LA-1.2

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Complex Numbers

Complex numbers extend the reals to ensure every polynomial has roots. They are indispensable for spectral theory in linear algebra—eigenvalues of real matrices may be complex, and the spectral theorem is most elegant over ℂ.

3-4 hours Foundation Level 10 Objectives

Learning Objectives

Construct the complex numbers algebraically as ordered pairs
Convert between rectangular and polar forms
Apply Euler's formula for complex exponentials
Use De Moivre's theorem to compute powers
Find all nth roots of complex numbers
Understand why ℂ is algebraically closed
Apply complex conjugates to simplify expressions
Prove basic properties using complex number algebra
Understand the geometric interpretation of complex operations
Apply complex numbers to solve polynomial equations

Prerequisites

Field axioms from LA-1.1 (Algebraic Structures)
Trigonometric functions (sine, cosine, tangent)
Basic properties of exponential functions
Polar coordinates (helpful but not required)
Quadratic formula and polynomial roots

Historical Context

Complex numbers emerged from attempts to solve polynomial equations. In the 16th century, Cardano and Bombelli encountered square roots of negative numbers while solving cubic equations—even for equations with real solutions!

Euler (18th century) introduced the notation $i$ for $\sqrt{-1}$ and discovered the remarkable formula $e^{i\theta} = \cos\theta + i\sin\theta$ .

Gauss gave the first rigorous proofs of the Fundamental Theorem of Algebra and popularized the geometric interpretation of complex numbers as points in a plane. Hamilton later provided the formal construction as ordered pairs.

1. Algebraic Construction of ℂ

We can construct the complex numbers rigorously as ordered pairs of real numbers with specially defined operations. This shows that $\mathbb{C}$ is not mysterious—it's simply $\mathbb{R}^2$ with a clever multiplication.

Definition 1.1: Complex Numbers

The complex numbers $\mathbb{C}$ is the set $\mathbb{R} \times \mathbb{R}$ with operations:

Addition: $(a, b) + (c, d) = (a + c, b + d)$
Multiplication: $(a, b) \cdot (c, d) = (ac - bd, ad + bc)$

We write $i = (0, 1)$ and identify $a \in \mathbb{R}$ with $(a, 0) \in \mathbb{C}$ . Then $(a, b) = a + bi$ .

Theorem 1.1: ℂ is a Field

$(\mathbb{C}, +, \cdot)$ as defined above is a field with:

Additive identity: $0 = (0, 0)$
Multiplicative identity: $1 = (1, 0)$
Additive inverse of $(a, b)$ : $(-a, -b)$
Multiplicative inverse of $(a, b) \neq 0$ : $\left(\frac{a}{a^2+b^2}, \frac{-b}{a^2+b^2}\right)$

Proof of Theorem 1.1:

Most axioms follow from direct calculation. The key is the multiplicative inverse. For $z = (a, b) \neq (0, 0)$ , we need $z \cdot w = 1$ .

Let $w = (c, d)$ . Then:

(a, b) \cdot (c, d) = (ac - bd, ad + bc) = (1, 0)

This gives the system:

ac - bd = 1, \quad ad + bc = 0

Solving: $c = \frac{a}{a^2 + b^2}$ , $d = \frac{-b}{a^2 + b^2}$ .

This is well-defined since $a^2 + b^2 > 0$ when $(a, b) \neq (0, 0)$ .

∎

Remark 1.1: The Imaginary Unit

Note that $i^2 = (0, 1) \cdot (0, 1) = (0 \cdot 0 - 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0) = (-1, 0) = -1$ .

This is the defining property: we have constructed a field containing $\mathbb{R}$ in which $x^2 + 1 = 0$ has a solution.

Example 1.4: Complex Arithmetic

Compute $(2 + 3i) + (4 - i)$ and $(2 + 3i)(4 - i)$ .

Addition:

(2 + 3i) + (4 - i) = (2 + 4) + (3 - 1)i = 6 + 2i

Multiplication:

(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i

Theorem 1.4: Powers of i

The powers of $i$ cycle with period 4:

i^0 = 1, \quad i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \ldots

In general, $i^n = i^{n \mod 4}$ .

