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.

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:

$Z=r*\exp(\Theta *i)$ talɣa tasusmirt s usumres n tanfalit n Euler
z = (r, θ) Talɣa tasfaylut
z = r (cosθ + i sinθ) = r cis (θ) (s tuggza cis^[1])

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 {a}{r}}\right)&{\text{ma }}b\geq 0\\-\arccos \left({\tfrac {a}{r}}\right)&{\text{ma }}b<0,\end{cases}}\quad \theta ={\begin{cases}\arcsin \left({\tfrac {b}{r}}\right)&{\text{ma}}a\geq 0\\\pi -\arcsin \left({\tfrac {b}{r}}\right)&{\text{ma }}a<0\end{cases}}\quad {\text{ed}}\quad \theta ={\begin{cases}\arctan \left({\tfrac {b}{a}}\right)&{\text{ma}}a>0\\\arctan \left({\tfrac {b}{a}}\right)+\pi &{\text{ma}}a<0\\\mathrm {sgn} (b){\tfrac {\pi }{2}}&{\text{ma }}a=0.\end{cases}}

Timhal d wassaɣen

Timernit

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.

Afaris

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 i $i^{2}=-1$

Taẓunt ${\ce {z1 / z2}}$

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.

Aneftay ${\ce {\bar {Z}}}$

Tagensest tudlift n uneftay n umḍan asemlal

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

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.


Tuggza	Alnamek	Ameskar	Azemz
℞. m. 15	amḍan ur yezmir ad yili acku amkuẓ-is (tazmert tis 2 ) tegda-d $-15$	Cardan	1545
"tasugnant"	Yal tasmekta yesɛan aẓar amkuẓ n umḍan azedran	Descartes	1637
$i$	${\sqrt {-1}}$	Euler	1777
azal amagdez r	azal amagdez n umḍan asemlal a +bi id ${\sqrt {a^{2}+b^{2}}}$	Argand	1806
$\|z\|$	Azal amagdez n Z	Karl Weierstrass
anaftay ${\ce {\bar {Z}}}$	Anaftay n $a+bi$ id $a-bi$	Cauchy	1821
amḍan asemlal $Z$	a + ib	Gauss	1831
asugnan kan	$ib$ ihi $a=0$
"Norme"	amkuẓ n wazal amagdez
Tiɣiret $\theta$	Tiɣmert gar amaway n 1 akked n Z	Cauchy	1838
Awṣil	awṣil w wagaz A n yimsidag $(a,b)$ id umḍan asemlal a+bi	Cauchy	1847

↑ (en) 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

[1] (en) 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

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