número complejo

1. Mat.

a + bi motako zenbakia, non
i=1
eta a eta b zenbaki errealak baitira. a-ri parte erreal deritzo, eta bi-ri parte irudikari.

1. Mat.
a + bi motako zenbakia, non
i=1
eta a eta b zenbaki errealak baitira. a-ri parte erreal deritzo, eta bi-ri parte irudikari.

Zenbaki konplexuak Edit

Egilea: Elhuyar

ZENBAKI KONPLEXUAK

Zenbaki konplexuen multzoa C letraz adierazten da. b = 0 denean, zenbakiak ez du parte irudikaririk; orduan, zenbaki erreala da. Beraz, zenbaki errealak zenbaki konplexuen azpimultzo bat dira (R CMathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabgkOimlaadoeaaaa@3988@).

a = 0 denean, zenbakiak ez du parte errealik; orduan, zenbaki irudikari purua da. z = a + b i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2da9iaadggacqGHRaWkcaWGIbGaamyAaaaa@3B8F@ zenbaki konplexua emanik, z ¯ = a b i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOEayaaraGaeyypa0JaamyyaiabgkHiTiaadkgacaWGPbaaaa@3BB2@ zenbakiari zenbaki konplexu konjugatu deritzo. z = a + b i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2da9iaadggacqGHRaWkcaWGIbGaamyAaaaa@3B8F@ zenbaki konplexuak bi ardatz perpendikularrak erabiliz irudika daitezke planoan (Arganden diagrama, Gaussen planoa edo plano konplexua). Parte erreala x koordenatua da, eta parte irudikaria y koordenatua da. a + b i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgUcaRiaadkgacaWGPbaaaa@398A@ zenbaki konplexua, hala, ( a,b ) puntuaren bidez adieraz daiteke. Bestalde, jatorritik puntu horretaraino doan bektore baten bidez ere adieraz daiteke. Horrek aukera ematen du zenbaki konplexuak beste era batera adierazteko, z = ρ θ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2da9iabeg8aYnaaBaaaleaacqaH4oqCaeqaaaaa@3B94@ forman, non ρ bektorearen luzera baita, θ bektorearen eta OX ardatzaren noranzko positiboaren arteko angelua, eta | z | = ρ eta arg( z ) = θ idazten dira. ρ balioa zenbaki konplexuaren modulua da, eta θ zenbakiaren argumentua. Zenbaki konplexuak hala adierazteko moduari forma polar deritzo.

Zenbaki konplexuekin batuketak (edo kenketak) egin daitezke, haien parte errealak eta irudikariak zein bere aldetik batuz (edo kenduz). Adibidez:

(4 + 3i) + (6 + 2i) = 10 + 5i.

Biderketak egiteko, garatu egin behar dira:

(a+bi)(c+di)=(acbd)+(ad+bc)i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacqGHRaWkcaWGIbGaaeyAaiaacMcacaGGOaGaam4yaiabgUcaRiaadsgacaqGPbGaaiykaiabg2da9iaacIcacaWGHbGaam4yaiabgkHiTiaadkgacaWGKbGaaiykaiabgUcaRiaacIcacaWGHbGaamizaiabgUcaRiaadkgacaWGJbGaaiykaiaabMgaaaa@4E6A@ ,

i 2 = 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAamaaCaaaleqabaGaaGOmaaaakiabg2da9iabgkHiTiaaigdaaaa@3A7C@ delako.

Zatiketa egiteko, formula hau dugu:

a+bi c+di = ac+bd c 2 + d 2 + bcad c 2 + d 2 i. MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbGaey4kaSIaamOyaiaabMgaaeaacaWGJbGaey4kaSIaamizaiaabMgaaaGaeyypa0ZaaSaaaeaacaWGHbGaam4yaiabgUcaRiaadkgacaWGKbaabaGaam4yamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadsgadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGIbGaam4yaiabgkHiTiaadggacaWGKbaabaGaam4yamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadsgadaahaaWcbeqaaiaaikdaaaaaaOGaaeyAaaaa@5268@

Adierazpen polarra erabiliz gero, biderketa eta zatiketa honela geratzen dira:  z = ρ θ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2da9iabeg8aYnaaBaaaleaacqaH4oqCaeqaaaaa@3B94@ , w = ρ ' θ ' MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daiabg2da9iabeg8aYjaacEcadaWgaaWcbaGaeqiUdeNaai4jaaqabaaaaa@3CE7@ badira, orduan z w = ( ρ ρ ' ) ( θ + θ ' ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabgwSixlaadEhacqGH9aqpcaGGOaGaeqyWdiNaeyyXICTaeqyWdiNaai4jaiaacMcadaWgaaWcbaGaaiikaiabeI7aXjabgUcaRiabeI7aXjaacEcacaGGPaaabeaaaaa@4984@ eta z w = ( ρ ρ ' ) ( θ θ ' ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWG6baabaGaam4DaaaacqGH9aqpdaqadaqaamaalaaabaGaeqyWdihabaGaeqyWdiNaai4jaaaaaiaawIcacaGLPaaadaWgaaWcbaGaaiikaiabeI7aXjabgkHiTiabeI7aXjaacEcacaGGPaaabeaaaaa@454B@ .