Example 1.5: Computing Powers of i

Compute $i^{100}$ and $i^{2023}$ .

$i^{100} = i^{4 \cdot 25} = (i^4)^{25} = 1^{25} = 1$
$i^{2023} = i^{4 \cdot 505 + 3} = (i^4)^{505} \cdot i^3 = 1 \cdot (-i) = -i$

Definition 1.4: ℂ as a Vector Space

$\mathbb{C}$ is a 2-dimensional vector space over $\mathbb{R}$ with basis $\{1, i\}$ . Every $z \in \mathbb{C}$ can be written uniquely as:

z = a \cdot 1 + b \cdot i = a + bi \quad (a, b \in \mathbb{R})

Remark 1.3: ℂ as 1-dimensional over itself

As a vector space over itself, $\mathbb{C}$ is 1-dimensional. This distinction matters: $\dim_{\mathbb{R}}(\mathbb{C}) = 2$ but $\dim_{\mathbb{C}}(\mathbb{C}) = 1$ .

2. Conjugate and Modulus

Definition 1.2: Complex Conjugate and Modulus

For $z = a + bi \in \mathbb{C}$ :

The complex conjugate is $\bar{z} = a - bi$
The modulus (or absolute value) is $|z| = \sqrt{a^2 + b^2}$
The real part is $\text{Re}(z) = a$
The imaginary part is $\text{Im}(z) = b$

Theorem 1.2: Properties of Conjugation and Modulus

For all $z, w \in \mathbb{C}$ :

$\overline{z + w} = \bar{z} + \bar{w}$ and $\overline{zw} = \bar{z} \cdot \bar{w}$
$z \cdot \bar{z} = |z|^2$
$|zw| = |z| \cdot |w|$
$z + \bar{z} = 2\text{Re}(z)$ and $z - \bar{z} = 2i \cdot \text{Im}(z)$
$\bar{\bar{z}} = z$
$z \in \mathbb{R} \iff \bar{z} = z$

Proof of Theorem 1.2:

Property 2: Let $z = a + bi$ . Then:

z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2 = |z|^2

Property 3: Using Property 2:

|zw|^2 = (zw)(\overline{zw}) = zw \bar{z} \bar{w} = (z\bar{z})(w\bar{w}) = |z|^2 |w|^2

Taking square roots (all quantities are non-negative) gives $|zw| = |z||w|$ .

∎

Example 1.1: Division Using Conjugates

To compute $\frac{2 + 3i}{1 - 2i}$ , multiply by $\frac{\overline{1-2i}}{\overline{1-2i}}$ :

\frac{2 + 3i}{1 - 2i} = \frac{(2 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{2 + 4i + 3i + 6i^2}{1 + 4} = \frac{2 + 7i - 6}{5} = \frac{-4 + 7i}{5}

So the answer is $\frac{-4}{5} + \frac{7}{5}i$ .

Example 1.2: Computing Modulus

Find $|3 - 4i|$ and $|2 + 2i|$ .

$|3 - 4i| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
$|2 + 2i| = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}$

Theorem 1.3: Conjugate of Real Polynomial Roots

If $p(x)$ is a polynomial with real coefficients and $z = a + bi$ is a root, then $\bar{z} = a - bi$ is also a root.

Proof of Theorem 1.3:

Let $p(z) = a_n z^n + \cdots + a_1 z + a_0$ with $a_i \in \mathbb{R}$ .

If $p(z) = 0$ , then:

\overline{p(z)} = \overline{a_n z^n + \cdots + a_0} = a_n \bar{z}^n + \cdots + a_0 = p(\bar{z})

Since $p(z) = 0$ , we have $p(\bar{z}) = \overline{p(z)} = \bar{0} = 0$ .

∎

Remark 1.2: Complex Roots Come in Pairs

This theorem explains why quadratics with real coefficients either have two real roots or two complex conjugate roots. For example, $x^2 + 1 = 0$ has roots $\pm i$ .

Definition 1.3: Multiplicative Inverse

For $z = a + bi \neq 0$ , the multiplicative inverse is:

z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2}

Example 1.3: Finding Inverses

Find $(3 + 4i)^{-1}$ .

