Kompleksni brojevi su u prvobitnoj predstavi izrazi oblika , gde su a i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ realni brojevi, $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" alt="{displaystyle i}" width="345.5" height="936.9">$ jedan simbol.

Sabiranje, množenje i deljenje kompleksnih brojeva definiše se formulama:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5c00b65128344fc70ab38864c98bc7010b736e" alt="{displaystyle (a+ib)+(x+iy)=(a+x)+(b+y)i,}" width="474" height="400">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10a5737edf7428aca4ee56193dda06e37e4a4ac" alt="{displaystyle (a+ib)(x+iy)=(ax-by)+(ay+bx)i,}" width="474" height="400">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f89a7d8569cfa015fc3ba95abab21a8880400804" alt="{displaystyle {frac {a+bi}{x+yi}}={frac {ax+by}{x^{2}+y^{2}}}+{frac {bx-ay}{x^{2}+y^{2}}}cdot i}" width="474" height="400">$

U kompleksnom broju $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef52538a0077e8dff45f937a232364d68ca6124" alt="{displaystyle z=a+bi}" width="474" height="400">$ broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ se naziva realni deo, piše se $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e1590f9ee22e1e921596365e6828e090a00fd6" alt="{displaystyle a=Re(z)}" width="474" height="400">$ , a broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ je imaginarni deo, piše se $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/779eb8e1a1e6d897313da6f8c013f84fbd48fefb" alt="{displaystyle b=Im(z)}" width="474" height="400">$ .

Kompleksan broj čiji je realni deo jednak nuli naziva se čisto imaginarni broj.

Realni brojevi predstavljaju poseban slučaj kompleksnih brojeva (kad je koeficijent uz $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" alt="{displaystyle i}" width="345.5" height="936.9">$ jednak nuli). Iako se kompleksnim brojevima ne izražavaju količine, kao što je to slučaj s realnim brojevima, njihovo uvođenje koristi u rešavanju problema sastavljenih u terminima realnih brojeva, na primer, problema o prolazu struje kroz provodnik, o profilu krila aviona (koristeći funkcije Žukovskog), itd.

Nije manje važna ni primena kompleksnih brojeva na čisto matematičke probleme. Tako na primer, za nalaženje korena kubne jednačine potrebne su operacije s kompleksnim brojevima. Istorijski, kompleksni brojevi su uvedeni radi rešavanja kvadratne jednačine. Činjenica da kompleksni brojevi ne izražavaju veličine dala je povod za idealističko tumačenje kompleksnih brojeva (G. Lajbnic). velika zasluga u smislu materijalističkog tumačenja kompleksnih brojeva pripada L. Ojleru. Kompleksni broj se aksiomatski definiše kao uređen par realnih brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" alt="{displaystyle (a,b)}" width="474" height="400">$ . Formule sabiranja, množenja, deljenja se postuliraju ovako:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfe40eaf225235b9a731e9677b34cf6a50716a7" alt="{displaystyle (a,b)+(x,y)=(a+x,b+y),}" width="474" height="400">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1549621cab7fa34847bf81f36552dca93f32db5d" alt="{displaystyle (a,b)cdot (x,y)=(ax-by,ay+bx)}" width="474" height="400">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f304cef1ca421c4ecb375d37fdc3453da50a30b9" alt="{displaystyle {frac {(a,b)}{(x,y)}}=({frac {ax+by}{x^{2}+y^{2}}},{frac {bx-ay}{x^{2}+y^{2}}})}" width="474" height="400">$ .

Par $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce2bfd1f48989c5da86e37af63bf1cb226cc43c" alt="{displaystyle (0;1)}" width="474" height="400">$ se naziva imaginarna jedinica i označava simbolom $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" alt="{displaystyle i}" width="345.5" height="936.9">$ . Iz poslednjih formula proizilazi da je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" alt="{displaystyle i^{2}=-1}" width="3412.5" height="1223.9">$ . Operacije sa kompleksnim brojevima zadovoljavaju obične zakone komutativnosti, distributivnosti i asocijativnosti (kao i u slučaju realnih brojeva). Međutim, operacije s kompleksnim brojevima pod radikalima (korenima) donekle se razlikuju od analognih operacija s realnim brojevima. Tako je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acfc5177c5118e5db754036017311ee9c6882cfa" alt="{displaystyle -1=i^{2}={sqrt {-1}}cdot {sqrt {-1}}not ={sqrt {(-1)(-1)}}=1}" width="474" height="400">$ .Definicija

Definiciju kompleksnih brojeva kao uređenih parova dao je William R. Hamilton , irski matematičar (1805– 1865.) Ta se definicija temelji samo na osobini realnih brojeva, čime se izbjegava donekle nerazjašnjeni pojam broja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" alt="{displaystyle {sqrt {-1}}}" width="474" height="400">$ .

S druge strane, zapis oblika $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" alt="{displaystyle z=x+yi}" width="4441" height="1080.4">$ pogodniji je za računanje.

Oba oblika kompleksnog broja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" alt="{displaystyle z=x+yi}" width="4441" height="1080.4">$ i

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0831690f111c4808ec9ebfc50f9ffc0baa824db2" alt="{displaystyle z=(x,y)}" width="474" height="400">$ potpuno su ekvivalentna.

Skup kompleksnih brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" alt="{displaystyle mathbb {C} }" width="474" height="400">$ je skup svih brojeva oblika $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" alt="{displaystyle z=x+iy}" width="474" height="400">$ , gdje su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" alt="{displaystyle x,yin mathbb {R} }" width="474" height="400">$ .

Posebno je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73eca76ba2219e7e672cec65c810fff0ef1c33cb" alt="{displaystyle 0=0+i0}" width="474" height="400">$ .

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22426dd3e2e2a13791fc000ed593173c8b2f3544" alt="{displaystyle x=mathrm {Re} (z)}" width="474" height="400">$ je realni dio kompleksnog broja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" alt="{displaystyle z}" width="468.5" height="721.6">$ ,

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00055d39ddb316e1428f6ffcf2f1f645d4d1ae17" alt="{displaystyle y=mathrm {Im} z}" width="474" height="400">$ je imaginarni dio kompleksnog broja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" alt="{displaystyle z}" width="468.5" height="721.6">$ .

