{Z}Anaftay n   id   Cauchy1821amḍan asemlal a + ibGauss1831asugnan kan ihi "Norme"amkuẓ n wazal amagdezTiɣiret Tiɣmert gar amaway n 1 akked n ZCauchy1838Awṣilawṣil w wagaz A n yimsidag  id umḍan asemlal a+bi1847

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