Summary $5 That Will Change Your Life — The Only Complex Numbers Book You'll Ever Need

[MATH]
Terminale Mathematics
Complex
Numbers
A Complete Guide from A to Z iR
Baccalauréat Level · Terminal Year




What you will master: z



Algebraic Form · Geometric Form · Polar Form
Exponential Form · De Moivre's Theorem
Roots · Equations · Trigonometric Applications R




Author: ZAYD gg
store of BIGBOSSES
Academic Year 20242025

, About This Book
Terminale
Baccalauréat
This course has been carefully designed for (Year
13) students preparing for the examina-
tion. Every concept is explained from the ground up with
full mathematical rigor, intuitive explanations, worked ex-
amples, and tips for solving exam-style problems e
ciently.

Mathematics is the language with
which God has written the universe.
 Galileo Galilei




Author: ZAYD gg
Publisher: BIGBOSSES
Level: Terminale  Baccalauréat
Subject: Mathematics

, Contents
1 Introduction to Complex Numbers 7
1.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Why Do We Need Complex Numbers? . . . . . . . . . . . . . . . . . . . . 7
1.3 Powers of i  The Cycle of Four . . . . . . . . . . . . . . . . . . . . . . . 8


2 Algebraic Form of Complex Numbers 9
2.1 De
nition and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Operations on Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.3 Complex Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.4 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Modulus of a Complex Number . . . . . . . . . . . . . . . . . . . . . . . . 12


3 The Complex Plane (Argand Diagram) 13
3.1 Geometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 A
xes and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Loci in the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 14


4 Argument of a Complex Number 15
4.1 De
nition and Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15


5 Trigonometric (Polar) Form 17
5.1 The Trigonometric Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Multiplication and Division in Polar Form . . . . . . . . . . . . . . . . . . 18


6 Exponential Form and Euler's Formula 19
6.1 Euler's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 The Exponential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.3 Operations in Exponential Form . . . . . . . . . . . . . . . . . . . . . . . . 20
6.4 Trigonometric Formulas from Euler's Formula . . . . . . . . . . . . . . . . 20


7 De Moivre's Theorem 21
7.1 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7.2 Applications  Expanding cos(nθ) and sin(nθ) . . . . . . . . . . . . . . . 21


3

,Complex Numbers CONTENTS



8 Linearization of Trigonometric Expressions 23
8.1 The Linearization Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


9 Roots of Complex Numbers 25
9.1 The nth Roots of a Complex Number . . . . . . . . . . . . . . . . . . . . . 25
9.2 Cube Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
9.3 Square Roots of a Complex Number . . . . . . . . . . . . . . . . . . . . . . 26


10 Equations with Complex Numbers 27
10.1 Quadratic Equations in C . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
10.2 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . 27
10.3 Real Polynomials  Conjugate Root Theorem . . . . . . . . . . . . . . . . 27


11 Geometric Transformations via Complex Numbers 29
11.1 Basic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
11.2 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
11.3 Homothety and Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


12 Key Identities and Summary of Formulas 31
12.1 Algebraic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
12.2 Complete Formula Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
12.3 Standard Argument Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


13 Exam Methods and Problem-Solving Techniques 33
13.1 Method 1  Converting Between Forms . . . . . . . . . . . . . . . . . . . 33
13.2 Method 2  Proving Geometric Results . . . . . . . . . . . . . . . . . . . 33
13.3 Method 3  Detecting Special Triangles . . . . . . . . . . . . . . . . . . . 34
13.4 Method 4  Solving Equations E
ciently . . . . . . . . . . . . . . . . . . 34


14 Full Practice Exercises 35
15 Detailed Solutions 37
Solutions to Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Solutions to Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Solutions to Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Solutions to Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Solutions to Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Solutions to Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


16 Rapid Review: Tips for the Baccalauréat 39
16.1 The 10 Most Important Things to Remember . . . . . . . . . . . . . . . . 39
16.2 Common Mistakes to Avoid . . . . . . . . . . . . . . . . . . . . . . . . . . 39
16.3 Time-Saving Shortcuts on Exams . . . . . . . . . . . . . . . . . . . . . . . 40


17 Supplementary Bac-Style Problems 41
18 Trigonometric Applications 43
18.1 Sum-to-Product and Product-to-Sum via Euler . . . . . . . . . . . . . . . 43
18.2 Half-Angle and Double-Angle Formulas . . . . . . . . . . . . . . . . . . . . 43
18.3 Chebyshev Polynomials and Complex Numbers . . . . . . . . . . . . . . . 44