Algebarski oblik kompleksnog broja je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" alt="{displaystyle z=x+iy}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1fd3534163cb031d88b529c837e5747ee40fc" alt="{displaystyle x,yin mathbb {R} }" width="474" height="400">$

Trigonometrijski oblik kompleksnog broja je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0142894c622b82b5e24674171897be64ed3a58e2" alt="{displaystyle z=r(costheta +isintheta ),rgeq 0,theta in mathbb {R} }" width="474" height="400">$

pri čemu je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0464ac0638a3562dedcb8072f316fa7b3c8130d" alt="{displaystyle r=mid zmid }" width="474" height="400">$ modul

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf8938ea6e8e7dc9101ad810d6ede24e1f22fb9" alt="{displaystyle theta =ARgz}" width="474" height="400">$ argument

Eksponencijalni oblik kompleksnog broja je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730b328ed186a34305d96cc42d683b13bd578684" alt="{displaystyle z=r^{itheta }}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5e22684924225c2776564c74781fc68426c040" alt="{displaystyle rgeq 0,theta in mathbb {R} }" width="474" height="400">$

pri čemu je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0464ac0638a3562dedcb8072f316fa7b3c8130d" alt="{displaystyle r=mid zmid }" width="474" height="400">$ modul

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf8938ea6e8e7dc9101ad810d6ede24e1f22fb9" alt="{displaystyle theta =ARgz}" width="474" height="400">$ argument

Dva kompleksna broja su jednaka ako su im jednaki realni i imaginarni dijelovi.

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4278537b91b131f1bd24a0a9bf277bc0f357ad43" alt="{displaystyle z_{1}=z_{2},,leftrightarrow ,,(operatorname {Re} (z_{1})=operatorname {Re} (z_{2}),land ,operatorname {Im} (z_{1})=operatorname {Im} (z_{2})).}" width="474" height="400">$

Konjugirano kompleksni broj broja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" alt="{displaystyle z=x+iy}" width="474" height="400">$ je broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f3fcfb169c13babe58e60f75cbc06c185193dfa" alt="{displaystyle {bar {z}}=x-iy}" width="474" height="400">$ .

Modul ili apsolutna vrijednost kompleksnog broja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" alt="{displaystyle z}" width="468.5" height="721.6">$ je nenegativni realni broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d490417c03e6f7414c1884df3993bb45fd675d" alt="{displaystyle r=vert zvert ={sqrt {x^{2}+y^{2}}}}" width="474" height="400">$ .

Osobine sabiranja kompleksnih brojeva

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c032d4b9bc910954eaf98f3f8c0c9befa90dad2b" alt="{displaystyle (z_{1}+z_{2}=z_{2}+z_{1}}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7afb185f7c3bbbfb0676913e85e0b3cacb0bb9e8" alt="{displaystyle forall z_{1},z_{2}in mathbb {C} }" width="474" height="400">$ komutativnost sabiranja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0937d87046bfef3fdcc36bb91f4fc3d0259a5d81" alt="{displaystyle z_{1}+(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3},}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd7b986c57d136f05d7145b9d486d6ca46574c6" alt="{displaystyle forall z_{1},z_{2},z_{3}in mathbb {C} }" width="6150.6" height="1080.4">$ asocijativnost sabiranja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b080113f99cefcb8ef0a7383d26a3e688cd2063" alt="{displaystyle exists 0in mathbb {C} z+0=z}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15559442f85f771e939fda69abd74d170efffce" alt="{displaystyle forall zin mathbb {C} }" width="474" height="400">$ neutralni element 0(nula) za sabiranje

Kompleksni broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d8c9408d1d1ba50a5f0b02c96f367299bae0a0" alt="{displaystyle 0=(0,0)=0+0i}" width="474" height="400">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66876c24c5f3faf10d9a696ab598a3add324fe" alt="{displaystyle (forall zin mathbb {C} )(exists (-z)in mathbb {C} z+(-z)=0}" width="474" height="400">$ postojanje inverznog elemanta.

Kompleksni broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec170d46240e02b145b691a87759cc2246d5443" alt="{displaystyle -z=(-x,-y)=-x-yi}" width="474" height="400">$

Osobine množenja kompleksnih brojeva

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/070866aa46b1251f45742c4c87f4d912de102c34" alt="{displaystyle (z_{1}*z_{2}=z_{2}*z_{1}}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7afb185f7c3bbbfb0676913e85e0b3cacb0bb9e8" alt="{displaystyle forall z_{1},z_{2}in mathbb {C} }" width="474" height="400">$ komutativnost množenja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f2fd3bff554125ef1b62b5962ba7f07ce2ba1c" alt="{displaystyle z_{1}*(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3}}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd7b986c57d136f05d7145b9d486d6ca46574c6" alt="{displaystyle forall z_{1},z_{2},z_{3}in mathbb {C} }" width="6150.6" height="1080.4">$ asocijativnost množenja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257ca922e8d052f1642c8b2f765640624053c0ca" alt="{displaystyle exists 1in mathbb {C} z*1=z}" width="474" height="400">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f15559442f85f771e939fda69abd74d170efffce" alt="{displaystyle forall zin mathbb {C} }" width="474" height="400">$ neutralni element $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" alt="{displaystyle 1}" width="500.5" height="936.9">$ za množenje

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b8238e58481c881e331396678c19af75a571ce" alt="{displaystyle forall zin mathbb {C} )(zneq 0)(exists z'in mathbb {C} z*(-z)=1}" width="474" height="400">$ postojanje reciproćnog elemanta

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7ebf33eef8b53c0e44809c815bfeac5d1b6285" alt="{displaystyle z_{1}*(z_{2}+z_{3})=z_{1}*z_{2}+z_{1}*z_{3}}" width="13829.6" height="1223.9">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd7b986c57d136f05d7145b9d486d6ca46574c6" alt="{displaystyle forall z_{1},z_{2},z_{3}in mathbb {C} }" width="6150.6" height="1080.4">$ distributivnost množenja u odnosu na sabiranje

Realan proizvod dva kompleksna broja

U skupu kompleksnih brojeva skalarnom proizvodu vektora odgovara pojam realnog proizvoda kompleksnih brojeva koji je skalarni proizvod vektora koji su određeni kompleksnim brojevima koji se množe. Definicija

Realan proizvod kompleksnih brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ , u oznaci $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f270b24a930a6546b42f355ad905d2d7a26d4b3" alt="{displaystyle acirc b}" width="1903.9" height="936.9">$ ,je realan broj određen kao

