An introduction to
COMPLEX NUMBERS
Jan van de Craats
Last update: April 25, 2022
,Illustrations and LATEX typesetting: Jan van de Craats
Prof. dr. J. van de Craats is professor emeritus in mathematics
at the University of Amsterdam
This is an English translation of chapters 1, 2 and 3 of
Jan van de Craats: Complexe getallen voor wiskunde D
Translated by the author.
, How to use this book
This is an exercise book. Each chapter starts with exercises, printed on the left-
hand pages. Once you have finished an exercise, you can check your answer at
the end of the book. On the right-hand pages, the theory behind the exercises
is explained in a clear and concise manner. Use this information if and when
required.
Background knowledge may be obtained from:
Jan van de Craats en Rob Bosch: Basisboek wiskunde. Tweede editie
Pearson, Amsterdam, 2009, ISBN 978-90-430-1673-5 (in Dutch)
or from its English translation:
Jan van de Craats and Rob Bosch: All you need in maths!
Pearson, Amsterdam, 2014, ISBN 978-90-430-3285-8.
The Greek Alphabet
α A alpha ι I iota ρ P rho
β B beta κ K kappa σ Σ sigma
γ Γ gamma λ Λ lambda τ T tau
δ ∆ delta µ M mu υ Υ upsilon
e E epsilon ν N nu ϕ Φ phi
ζ Z zeta ξ Ξ xi χ X chi
η H eta o O omicron ψ Ψ psi
ϑ Θ theta π Π pi ω Ω omega
iii
source: https://staff.fnwi.uva.nl/j.vandecraats/
, Contents
1 Calculating with complex numbers 2
Square roots of negative numbers . . . . . . . . . . . . . . . . . . . . 3
The abc-formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . 7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The geometry of complex calculations 10
Complex numbers as vectors . . . . . . . . . . . . . . . . . . . . . . 11
Complex numbers on the unit circle . . . . . . . . . . . . . . . . . . 13
Formulas of Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
The (r, ϕ) notation for complex numbers . . . . . . . . . . . . . . . . 17
The complex functions e z , cos z and sin z . . . . . . . . . . . . . . . 19
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Roots and polynomials 22
What is a complex nth root? . . . . . . . . . . . . . . . . . . . . . . . 23
Why complex roots are multi-valued . . . . . . . . . . . . . . . . . . 25
On nth roots and nth -degree polynomials . . . . . . . . . . . . . . . 27
The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . 29
Real polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Symmary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Answers to the exercises 33
Index 37
COMPLEX NUMBERS
Jan van de Craats
Last update: April 25, 2022
,Illustrations and LATEX typesetting: Jan van de Craats
Prof. dr. J. van de Craats is professor emeritus in mathematics
at the University of Amsterdam
This is an English translation of chapters 1, 2 and 3 of
Jan van de Craats: Complexe getallen voor wiskunde D
Translated by the author.
, How to use this book
This is an exercise book. Each chapter starts with exercises, printed on the left-
hand pages. Once you have finished an exercise, you can check your answer at
the end of the book. On the right-hand pages, the theory behind the exercises
is explained in a clear and concise manner. Use this information if and when
required.
Background knowledge may be obtained from:
Jan van de Craats en Rob Bosch: Basisboek wiskunde. Tweede editie
Pearson, Amsterdam, 2009, ISBN 978-90-430-1673-5 (in Dutch)
or from its English translation:
Jan van de Craats and Rob Bosch: All you need in maths!
Pearson, Amsterdam, 2014, ISBN 978-90-430-3285-8.
The Greek Alphabet
α A alpha ι I iota ρ P rho
β B beta κ K kappa σ Σ sigma
γ Γ gamma λ Λ lambda τ T tau
δ ∆ delta µ M mu υ Υ upsilon
e E epsilon ν N nu ϕ Φ phi
ζ Z zeta ξ Ξ xi χ X chi
η H eta o O omicron ψ Ψ psi
ϑ Θ theta π Π pi ω Ω omega
iii
source: https://staff.fnwi.uva.nl/j.vandecraats/
, Contents
1 Calculating with complex numbers 2
Square roots of negative numbers . . . . . . . . . . . . . . . . . . . . 3
The abc-formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . 7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The geometry of complex calculations 10
Complex numbers as vectors . . . . . . . . . . . . . . . . . . . . . . 11
Complex numbers on the unit circle . . . . . . . . . . . . . . . . . . 13
Formulas of Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
The (r, ϕ) notation for complex numbers . . . . . . . . . . . . . . . . 17
The complex functions e z , cos z and sin z . . . . . . . . . . . . . . . 19
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Roots and polynomials 22
What is a complex nth root? . . . . . . . . . . . . . . . . . . . . . . . 23
Why complex roots are multi-valued . . . . . . . . . . . . . . . . . . 25
On nth roots and nth -degree polynomials . . . . . . . . . . . . . . . 27
The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . 29
Real polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Symmary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Answers to the exercises 33
Index 37