Solution:

(3 + 4i)^{-1} = \frac{3 - 4i}{3^2 + 4^2} = \frac{3 - 4i}{25} = \frac{3}{25} - \frac{4}{25}i

Verify: $(3 + 4i)(\frac{3}{25} - \frac{4}{25}i) = \frac{9 + 16}{25} + \frac{-12 + 12}{25}i = 1$ ✓

3. Polar Form and Euler's Formula

The polar form reveals the geometric meaning of complex numbers and makes multiplication and powers much simpler.

Definition 1.3: Polar Form

Every non-zero complex number $z$ can be written as:

z = r(\cos\theta + i\sin\theta)

where:

$r = |z|$ is the modulus
$\theta = \arg(z)$ is the argument (angle from positive real axis)

Theorem 1.3: Euler's Formula

For any $\theta \in \mathbb{R}$ :

e^{i\theta} = \cos\theta + i\sin\theta

Thus every non-zero complex number can be written as $z = re^{i\theta}$ .

Remark 1.2: Multiplication in Polar Form

If $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$ , then:

z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}

Multiplication scales by the product of moduli and rotates by the sum of arguments.

Example 1.2: Polar Form Conversion

Convert $z = 1 + i$ to polar form.

Modulus: $|z| = \sqrt{1^2 + 1^2} = \sqrt{2}$
Argument: $\theta = \arctan(1/1) = \pi/4$

Thus $z = \sqrt{2} e^{i\pi/4} = \sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})$ .

Example 1.3: Converting to Rectangular Form

Convert $z = 4e^{i\pi/3}$ to rectangular form.

Solution:

z = 4(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}) = 4(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = 2 + 2\sqrt{3}i

Definition 3.1: Principal Argument

The principal argument of $z$ , denoted $\text{Arg}(z)$ (capital A), is the unique value of $\arg(z)$ in the interval $(-\pi, \pi]$ .

Example 3.1: Finding Arguments

Find the principal argument of:

$z = 1$ : $\text{Arg}(1) = 0$
$z = i$ : $\text{Arg}(i) = \pi/2$
$z = -1$ : $\text{Arg}(-1) = \pi$
$z = -i$ : $\text{Arg}(-i) = -\pi/2$
$z = -1 - i$ : $\text{Arg}(-1-i) = -3\pi/4$

Theorem 3.1: Argument Properties

For nonzero complex numbers $z$ and $w$ :

$\arg(zw) = \arg(z) + \arg(w)$ (modulo $2\pi$ )
$\arg(z/w) = \arg(z) - \arg(w)$ (modulo $2\pi$ )
$\arg(\bar{z}) = -\arg(z)$
$\arg(z^n) = n \cdot \arg(z)$ (modulo $2\pi$ )

Remark 3.1: Why Polar Form is Useful

Polar form makes multiplication simple: multiply moduli and add arguments. This is much easier than expanding $(a + bi)(c + di)$ in rectangular form, especially for powers and roots.

Example 3.2: Multiplication in Polar Form

Multiply $z_1 = 2e^{i\pi/6}$ and $z_2 = 3e^{i\pi/4}$ .

Solution:

z_1 z_2 = 2 \cdot 3 \cdot e^{i(\pi/6 + \pi/4)} = 6e^{i(2\pi + 3\pi)/12} = 6e^{i5\pi/12}

Example 3.3: Division in Polar Form

Divide $z_1 = 8e^{i\pi/3}$ by $z_2 = 2e^{i\pi/6}$ .

Solution:

\frac{z_1}{z_2} = \frac{8}{2} e^{i(\pi/3 - \pi/6)} = 4e^{i\pi/6} = 4(\frac{\sqrt{3}}{2} + \frac{1}{2}i) = 2\sqrt{3} + 2i

4. De Moivre's Theorem and Roots of Unity

Theorem 1.4: De Moivre's Theorem

For any $\theta \in \mathbb{R}$ and $n \in \mathbb{Z}$ :

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

Equivalently, $(e^{i\theta})^n = e^{in\theta}$ .