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/289797253a113157c32fc37a50e59ef3107c1a92" alt="{displaystyle acirc b={frac {1}{2}}({overline {a}}b+a{overline {b}})}" width="8117.4" height="2228.5">$

Neka su A i B tačke određene kompleksnim brojevima $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4d15e3651c592dd3d1594f0fa9fdd390ac702e" alt="{displaystyle a=mid amid (cosvarphi +isinvarphi )}" width="9966.1" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc582fd89618e7a4a6af0a06cf0875789e0fd7a2" alt="{displaystyle b=mid bmid (cospsi +isinpsi )}" width="9760.1" height="1223.9">$ Lako je proveriti da je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95d415026e20b7c68f791f7a48225e82557af693" alt="{displaystyle acirc b=mid amid mid bmid (cosvarphi +isinpsi )=mid OAmid mid ABmid cos{widehat {AOB}}}" width="23124.7" height="1582.7">$

Osobine realnog proizvoda dva kompleksna broja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f020e0e674a4b57e733dec73b4c6db166833f7" alt="{displaystyle acirc a=mid amid ^{2}}" width="4878.4" height="1367.4">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aafdf8dde092f892242136e6e23df710a35c703" alt="{displaystyle acirc b=bcirc a}" width="5141.9" height="936.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebd84915cf1bee1a39a59afd234cad589d0ebc8" alt="{displaystyle {overline {acirc b}}=acirc b}" width="5191.4" height="1295.7">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f05d7235f56899f92c07c793bba380cae0142" alt="{displaystyle (alpha a)circ b=alpha (acirc b)=acirc (alpha b)}" width="12638.4" height="1223.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2428bef294fe82af630fb0dd55c44c608e8b2376" alt="{displaystyle (az))bz)=mid zmid ^{2}(acirc b)}" width="9506" height="1367.4">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd4ae3a95365dcee3c0297b26e9549ed2fd6ade" alt="{displaystyle acirc b=0<=>OAperp OB}" width="11000.6" height="936.9">$ (za tačke A i B kompleksne ravni određene kompleksnim brojevima $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ )

Realan proizvod kompleksnih brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ jednak je potenciji koordinantnog početka $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" alt="{displaystyle O}" width="763.5" height="936.9">$ kompleksne ravni u odnosu na krug čiji je prečnik $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" alt="{displaystyle AB}" width="1510" height="936.9">$ , gdje su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" alt="{displaystyle A}" width="750.5" height="936.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" alt="{displaystyle B}" width="759.5" height="936.9">$ tačke kompleksne ravni određene kompleksnim brojevima $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ .

Tačka $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" alt="{displaystyle M}" width="1051.5" height="936.9">$ je sredina duži AB određena kompleksnim brojem $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1325e0aa44cdaf4b2e765a44c7109e6b9ed74e77" alt="{displaystyle {frac {a+b}{2}}}" width="2541.9" height="2300.3">$ , potencija tačke $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" alt="{displaystyle O}" width="763.5" height="936.9">$ u odnosu na krug sa središtem u tački $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" alt="{displaystyle M}" width="1051.5" height="936.9">$ i poluprečnikom

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8514d8a2b6ca087e3ad5182380ad6c26c028295a" alt="{displaystyle r={frac {a-b}{2}}={frac {mid a-bmid }{2}}}" width="8760.5" height="2443.8">$ jednaka je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d9f0b753108785b1aaccbfba81b439dadc406f" alt="{displaystyle OM^{2}-r^{2}=mid {frac {a+b}{2}}mid -mid {frac {a-b}{2}}mid ={frac {(a+b)({overline {a}}+{overline {b}}}{4}}-{frac {(a-b)({overline {a}}-{overline {b}})}{4}}=acirc b}" width="32010.4" height="2730.8">$

Neka su tačke $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" alt="{displaystyle A}" width="750.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" alt="{displaystyle B}" width="759.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" alt="{displaystyle C}" width="760.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" alt="{displaystyle D}" width="828.5" height="936.9">$ taačke kompleksne ravni određene kompleksnim brojevima $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" alt="{displaystyle c}" width="433.5" height="721.6">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" alt="{displaystyle d}" width="523.5" height="936.9">$ . Tada su sledeća tvrđenja ekvivalentna:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377c7354b1305512ddca0fd2dee141324f7a76fa" alt="{displaystyle ABperp CD}" width="4433.1" height="936.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2355447a8948184685615fd016d089fbcb7ad1" alt="{displaystyle (a+b)circ (c+d)=0}" width="8699.4" height="1223.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f3b7b7991cb43e398955dd399f7dc0d1785af1" alt="{displaystyle {frac {b-a}{d-c}}in imathbb {R} {begin{Bmatrix}0end{Bmatrix}}}" width="6908" height="2443.8">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d4423c918b3541793383b2369e8dbf6953d59e" alt="{displaystyle Re({frac {b-a}{d-c}})=0}" width="6381.5" height="2443.8">$

Središte kružnice opisane oko trougla $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" alt="{displaystyle ABC}" width="2270.5" height="936.9">$ nalazi se u koordinantnom početku kompleksne ravni. Ako su tjemena $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" alt="{displaystyle A}" width="750.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" alt="{displaystyle B}" width="759.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" alt="{displaystyle C}" width="760.5" height="936.9">$ trougla $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55b44cfd965fbdc7a328d5db8a35a619db0971" alt="{displaystyle ABC}" width="2270.5" height="936.9">$ određena kompleksnim brojevima $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" alt="{displaystyle c}" width="433.5" height="721.6">$ respektivno, tada je ortocentar $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" alt="{displaystyle H}" width="888.5" height="936.9">$ tog trougla određen kompleksnim brojem $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a3f719f3d0aac4cf1904b445d04652f615c08f" alt="{displaystyle h=a+b+c}" width="5748.9" height="1008.6">$ .

Kompleksan proizvod dva kompleksna broja

Kompleksan proizvod dva kompleksna broja je analogan vektorskom proizvodu vektora.

Definicija

Kompleksan broj

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7337254fdd298766eca1b6f2e4e7a86695ab450" alt="{displaystyle atimes b={frac {{overline {a}}b-a{overline {b}}}{2}}}" width="7115.9" height="2587.3">$ nazivamo kompleksnim proizvodom kompleksnih brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ .

