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:

Tanekda

Talɣa tajebrant

Z = a + bi Anda:

Talɣa tasfaylut

Z = r(cos(ϕ) + i sin(ϕ)) Anda:

Tikwal amḍan asemlal yettwaru deg yiwet n talɣiwin igi:

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

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

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

Timhal d wassaɣen

Timernit

Ma yella Z 1 = a + b i {\displaystyle Z{\scriptstyle {\text{1}}}=a+bi} d Z 2 = c + d i {\displaystyle Z{\scriptstyle {\text{2}}}=c+di} sin n yimḍanen isemlalen:

Z 1 + Z 2 = ( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i {\displaystyle 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 1 = a + b i {\displaystyle Z{\scriptstyle {\text{1}}}=a+bi} d Z 2 = c + d i {\displaystyle Z{\scriptstyle {\text{2}}}=c+di} sin n yimḍanen isemlalen:

Z 1 ∗ Z 2 = ( a + b i ) ∗ ( c + d i ) = ( a c − b d ) + ( a d + b c ) i {\displaystyle 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 ) = a c + a d + b c + b d {\displaystyle (a+b)(c+d)=ac+ad+bc+bd} s usmekti bell i i 2 = − 1 {\displaystyle i^{2}=-1}

Taẓunt z 1 / z 2 {\displaystyle {\ce {z1 / z2}}}

Ma yella Z 1 = a + b i {\displaystyle Z{\scriptstyle {\text{1}}}=a+bi} d Z 2 = c + d i {\displaystyle Z{\scriptstyle {\text{2}}}=c+di} sin n yimḍanen isemlalen ( Z 2 ≠ 0 {\displaystyle Z{\scriptstyle {\text{2}}}\neq 0} ):

Z 1 / Z 2 = ( a + b i ) / ( c + d i ) = [ ( a + b i ) ( c − d i ) ) ] / [ ( c + d i ) ( c − d i ) ] = [ ( a c + b d ) + ( b c − a d ) i ] / ( c 2 + d 2 ) {\displaystyle 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 Z ¯ {\displaystyle {\ce {\bar {Z}}}}

Aneftay n umḍan asemlal Z = a + b i {\displaystyle Z=a+bi} id Z ¯ = a − bi {\displaystyle {\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.

a 2 + b 2 {\displaystyle {\sqrt {a^{2}+b^{2}}}}