Example 1.3: Computing Powers

Compute $(1 + i)^{10}$ .

We have $1 + i = \sqrt{2} e^{i\pi/4}$ , so:

(1 + i)^{10} = (\sqrt{2})^{10} e^{10 \cdot i\pi/4} = 32 e^{i \cdot 5\pi/2}

Since $e^{i \cdot 5\pi/2} = e^{i\pi/2} = i$ , we get:

(1 + i)^{10} = 32i

Definition 1.4: Roots of Unity

The $n$ th roots of unity are the solutions to $z^n = 1$ . They are:

\omega_k = e^{2\pi i k / n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \quad k = 0, 1, \ldots, n-1

The primitive $n$ th root of unity is $\omega = e^{2\pi i / n}$ .

Theorem 1.5: nth Roots of Any Complex Number

If $w = r e^{i\phi} \neq 0$ , then the $n$ solutions to $z^n = w$ are:

z_k = r^{1/n} e^{i(\phi + 2\pi k)/n}, \quad k = 0, 1, \ldots, n-1

Example 1.4: Cube Roots of -8

Find all cube roots of $-8$ .

Write $-8 = 8 e^{i\pi}$ . The cube roots are:

z_k = 2 e^{i(\pi + 2\pi k)/3}, \quad k = 0, 1, 2

$k = 0$ : $z_0 = 2e^{i\pi/3} = 2(\frac{1}{2} + \frac{\sqrt{3}}{2}i) = 1 + \sqrt{3}i$
$k = 1$ : $z_1 = 2e^{i\pi} = -2$
$k = 2$ : $z_2 = 2e^{i5\pi/3} = 1 - \sqrt{3}i$

Example 4.3: Computing (1 - i)^8

Compute $(1 - i)^8$ using polar form.

Solution: First convert to polar:

|1 - i| = \sqrt{2}, \quad \arg(1-i) = -\pi/4

So $1 - i = \sqrt{2}e^{-i\pi/4}$ .

(1 - i)^8 = (\sqrt{2})^8 e^{-8i\pi/4} = 16 e^{-2\pi i} = 16 \cdot 1 = 16

Theorem 4.2: Product of Roots of Unity

For $n \geq 1$ , the product of all $n$ th roots of unity is:

\prod_{k=0}^{n-1} e^{2\pi ik/n} = (-1)^{n+1}

Proof:

Using properties of exponents:

\prod_{k=0}^{n-1} e^{2\pi ik/n} = e^{2\pi i \sum_{k=0}^{n-1} k/n} = e^{2\pi i \cdot n(n-1)/(2n)} = e^{i\pi(n-1)}

When $n-1$ is even, this equals 1. When $n-1$ is odd, this equals -1.

So the product is $(-1)^{n-1} = (-1)^{n+1}$ .

∎

Corollary 1.1: Sum of Roots of Unity

For $n \geq 2$ :

\sum_{k=0}^{n-1} \omega^k = 0 \quad \text{where } \omega = e^{2\pi i/n}

Proof of Corollary 1.1:

This is a geometric series with ratio $\omega \neq 1$ :

\sum_{k=0}^{n-1} \omega^k = \frac{1 - \omega^n}{1 - \omega} = \frac{1 - 1}{1 - \omega} = 0

∎

Example 4.1: 5th Roots of Unity

The 5th roots of unity are $\omega_k = e^{2\pi ik/5}$ for $k = 0, 1, 2, 3, 4$ .

\omega_0 = 1, \quad \omega_1 = e^{2\pi i/5}, \quad \omega_2 = e^{4\pi i/5}, \quad \omega_3 = e^{6\pi i/5}, \quad \omega_4 = e^{8\pi i/5}

These form a regular pentagon on the unit circle!

Theorem 4.1: Properties of Roots of Unity

Let $\omega = e^{2\pi i/n}$ be a primitive $n$ th root of unity.

The $n$ th roots of unity are $1, \omega, \omega^2, \ldots, \omega^{n-1}$
They form a cyclic group of order $n$ under multiplication
They are evenly spaced on the unit circle
$\omega^n = 1$ and $\omega^k \neq 1$ for $0 < k < n$
$\overline{\omega^k} = \omega^{n-k} = \omega^{-k}$

Example 4.2: Square Roots of i

Problem: Find all square roots of $i$ .