Neka su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" alt="{displaystyle A}" width="750.5" height="936.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" alt="{displaystyle B}" width="759.5" height="936.9">$ tačke određene kompleksnim brojevima $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4d15e3651c592dd3d1594f0fa9fdd390ac702e" alt="{displaystyle a=mid amid (cosvarphi +isinvarphi )}" width="9966.1" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df738f785a440c8745f6503a5a8211d45798a8ab" alt="{displaystyle a=mid bmid (cospsi +isinpsi )}" width="9860.1" height="1223.9">$ Lako je provjeriti da je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76bd1c387e92420784971bb1e47b9b9e50440620" alt="{displaystyle mid atimes bmid =mid amid mid bmid sin(varphi -psi )=mid OAmid mid ABmid sin{widehat {AOB}}=2P_{AOB}}" width="26437.4" height="1582.7">$

Neka su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" alt="{displaystyle c}" width="433.5" height="721.6">$ kompleksni brojevi. Tada kompleksan proizvod dva kompleksna broja ima sledeće osobine

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66691e5fc0226fd4eb76f458e4ca9bca9ed62756" alt="{displaystyle {overline {atimes b}}=-atimes b}" width="6525.9" height="1367.4">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dccef36d56bb4279a618efa9e40ee864a4c1e3ba" alt="{displaystyle atimes b=0<=>a=0lor b=0lor a=lambda b}" width="16636.1" height="936.9">$ gdje je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f43c1022dd80d193e60b06042c409263f508a2c8" alt="{displaystyle lambda in mathbb {R} {begin{Bmatrix}0end{Bmatrix}}}" width="4604.1" height="1223.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1421a30ec49b8af13c476bde94f1d19516ce9a9b" alt="{displaystyle atimes b=-btimes a}" width="6476.4" height="1008.6">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/954a30e7bb8b2bf942977e706d2a807247c122e7" alt="{displaystyle alpha (atimes b)=(alpha a)times b=atimes (alpha b)}" width="13472.4" height="1223.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d826309c98a2f2572fb7e1d6ace61ac3fe746b" alt="{displaystyle forall alpha in mathbb {R} }" width="3142.6" height="936.9">$ )

Ako su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24914ddcb156090cc3b746d9524797c4af7bad6" alt="{displaystyle A(a)}" width="2059" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13eaa65d1d08a38a36c81b12382eaf72f36826e1" alt="{displaystyle B(b)}" width="1968" height="1223.9">$ dvije različite tačke različite od $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79d3a3005d4429de8753d6f79d1ed5ce33a696c6" alt="{displaystyle O(0)}" width="2043" height="1223.9">$ , tada je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c7e0f7d652fcc1beaf0b176f7555a791316112" alt="{displaystyle atimes b=0}" width="4016.5" height="936.9">$ onda i samo onda ako su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" alt="{displaystyle O}" width="763.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" alt="{displaystyle A}" width="750.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" alt="{displaystyle B}" width="759.5" height="936.9">$ kolinearne tačke.

Neka su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e940149401f82fc27e02e293e0639467a4a86d1" alt="{displaystyle A(a}" width="1669.5" height="1223.9">$ ) i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3001fb4f6f071d0b14f37ab739c0b114b8c0dc3a" alt="{displaystyle B(b}" width="1578.5" height="1223.9">$ ) dvije različite tačke u kompleksnoj ravni različite od koordinantnog početka. Kompleksan proizvod brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" alt="{displaystyle a}" width="529.5" height="721.6">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" alt="{displaystyle b}" width="429.5" height="936.9">$ ima sljedeći geometrijski smisao

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c47a691e14681056599b6d7c9dbcb848f1145e4" alt="{displaystyle atimes b={begin{cases}2iP_{AOB} za trougao AOB pozitivno orijentisan\-2iP_{AOB} za trougao AOB negativno orijentisanend{cases}}}" width="25876.7" height="2659.1">$ Neka su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24914ddcb156090cc3b746d9524797c4af7bad6" alt="{displaystyle A(a)}" width="2059" height="1223.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13eaa65d1d08a38a36c81b12382eaf72f36826e1" alt="{displaystyle B(b)}" width="1968" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0977ae3174fb0e4b336b2ee0b1170a378607443b" alt="{displaystyle C(c)}" width="1973" height="1223.9">$ tri različite tačke u kompleksnoj ravni. Tada je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d20cc46745d7ca9938ab6d989c5f83f65b87f0" alt="{displaystyle P_{ABC}={begin{cases}{frac {1}{2}}(atimes b+btimes c+ctimes a) ako je ABC pozitivno orjentisan\{frac {1}{2}}(atimes b+btimes c+ctimes a) ako je ABC negativno orjentisanend{cases}}}" width="30219.2" height="3233.2">$

Neka su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24914ddcb156090cc3b746d9524797c4af7bad6" alt="{displaystyle A(a)}" width="2059" height="1223.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13eaa65d1d08a38a36c81b12382eaf72f36826e1" alt="{displaystyle B(b)}" width="1968" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0977ae3174fb0e4b336b2ee0b1170a378607443b" alt="{displaystyle C(c)}" width="1973" height="1223.9">$ tri različite tačke u kompleksnoj ravni. Tada su sljedeća tvrđenja ekvivalentna

Tačke $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" alt="{displaystyle A}" width="750.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" alt="{displaystyle B}" width="759.5" height="936.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" alt="{displaystyle C}" width="760.5" height="936.9">$ su kolinearne
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/865ce5c4247f499a468052cbe4b483eaf7a0a4a7" alt="{displaystyle (b-a)times (c-a)=0}" width="8983.4" height="1223.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef646147172bf26b5b6b32a10d095e7b9fd286b" alt="{displaystyle atimes b+btimes c+ctimes a=0}" width="10734.3" height="1008.6">$

Neka su $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24914ddcb156090cc3b746d9524797c4af7bad6" alt="{displaystyle A(a)}" width="2059" height="1223.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13eaa65d1d08a38a36c81b12382eaf72f36826e1" alt="{displaystyle B(b)}" width="1968" height="1223.9">$ , $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0977ae3174fb0e4b336b2ee0b1170a378607443b" alt="{displaystyle C(c)}" width="1973" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/febc920410b8ca7d0e942ff7016b6d83e57e5528" alt="{displaystyle D(d)}" width="2131" height="1223.9">$ četiri tačke od kojih nikoje tri nisu kolinearne. Tada je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4604e0c72f8efbce903322cf4cb947e89e11257e" alt="{displaystyle ABparallel CD}" width="4155.1" height="1223.9">$ onda i samo onda ako je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d9a1b3d59c2d6b17511c1d85da57fa80b0aa19" alt="{displaystyle (b-a)times (d-c)=0}" width="8977.4" height="1223.9">$

