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3.2 Übungen

Aus Online Mathematik Brückenkurs 2

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Version vom 13:43, 10. Mär. 2009

Theorie

Übungen

Übung 3.2:1

Given the complex numbers \displaystyle \,z=2+i\,, \displaystyle \,w=2+3i\, and \displaystyle \,u=-1-2i\,. Mark the following numbers on the complex plane:

a)	\displaystyle z\, and \displaystyle \,w	b)	\displaystyle z+u\, and \displaystyle \,z-u
c)	\displaystyle 2z+w	d)	\displaystyle z-\overline{w}+u

Antwort

Lösung a

Lösung b

Lösung c

Lösung d

Übung 3.2:2

Draw the following sets in the complex number plane

a)	\displaystyle 0\le \mbox{Im}\, z \le 3	b)	\displaystyle 0 \le \mbox{Re} \, z \le \mbox{Im}\, z \le 3
c)	\displaystyle \|z\|=2	d)	\displaystyle \|z-1-i\|=3
e)	\displaystyle \mbox{Re}\, z = i + \bar z	f)	\displaystyle 2<\|z-i\|\le3

Antwort

Lösung a

Lösung b

Lösung c

Lösung d

Lösung e

Lösung f

Übung 3.2:3

The complex numbers \displaystyle \,1+i\,, \displaystyle \,3+2i\, and \displaystyle \,3i\, constitute three corners of a square in the complex number plane. Determine the square's fourth corner.

Antwort

Antwort 3.2:3

Lösung

Lösung 3.2:3

Übung 3.2:4

Determine the magnitude of

a)	\displaystyle 3+4i	b)	\displaystyle (2-i) + (5+3i)
c)	\displaystyle (3-4i)(3+2i)	d)	\displaystyle \displaystyle\frac{3-4i}{3+2i}

Antwort

Lösung a

Lösung b

Lösung c

Lösung d

Übung 3.2:5

Determine the argument of

a)	\displaystyle -10	b)	\displaystyle -2+2i
c)	\displaystyle (\sqrt{3} +i)(1-i)	d)	\displaystyle \displaystyle\frac{i}{1+i}

Antwort

Lösung a

Lösung b

Lösung c

Lösung d

Übung 3.2:6

Write the following numbers in polar form

a)	\displaystyle 3	b)	\displaystyle -11i
c)	\displaystyle -4-4i	d)	\displaystyle \sqrt{10} + \sqrt{30}\,i
e)	\displaystyle \displaystyle\frac{1+i\sqrt{3}}{1+i}	f)	\displaystyle \displaystyle\frac{(2+2i)(1+i\sqrt{3}\,)}{3i(\sqrt{12} -2i)}