Guided Notes 7 — Complex Numbers (Komplexe
Zahlen)
Algebraic form, polar form, and roots
by Ugorotti
Learning goals (Lernziele). After this booklet you should be able to:
•work confidently with complex numbers (komplexe Zahlen),
•switch between algebraic form (algebraische Form) and polar form (Polarform),
•compute modulus (Betrag) and argument (Argument),
•apply Euler’s formula (Eulersche Formel),
•compute powers and roots of complex numbers (Potenzen und Wurzeln).
1. Definition and algebraic form (algebraische Form)
Complex number (komplexe Zahl). The set of complex numbers is
C:= {a+bi |a, b ∈R, i2=−1}.
Here:
ℜ(z)=a(real part / Realteil),ℑ(z)=b(imaginary part / Imagin¨arteil).
Example. Let z= 3 −2i. Then
ℜ(z)=3,ℑ(z)=−2.
Equality criterion (Gleichheitskriterium). Two complex numbers are equal if and only if
their real and imaginary parts are equal:
a+bi =c+di ⇐⇒ a=cand b=d.
2. Arithmetic operations (Rechenoperationen)
Addition and multiplication. Let z1=a+bi and z2=c+di. Then
z1+z2= (a+c) + (b+d)i,
z1z2= (ac −bd)+(ad +bc)i.
Example.
(1 + 2i)(3 −i) = 3 −i+ 6i−2i2= 5 + 5i.
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