Dijeljenje kompleksnih brojeva

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d957a1de2bf79c7bf60b9af6981885541eb0fb1" alt="{displaystyle displaystyle {frac {z_{1}}{z_{2}}}displaystyle ={frac {x_{1}+iy_{1}}{x_{2}+iy_{2}}}cdot {frac {x_{2}-iy_{2}}{x_{2}-iy_{2}}}displaystyle ={frac {x_{1}x_{2}+y_{1}y_{2}}{x_{2}^{2}+y_{2}^{2}}}+i{frac {y_{1}x_{2}-x_{1}y_{2}}{x_{2}^{2}+y_{2}^{2}}},quad {textrm {za}}quad z_{2}neq 0}" width="31230.7" height="2730.8">$

U svakom skupu brojeva dijeljenje se definiše kao množenje inverznim elementom. Uvjerimo se da za

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3153dc25bf8da8c350e162c3479da2c5b2cab385" alt="{displaystyle forall zin mathbb {C} )(zneq 0)exists z'in mathbb {C} }" width="9708.4" height="1295.7">$

Neka je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9539ff667699a7f1777e6eb87b066d40937486e" alt="{displaystyle z=x+yineq 0}" width="6275.6" height="1152.1">$ bilo koji. Onda je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0588b9cdb5d3c67dcadb01c675cdec31d6aa64ac" alt="{displaystyle x^{2}+y^{2}neq 0}" width="5037.5" height="1367.4">$ pa je dobro definisan broj

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac504b03625c86620520fdcb3ee209d0413e8d31" alt="{displaystyle z'={frac {x}{x^{2}+y^{2}}}+{frac {-y}{x^{2}+y^{2}}}i}" width="10792.6" height="2515.6">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c485ccc5cd0b462105f3d2a24160fd2b2d9adc" alt="{displaystyle {frac {1}{z}}={frac {bar {z}}{z{bar {z}}}}={frac {bar {z}}{x^{2}+y^{2}}}={frac {x}{x^{2}+y^{2}}}-{frac {y}{x^{2}+y^{2}}}i.}" width="18784.9" height="2515.6">$

imamo

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13cc4b7ca66e49c0055588c2d347fc7bb6e0a12d" alt="{displaystyle z'*z=z*z'=1}" width="7524" height="1080.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/368bc6c2891aeab47ae896ba577cbe30cd7fa959" alt="{displaystyle z'=z^{-1}={frac {1}{z}}}" width="5766.7" height="2228.5">$

Konjugovano kompleksni brojevi

Kompleksan broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f75db1c6a6bf009b415fbd53cea00ed5dbe699" alt="{displaystyle {overline {z}} =x-yi=r^{-itheta }}" width="7754.8" height="1295.7">$ nazivamo konjugovanim broju $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cb5d53af83eacb2c993c25a9ff8c6aaadcda1d" alt="{displaystyle z=x+yi=r^{itheta }}" width="6902.8" height="1295.7">$ .

Brojevi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" alt="{displaystyle z}" width="468.5" height="721.6">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64281d029a1d4bef9545644f01821c713f876f76" alt="{displaystyle {overline {z}}}" width="520" height="1008.6">$ čine par konjugovanik brojeva. Njihovim sabiranjem i oduzimanjem dobijamo

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875f77607ae6e3f5ae58a0e074496e0028e8a08b" alt="{displaystyle Rez={frac {1}{2}}(z+{overline {z}})}" width="6879.5" height="2228.5">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1d163e327b54b7d6c9b8617f3e8f55207a8562" alt="{displaystyle Imz={frac {1}{2i}}(z-{overline {z}})}" width="7382" height="2228.5">$

Lako se provjerava da vrijedi

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fc98456ab06986923239947d8de18d56e79b3d1" alt="{displaystyle {overline {z_{1}+z_{2}}}={overline {z_{1}}}+{overline {z_{2}}}}" width="7606.1" height="1295.7">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a24d8848b806f626abfc57784e88333b7d0a74" alt="{displaystyle {overline {z_{1}-z_{2}}}={overline {z_{1}}}-{overline {z_{2}}}}" width="7606.1" height="1295.7">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e3c9f9c57f8196298326e4796eb09a7aae3453" alt="{displaystyle {overline {z_{1}*z_{2}}}={overline {z_{1}}}*{overline {z_{2}}}}" width="7050.1" height="1223.9">$
$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7e164222b80ca114e3ec61c56ba24f09b6cc71" alt="{displaystyle {overline {({frac {z_{1}}{z_{2}}})}}={frac {overline {z_{1}}}{overline {z_{1}}}}}" width="4770.9" height="2659.1">$ ^[6]

Neka je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc21a98d2457388610e7e395cce83adeb66ab1be" alt="{displaystyle z=r(costheta +isintheta )=r cistheta }" width="12098.1" height="1223.9">$ trigonometrijski oblik kompleksnog broja. Tada je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab761b35d475cc808365d25fe639546cf81fc2ae" alt="{displaystyle z^{2}=z*z}" width="4139.4" height="1152.1">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d29cdfa281fef4ad9dada6ee0f788a9f9497342" alt="{displaystyle z^{2}=r cistheta *r cistheta =r^{2} cis(theta +theta )=r^{2} cis2theta }" width="19428.2" height="1367.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ec98e8fb0fbea3120635a53fe385a006006bd4" alt="{displaystyle z^{3}=r^{2} cis2theta *r cistheta =r^{3} cis(2theta +theta )=r^{3} cis3theta }" width="20883.1" height="1367.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775d80781d2fe500c9b1f432a80b9824131bd73e" alt="{displaystyle z^{4}=r^{3} cis3theta *r cistheta =r^{4} cis(3theta +theta )=r^{4} cis4theta }" width="20883.1" height="1367.4">$

Na ovaj način dobijamo opšti oblik De Moavrova teorema koji ima važnu ulogu u elektrotehnici

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53220af0147bbf2511d469bdb5e6f838d7a8e5d8" alt="{displaystyle z^{n}=r^{n} cisntheta }" width="5872.7" height="1008.6">$ ili

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be260f91432657b9b4d778a88f24469d90a828d3" alt="{displaystyle (costheta +isintheta )^{n}=cosntheta +isinntheta (nin Z)}" width="17787.6" height="1223.9">$

Stepenovanje kompleksnog broja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef56b44140f69838c91b28012fffae3abb0c125" alt="{displaystyle z^{n}=r^{n}(cosntheta +isinntheta )=r^{n}e^{intheta }}" width="14473.3" height="1367.4">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedc99ee9cd35f9c34b0f743f8c26b34359ca2fe" alt="{displaystyle nin N}" width="2712.1" height="936.9">$ .