4 ZAYD gg | BIGBOSSES

,CONTENTS Complex Numbers




18.4 Expressing Sums of Cosines and Sines . . . . . . . . . . . . . . . . . . . . . 44


19 Geometry with Complex Numbers  Deep Dive 47
19.1 Cross-Ratio and Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
19.2 Angles Between Lines  The Argument as Angle Measure . . . . . . . . . 47
19.3 The Nine Classic Geometry Theorems via Complex Numbers . . . . . . . . 47
19.3.1 Theorem 1  Midpoint and Centroid . . . . . . . . . . . . . . . . . 48
19.3.2 Theorem 2  Equilateral Triangle Characterisation . . . . . . . . . 48
19.3.3 Theorem 3  Rotation Maps One Triangle to Another . . . . . . . 48
19.4 Worked Geometry Example  Full Solution . . . . . . . . . . . . . . . . . 48


20 Complete Worked Baccalauréat Problems 51
20.1 Bac Problem 1  Geometric Complex Numbers (Full Solution) . . . . . . 51
20.2 Bac Problem 2  Polynomial Factorisation (Full Solution) . . . . . . . . . 52
20.3 Bac Problem 3  Linearisation & Integration (Full Solution) . . . . . . . . 53


21 The Complex Exponential and Advanced Topics 55
21.1 The Complex Exponential Function . . . . . . . . . . . . . . . . . . . . . . 55
21.2 The Complex Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
21.3 Complex Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
21.4 Summary  Three Representations of Any Complex Number . . . . . . . 56


22 Glossary and Index of Key Terms 59




5 ZAYD gg | BIGBOSSES

,Complex Numbers CONTENTS




6 ZAYD gg | BIGBOSSES

, 1 CHAPTER
Introduction to Complex Numbers
Ÿ1.1 A Brief History
The story of complex numbers is one of the most fascinating journeys in the history of
mathematics. For centuries, mathematicians refused to accept the idea of the square root
of a negative number. It was considered impossible  even absurd.


16th century 1572 1637 1806
Cardano introduces Rafael Bombelli Descartes coins Argand and
√
−1 in formulas formalizes rules imaginary geometric



The key insight came from trying to solve cubic equations. Even when all three roots
are real, Cardano's formula sometimes passes through the impossible square root of a
negative number before giving a perfectly real answer. Mathematicians were forced to
take these numbers seriously.




Ÿ1.2 Why Do We Need Complex Numbers?
Consider the equation:

x2 + 1 = 0


Fundamental Theorem of
In R, there is no solution, since x2 ≥ 0 for all real x. Yet polynomials of degree n should


Algebra extend R
have exactly n roots (counting multiplicity)  this is the
. To
x this, we to a larger set: the complex numbers C.

❖ De
nition  The Imaginary Unit
The imaginary unit i is de
ned by:

√
i2 = −1 equivalently, i= −1

With this, the equation x2 + 1 = 0 has two solutions: x=i and x = −i.



Ÿ1.3 Powers of i  The Cycle of Four
7

,Complex Numbers CHAPTER 1. INTRODUCTION TO COMPLEX NUMBERS




➜ The Cyclic Pattern of Powers of i
i0 = 1, i1 = i, i2 = −1, i3 = −i, i4 = 1, ...
The pattern repeats every 4 steps. In general:


in = in mod 4

✩ Trick  Computing i Quickly n


n
Divide the exponent remainder r ∈ {0, 1, 2, 3}
by 4. The tells you:


r = 0 ⇒ in = 1, r = 1 ⇒ in = i, r = 2 ⇒ in = −1, r = 3 ⇒ in = −i.

Example: i 47
47 = 4 × 11 + 3
. Since , remainder is 3, so i47 = −i.

✎ Mini Exercise 1.1
Compute each power of i:

1. i 10
2. i 25
3. i100
4. i−3


Answers: (1) −1 (2) i (3) 1 (4) i




8 ZAYD gg | BIGBOSSES

, 2 CHAPTER
Algebraic Form of Complex Numbers
Ÿ2.1 De
nition and Notation
❖ De
nition  Algebraic Form
A complex number z is an expression of the form:


z = a + bi

where a, b ∈ R and i2 = −1.

ˆ a = Re(z) is the real part zof


ˆ b = Im(z) is the imaginary part of z

ˆ The set of all complex numbers is denoted C




z = |{z}
a + b ·i
real part imaginary part
|{z}



a = Re(z) ∈bR= Im(z) ∈ R




☞ Common Mistake
The imaginary part of z = a + bi is b, not bi .
For example, Im(3 + 5i) = 5, not 5i.



Ÿ2.2 Special Cases
9