Analiza I

1.7. KOMPLEKSNA ˇ STEVILA C 25
1.
2.
3.
z +w = (a+ib)+(c+id)
= a+c+i(b+d)
z +w = a+c−i(b+d)
z +w = (a−ib)+(c−id)
= a+c−i(b+d)
z ·w = (a+ib)(c+id)
= ac+iad+ibc−bd
= (ac−bd)+i(ad+bc)
z ·w = (ac−bd)−i(ad+bc)
z ·w = (a−ib)(c−id)
= (ac−bd)−i(ad+bc)
z +z
z +z = a+ib+a−ib = 2a ⇒ a = Rez =
2
z −z
z −z = a+ib−a+ib = 2ib ⇒ b = Imz =
2i
Trditev 10 Naj bosta z,w kompleksni ˇstevili. Tedaj je:
1. |z| > 0 razen, ko je z = 0,
2. |z| = |z|,
3. |z ·w| = |z| |w|,