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512f14ab77122f8a4f507a3c8da9e6700775cfde" alt="{displaystyle z^{m}z^{n}=z^{m+n}}" width="5684.5" height="1080.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c04e92a6ae3242544f7d70e36696efa42b44bb7" alt="{displaystyle (z_{1}z_{2})^{n}=(z_{1}z_{2})^{n}}" width="7618.9" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab2d48163b2737c04d684cdc48d22db95317dba" alt="{displaystyle (z^{m})^{n}=z^{mn}}" width="5443.6" height="1223.9">$

Korjenovanje kompleksnog broja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf0e60c4d3392df53c167b88044feef6328bd51" alt="{displaystyle {sqrt[{n}]{z}}={begin{Bmatrix}u_{0},u_{1}...u_{n}end{Bmatrix}}}" width="8891.2" height="1295.7">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedc99ee9cd35f9c34b0f743f8c26b34359ca2fe" alt="{displaystyle nin N}" width="2712.1" height="936.9">$

gdje je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1878321155867694396400b6e51365be407648" alt="{displaystyle u_{k}={sqrt[{n}]{r}}(cos{frac {sqrt[{n}]{r}}{n}}+isin{frac {theta +2kpi }{n}})}" width="14104.7" height="2515.6">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfe053d856f4ab667c8ae64781e99c232c2aba7" alt="{displaystyle k=0,1,...(n-1)}" width="8185.3" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133d35aa2d3bede7963901515ab63b4928ba11f2" alt="{displaystyle u_{k}={sqrt[{n}]{r}}e^{i(theta +2kpi )/2}}" width="7740.4" height="1510.9">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfe053d856f4ab667c8ae64781e99c232c2aba7" alt="{displaystyle k=0,1,...(n-1)}" width="8185.3" height="1223.9">$

Kvadratni korjen imaginarnog broja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0512e8a48647211acdbd5bd118ee725919d29f" alt="{displaystyle {sqrt {i}}={frac {1}{2}}{sqrt {2}}+i{frac {1}{2}}{sqrt {2}}={frac {sqrt {2}}{2}}(1+i).}" width="14625" height="2515.6">$

Ovaj rezultat možemo dobiti na sljedeći način

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/033378963eb0ede8d3e02ece898ffc1d445e008f" alt="{displaystyle i=(a+bi)^{2}!}" width="5439.9" height="1367.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929217ab8b3f537ed2d1c5ceee55e0ece39ca957" alt="{displaystyle i=a^{2}+2abi-b^{2}.!}" width="7996.8" height="1223.9">$

Dobijamo dvije jednačine

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e71e42155b014c9f91d8279c3c02f9324fc066" alt="{displaystyle {begin{cases}2ab=1!\a^{2}-b^{2}=0!end{cases}}}" width="5831.7" height="2659.1">$

čija su rješenja

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e76d88af4c5f1f7c3eb7fe1d6d5f5472e35b494" alt="{displaystyle a=b=pm {frac {1}{sqrt {2}}}.}" width="6378.1" height="2659.1">$

Izbor glavnog korjena daje

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee710b2f80516624c2569ecce4cdf7f7e5d6d0f" alt="{displaystyle a=b={frac {1}{sqrt {2}}}.}" width="5599.6" height="2659.1">$

Rezultat možemo dobiti pomoću Moavrova formule

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9953f8002884039314461306d40b398751c2e0e5" alt="{displaystyle i=cos left({frac {pi }{2}}right)+isin left({frac {pi }{2}}right)}" width="10240.7" height="2085">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0aa31b7a8a10455a9c6c7f8b98508b0e0bc104" alt="{displaystyle {begin{aligned}{sqrt {i}}&=left(cos left({frac {pi }{2}}right)+isin left({frac {pi }{2}}right)right)^{frac {1}{2}}\&=cos left({frac {pi }{4}}right)+isin left({frac {pi }{4}}right)\&={frac {1}{sqrt {2}}}+ileft({frac {1}{sqrt {2}}}right)={frac {1}{sqrt {2}}}(1+i).\end{aligned}}}" width="15587.2" height="7538.7">$

Apsolutna vrijednost argumenta

Apsolutna vrijednost (ili modul ili veličine) kompleksnog broja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639f77c05613faa61f43ee28a4d5ca7c35fffa38" alt="{displaystyle z=x+yi}" width="4441" height="1080.4">$ je

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24dfa385d25cce7ee5aa589ef3900045825957da" alt="{displaystyle textstyle r=|z|={sqrt {x^{2}+y^{2}}}.,}" width="8793.7" height="1510.9">

Kvadrat apsolutne vrijednosti je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a7ca7677ac0764990aeda2bb55a57275eeea0e" alt="{displaystyle textstyle |z|^{2}=z{bar {z}}=x^{2}+y^{2}.,}" width="8822.2" height="1439.2">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c19a62bfc1ee4ad9d5c4af8e98733608cc8a92b" alt="{displaystyle varphi =arg(z)={begin{cases}arctan({frac {y}{x}})&{mbox{if }}x>0\arctan({frac {y}{x}})+pi &{mbox{if }}x<0{mbox{ and }}ygeq 0\arctan({frac {y}{x}})-pi &{mbox{if }}x<0{mbox{ and }}y<0\{frac {pi }{2}}&{mbox{if }}x=0{mbox{ and }}y>0\-{frac {pi }{2}}&{mbox{if }}x=0{mbox{ and }}y<0\{mbox{indeterminate }}&{mbox{if }}x=0{mbox{ and }}y=0.end{cases}}}" width="22458.3" height="8399.8">$