Solution: Write $i = e^{i\pi/2}$ .

z_k = e^{i(\pi/2 + 2\pi k)/2} = e^{i\pi(1 + 4k)/4}, \quad k = 0, 1

$k = 0$ : $z_0 = e^{i\pi/4} = \frac{1}{\sqrt{2}}(1 + i)$
$k = 1$ : $z_1 = e^{i5\pi/4} = \frac{1}{\sqrt{2}}(-1 - i)$

Verify: $\left(\frac{1+i}{\sqrt{2}}\right)^2 = \frac{1 + 2i - 1}{2} = i$ ✓

Remark 4.1: Applications of Roots of Unity

Roots of unity appear in:

Discrete Fourier Transform: Uses $n$ th roots of unity
Cyclotomic polynomials: Minimal polynomials of primitive roots
Filter design: Poles/zeros on the unit circle
Group theory: Cyclic groups of finite order

5. The Fundamental Theorem of Algebra

Theorem 1.6: Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has at least one root in $\mathbb{C}$ .

Equivalently, $\mathbb{C}$ is algebraically closed.

Remark 1.3: Importance for Linear Algebra

This theorem guarantees that the characteristic polynomial of any matrix over $\mathbb{C}$ factors completely into linear factors. Hence:

Every complex matrix has at least one eigenvalue
Every complex matrix can be put into Jordan normal form
The spectral theorem has its cleanest statement over $\mathbb{C}$

Corollary 1.2: Complete Factorization

Every polynomial $p(z) = a_n z^n + \cdots + a_1 z + a_0$ with $a_n \neq 0$ can be written as:

p(z) = a_n(z - r_1)(z - r_2) \cdots (z - r_n)

where $r_1, \ldots, r_n \in \mathbb{C}$ are the roots (counting multiplicity).

Example 5.1: Factoring Over ℂ

Factor $p(x) = x^4 + 4$ over $\mathbb{C}$ .

Solution: Find roots by solving $x^4 = -4 = 4e^{i\pi}$ :

x_k = \sqrt{2} e^{i(\pi + 2\pi k)/4}, \quad k = 0, 1, 2, 3

The roots are:

$x_0 = \sqrt{2} e^{i\pi/4} = 1 + i$
$x_1 = \sqrt{2} e^{i3\pi/4} = -1 + i$
$x_2 = \sqrt{2} e^{i5\pi/4} = -1 - i$
$x_3 = \sqrt{2} e^{i7\pi/4} = 1 - i$

x^4 + 4 = (x - 1 - i)(x + 1 - i)(x + 1 + i)(x - 1 + i)

Theorem 5.1: Real Polynomials Factor into Linear and Quadratic Factors

Every polynomial with real coefficients factors over $\mathbb{R}$ into a product of linear factors and irreducible quadratic factors.

Proof:

By the Fundamental Theorem, $p(x)$ factors completely over $\mathbb{C}$ . Since coefficients are real, complex roots come in conjugate pairs: if $r$ is a root, so is $\bar{r}$ .

Each conjugate pair gives a real quadratic factor:

(x - r)(x - \bar{r}) = x^2 - (r + \bar{r})x + r\bar{r} = x^2 - 2\text{Re}(r)x + |r|^2

This is a quadratic with real coefficients and negative discriminant (irreducible over $\mathbb{R}$ ).

∎

Example 5.2: Factoring Over ℝ

Factor $x^4 + 4$ over $\mathbb{R}$ .

Solution: Pair conjugate roots:

(x - (1+i))(x - (1-i)) = x^2 - 2x + 2

(x - (-1+i))(x - (-1-i)) = x^2 + 2x + 2

Therefore:

x^4 + 4 = (x^2 - 2x + 2)(x^2 + 2x + 2)

6. Geometric Interpretation

Complex numbers have a beautiful geometric interpretation as points (or vectors) in the Euclidean plane, often called the complex plane or Argand diagram.

