List of results

Theorems and lemmas 23 Definitions 3 Examples 6 Exercises 39

Complex Numbers

1.1 The Complex Numbers as Pairs of Reals

Definition 1.1.1 Complex Numbers
Proposition 1.1.1 The Real Numbers inside the Complex Numbers
Example First Computations in $\mathbb{C}$
Proposition 1.1.2 Arithmetic Laws in $\mathbb{C}$
Proposition 1.1.3 Laws of Powers in a Commutative Ring
Lemma 1.1.4 Difference of Powers
Exercise A1
Exercise B1: Powers of $i$
Exercise B2
Exercise B3: The zero divisors of the minimal change
Exercise C1: There is only one way to adjoin a square root of minus one

1.2 Conjugate, Modulus, and Division

Definition 1.2.1 Conjugate and Modulus
Proposition 1.2.1 The Modulus Extends the Absolute Value
Theorem 1.2.2 Properties of Conjugation and Modulus
Proposition 1.2.3 Inverse of a Complex Number
Theorem 1.2.4 The Complex Numbers Form a Field
Example Dividing Complex Numbers
Exercise A1
Exercise B1: Arithmetic
Exercise B2
Exercise B3
Exercise C1: Exit check: the complex numbers form a field
Exercise C2: Gaussian rationals

1.3 The Triangle Inequality

Lemma 1.3.1 Real and Imaginary Parts against the Modulus
Theorem 1.3.2 Triangle Inequality for Complex Numbers
Corollary 1.3.3 Variants of the Triangle Inequality
Exercise A1
Exercise B1: The equality case
Exercise B2: A distance that does not change
Exercise B3
Exercise C1: Exit check: the triangle inequality
Exercise C2: Inside the unit disc

1.4 Square Roots, Quadratic Equations, and the Missing Order

Proposition 1.4.1 Square Roots of Complex Numbers
Example Square Roots of $5 - 12i$
Corollary 1.4.2 Quadratic Formula over the Complex Numbers
Proposition 1.4.3 The Complex Numbers Cannot Be Ordered
Exercise A1
Exercise B1
Exercise B2: Put the proof in order
Exercise B3
Exercise C1: Find the error
Exercise C2: A second proof that there is no order

1.5 The Unit Circle and Polar Form

Proposition 1.5.1 The Unit Circle
Axiom 1.5.2 What This Volume Borrows about Cosine and Sine
Definition 1.5.1 Polar Form
Theorem 1.5.3 Multiplication in Polar Form
Proposition 1.5.4 Existence and Uniqueness of Polar Form
Theorem 1.5.5 De Moivre’s Formula
Example Special Angles
Example A Power via Polar Form
Exercise A1
Exercise B1
Exercise B2: Symmetries of Cosine and Sine
Exercise B3
Exercise C1: Triple Angles and Negative Powers
Exercise C2: Points on the Circle without Angles

1.6 Higher Roots, Roots of Unity, and a Look Ahead

Lemma 1.6.1 Real n-th Roots
Theorem 1.6.2 n-th Roots of a Complex Number
Example Roots of Unity
Proposition 1.6.3 Sums of Roots of Unity
Proposition 1.6.4 Real Coefficients and Conjugate Roots
Exercise A1
Exercise B1
Exercise B2
Exercise B3
Exercise C1: Exit check: n-th roots
Exercise C2: Find the error
Exercise R1
Exercise R2
Exercise R3
Exercise R4