Množenje i dijeljenje u polarnom obliku

Iz trigonometrijskih identiteta

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a4639e102eae5016667100fbd269f503f5a4e8" alt="{displaystyle cos(a)cos(b)-sin(a)sin(b)=cos(a+b)}" width="17362.8" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a317e0c298766a097f6c8792589e76a1458156e4" alt="{displaystyle cos(a)sin(b)+sin(a)cos(b)=sin(a+b)}" width="17252.8" height="1223.9">$

imamo

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf7c4db38d7a7165ef4cabdabd904cd30ccad95" alt="{displaystyle z_{1}z_{2}=r_{1}r_{2}(cos(varphi _{1}+varphi _{2})+isin(varphi _{1}+varphi _{2})).,}" width="18949.5" height="1223.9">$

Primjer

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c716ac35f98c3f67832554a88863765d5826238e" alt="{displaystyle (2+i)(3+i)=5+5i.,}" width="10044.6" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b6dbcdc93c63606ba43a9d5757af4bd9364eb3" alt="{displaystyle {frac {pi }{4}}=arctan {frac {1}{2}}+arctan {frac {1}{3}}}" width="11112.8" height="2228.5">$

Dijeljenje

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88491695b38b8b7c773d06d58e925c90e5e4aa08" alt="{displaystyle {frac {z_{1}}{z_{2}}}={frac {r_{1}}{r_{2}}}left(cos(varphi _{1}-varphi _{2})+isin(varphi _{1}-varphi _{2})right).}" width="18011.3" height="2156.8">$

Trigonometrijski oblik

Ponekad je kompleksne brojeve pogodno pisati u trigonometrijskom obliku:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7a8683d333ee55e56137e3a81882de44a06861" alt="{displaystyle a+bi=rho (cos phi +isin phi ),}" width="11155.1" height="1223.9">$ ,

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d15e943d2803b86b2f3fe74cfa4d3a8e83c0d41" alt="{displaystyle rho ={sqrt {a^{2}+b^{2}}}, phi =arctan {frac {b}{a}}}" width="12407.7" height="2300.3">$ , za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" alt="{displaystyle a>0}" width="2364.1" height="936.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/667bb2b83357c87ef01d5a915c782d94f84c52e7" alt="{displaystyle phi =pi +arctan {frac {b}{a}}}" width="7567.2" height="2300.3">$ za $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c" alt="{displaystyle a<0}" width="2364.1" height="936.9">$ ; kada je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" alt="{displaystyle a=0}" width="2364.1" height="936.9">$ onda je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a633e700dd0b05a98f93102f35421b3ccbd9fa" alt="{displaystyle phi ={frac {pi }{2}}}" width="2864.1" height="2013.3">$ , ako je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" alt="{displaystyle b>0}" width="2264.1" height="936.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27c327eced4f372f8635ce9d0cf33a2e9d44c3a4" alt="{displaystyle phi =-{frac {pi }{2}}}" width="3642.6" height="2013.3">$ , ako je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1319de18afa795d6489f49303614c84472f6d1ed" alt="{displaystyle b<0}" width="2264.1" height="936.9">$ . Broj $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" alt="{displaystyle rho }" width="517.5" height="936.9">$ se naziva moduo kompleksnog broja, a $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" alt="{displaystyle phi }" width="596.5" height="1080.4">$ je argument kompleksnog broja. Množiti kompleksne brojeve je veoma pogodno baš u ovom obliku: u množenju kompleksnih brojeva množe se njihovi moduli, a argumenti se sabiraju. Iz ovog pravila proizilazi Moavrova formula:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c39bcdb3049921f9675a2d3359028a70df03" alt="{displaystyle (cos phi +isin phi )^{n}=cos nphi +isin nphi ,}" width="15666.2" height="1223.9">$ .

Kompleksni brojevi se često predstavljaju vektorima u kompleksnoj ravni (slika dole). Geometrijski smisao brojeva $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2d69774f9f74dbf951bf32a94686cd5bdadc03" alt="{displaystyle a,b,rho ,phi }" width="3408.5" height="1152.1">$ vidi se na crtežu. U sabiranju kompleksnih brojeva njihovi vektori se sabiraju po pravilu paralelograma.

Dužina vektora $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" alt="{displaystyle rho }" width="517.5" height="936.9">$ je moduo, ili modul kompleksnog broja, a kao što se vidi na gornjoj slici, može se dobiti pomoću Pitagorine teoreme. Modul, intenzitet kompleksnog broja često označavamo kao apsolutnu vrednost, tj. udaljenost broja od ishodišta koordinatnog sistema: $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62600de890c2cbf065ce21e457233e3d5576e65" alt="{displaystyle |z|=rho ={sqrt {a^{2}+b^{2}}}}" width="8301.4" height="1510.9">$ .

Kompleksni brojevi u trigonometrijskom obliku su usko povezani s eksponencijalnom funkcijom imaginarnog argumenta. Važi sledeća Ojlerova formula:

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a65ebf8123411426b87c78fc485ee3a8effa9116" alt="{displaystyle e^{iphi }=cos phi +isin phi ,}" width="8563.8" height="1295.7">$ ;

tj.

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d904caa968e37478a3894a9bd7b5ff238817885e" alt="{displaystyle e^{inphi }=(cos phi +isin phi )^{n},}" width="10292" height="1367.4">$ ;

pomoću nje se definiše stepenovanje kompleksnih brojeva, logaritam kompleksnog broja i dr.

Kompleksni brojevi obrazuju algebarsko zatvoreno polje. Polje kompleksnih brojeva je proširenje polja realnih brojeva pridruživanjem ovom polju elementa $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" alt="{displaystyle i}" width="345.5" height="936.9">$ , takvog da je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" alt="{displaystyle i^{2}=-1}" width="3412.5" height="1223.9">$ .

Množenje

Množenje kompleksnih brojeva u trigonometrijskom obliku je slično množenju kompleksnih brojeva u standardnom obliku.

Neka su zadani kompleksni brojevi

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6dd23658225a717ac8781c11a660fb1ea11d9f" alt="{displaystyle z_{1}=r_{1}(cosvarphi _{1}+isinvarphi _{1})}" width="10527.1" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f2b7c9f30bfc56c5ece73f8b3a01046d8dc7a9" alt="{displaystyle z_{2}=r_{2}(cosvarphi _{2}+isinvarphi _{2})}" width="10527.1" height="1223.9">$

onda je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc87d1e0d3658a1040693dd9dfafdafa56f3a1ef" alt="{displaystyle z_{1}z_{2}=r_{1}(cosvarphi _{1}+isinvarphi _{1})r_{2}(cosvarphi _{2}+isinvarphi _{2})=}" width="20776.5" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f524466fa0e95ae7c907fda43f69dc834d379093" alt="{displaystyle r_{1}r_{2}(cosvarphi _{1}cosvarphi _{2}+icosvarphi _{1}sinvarphi _{2}+icosvarphi _{2}sinvarphi _{1}+i^{2}sinvarphi _{1}sinvarphi _{2})=}" width="28888.6" height="1367.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1831fb45af88995aa915e884d5a6134bd445fdbc" alt="{displaystyle r_{1}r_{2}((cosvarphi _{1}cosvarphi _{2}-sinvarphi _{1}sinvarphi _{2})+i(cosvarphi _{1}sinvarphi _{2}cosvarphi _{2}sinvarphi _{1})=}" width="27689.2" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f5022dbf747d421b35e5b76bee9e38a09ba5f7" alt="{displaystyle r_{1}r_{2}(cos(varphi _{1}+varphi _{2})+isin(varphi _{1}+varphi _{2})}" width="15010.3" height="1223.9">$

Dijeljenje

Neka su zadani kompleksni brojevi

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6dd23658225a717ac8781c11a660fb1ea11d9f" alt="{displaystyle z_{1}=r_{1}(cosvarphi _{1}+isinvarphi _{1})}" width="10527.1" height="1223.9">$ i $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f2b7c9f30bfc56c5ece73f8b3a01046d8dc7a9" alt="{displaystyle z_{2}=r_{2}(cosvarphi _{2}+isinvarphi _{2})}" width="10527.1" height="1223.9">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcc4ce69974d207d54a03e16ad63e184b9b8e4a" alt="{displaystyle {frac {z_{1}}{z_{2}}}={frac {r_{1}(cosvarphi _{1}+isinvarphi _{1})}{r_{2}(cosvarphi _{2}+isinvarphi _{2})}}={frac {r_{1}(cosvarphi _{1}+isinvarphi _{1})}{r_{2}(cosvarphi _{2}+isinvarphi _{2})}}*{frac {r_{1}(cosvarphi _{2}-isinvarphi _{2})}{r_{2}(cosvarphi _{2}-isinvarphi _{2})}}}" width="30793.5" height="2802.6">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4bee3d690032512513caf377b39054fdbf88f2" alt="{displaystyle {frac {r_{1}}{r_{2}}}*{frac {cosvarphi _{1}cosvarphi _{2}-icosvarphi _{1}sinvarphi _{2}+icosvarphi _{2}sinvarphi _{1}-i^{2}sinvarphi _{1}sinvarphi _{2}}{cos^{2}varphi _{2}+sin^{2}varphi _{2}}}}" width="27812.8" height="2730.8">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b741bb4471369646a803c267eac211292dac0dc" alt="{displaystyle {frac {r_{1}}{r_{2}}}(cos(varphi _{1}-varphi _{2})+isin(varphi _{1}-varphi _{2}))={frac {r_{1}}{r_{2}}}(cis(varphi _{1}-varphi _{2})}" width="23310.6" height="2156.8">$

De Moavrova formula

Neka je $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc21a98d2457388610e7e395cce83adeb66ab1be" alt="{displaystyle z=r(costheta +isintheta )=r cistheta }" width="12098.1" height="1223.9">$ trigonometrijski oblik kompleksnog broja. Tada je

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab761b35d475cc808365d25fe639546cf81fc2ae" alt="{displaystyle z^{2}=z*z}" width="4139.4" height="1152.1">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d29cdfa281fef4ad9dada6ee0f788a9f9497342" alt="{displaystyle z^{2}=r cistheta *r cistheta =r^{2} cis(theta +theta )=r^{2} cis2theta }" width="19428.2" height="1367.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ec98e8fb0fbea3120635a53fe385a006006bd4" alt="{displaystyle z^{3}=r^{2} cis2theta *r cistheta =r^{3} cis(2theta +theta )=r^{3} cis3theta }" width="20883.1" height="1367.4">$

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775d80781d2fe500c9b1f432a80b9824131bd73e" alt="{displaystyle z^{4}=r^{3} cis3theta *r cistheta =r^{4} cis(3theta +theta )=r^{4} cis4theta }" width="20883.1" height="1367.4">$

Na ovaj način dobijamo opšti oblik De Moavrova teorema koji ima važnu ulogu u elektrotehnici

$<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c39bcdb3049921f9675a2d3359028a70df03" alt="{displaystyle (cos phi +isin phi )^{n}=cos nphi +isin nphi ,}" width="15666.2" height="1223.9">$

Izvor: Wikipedia

[6]

www.svafizika.org

Šta je to kompleksan broj?

Kompleksan broj

Osobine sabiranja kompleksnih brojeva

Osobine množenja kompleksnih brojeva

Realan proizvod dva kompleksna broja

Kompleksan proizvod dva kompleksna broja

Dijeljenje kompleksnih brojeva

Konjugovano kompleksni brojevi

Stepenovanje kompleksnog broja

Korjenovanje kompleksnog broja

Kvadratni korjen imaginarnog broja

Apsolutna vrijednost argumenta

Množenje i dijeljenje u polarnom obliku

Trigonometrijski oblik

De Moavrova formula

Izvor: Wikipedia

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Stranica o prirodi i svemu vezanom za prirodu.

Kompleksan broj

Osobine sabiranja kompleksnih brojeva

Osobine množenja kompleksnih brojeva

Realan proizvod dva kompleksna broja

Kompleksan proizvod dva kompleksna broja

Dijeljenje kompleksnih brojeva

Konjugovano kompleksni brojevi

Stepenovanje kompleksnog broja

Korjenovanje kompleksnog broja

Kvadratni korjen imaginarnog broja

Apsolutna vrijednost argumenta

Množenje i dijeljenje u polarnom obliku

Trigonometrijski oblik

De Moavrova formula

Izvor: Wikipedia

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Stranica o prirodi i svemu vezanom za prirodu.

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