Definition 6.1: The Complex Plane

The complex plane identifies $z = a + bi$ with the point $(a, b) \in \mathbb{R}^2$ .

The real axis is the horizontal axis (where $b = 0$ )
The imaginary axis is the vertical axis (where $a = 0$ )
The modulus $|z|$ is the distance from the origin
The argument $\arg(z)$ is the angle from the positive real axis

Theorem 6.1: Geometric Meaning of Operations

Addition: $z + w$ corresponds to vector addition (parallelogram law)
Multiplication by $e^{i\theta}$ : Rotates by angle $\theta$ counterclockwise
Multiplication by $r$ : Scales by factor $r$
Conjugation: Reflects across the real axis
Inversion $1/z$ : Inverts modulus and negates argument

Example 6.1: Rotation by 90°

Multiplying by $i$ rotates by $90°$ :

i \cdot (a + bi) = ai + bi^2 = -b + ai

The point $(a, b)$ becomes $(-b, a)$ —exactly a $90°$ counterclockwise rotation!

Theorem 6.2: Triangle Inequality

For all $z, w \in \mathbb{C}$ :

|z + w| \leq |z| + |w|

Equality holds if and only if $z$ and $w$ have the same argument (point in the same direction from the origin).

Proof of Theorem 6.2:

We compute $|z + w|^2$ :

|z + w|^2 = (z + w)(\bar{z} + \bar{w}) = |z|^2 + z\bar{w} + \bar{z}w + |w|^2

Note that $z\bar{w} + \bar{z}w = 2\text{Re}(z\bar{w}) \leq 2|z\bar{w}| = 2|z||w|$ .

|z + w|^2 \leq |z|^2 + 2|z||w| + |w|^2 = (|z| + |w|)^2

Taking square roots gives the triangle inequality.

∎

Corollary 6.1: Reverse Triangle Inequality

For all $z, w \in \mathbb{C}$ :

\big||z| - |w|\big| \leq |z - w|

Example 6.2: Circles in the Complex Plane

The equation $|z - z_0| = r$ describes a circle:

Center: $z_0$
Radius: $r$

For example, $|z - 2 - i| = 3$ is a circle centered at $2 + i$ with radius 3.

Example 6.3: Lines in the Complex Plane

The equation $\text{Re}(\bar{a}z) = c$ describes a line.

For $a = 1 + i$ and $c = 2$ :

\text{Re}((1-i)(x+iy)) = \text{Re}(x + y + i(y - x)) = x + y = 2

This is the line $x + y = 2$ .

Definition 6.2: Unit Circle

The unit circle is the set $\{z \in \mathbb{C} : |z| = 1\}$ . Points on the unit circle have the form $e^{i\theta} = \cos\theta + i\sin\theta$ .

Remark 6.1: Complex Numbers and 2D Geometry

Complex numbers elegantly encode 2D geometry:

Distance between $z$ and $w$ : $|z - w|$
Midpoint of $z$ and $w$ : $\frac{z + w}{2}$
Rotation by $\theta$ : multiply by $e^{i\theta}$
Reflection across real axis: take conjugate

Example 6.4: Rotation of a Point

Rotate the point $z = 3 + 4i$ by $90°$ counterclockwise around the origin.

Solution:

z' = e^{i\pi/2} \cdot z = i(3 + 4i) = 3i + 4i^2 = -4 + 3i

The point $(3, 4)$ becomes $(-4, 3)$ .

7. Connection to Linear Algebra

Complex numbers are deeply connected to linear algebra in several ways.

Eigenvalues: Even real matrices may have complex eigenvalues. For example, rotation matrices have eigenvalues $e^{\pm i\theta}$ .
Characteristic polynomial: Factors completely over $\mathbb{C}$ , guaranteeing eigenvalues exist.
Spectral theorem: Normal matrices are unitarily diagonalizable over $\mathbb{C}$ .
Jordan form: Every matrix has a Jordan normal form over $\mathbb{C}$ .
Inner products: Complex inner products use conjugation: $\langle u, v \rangle = \sum \bar{u}_i v_i$ .

Example 7.1: Rotation Matrix Eigenvalues

The 2D rotation matrix by angle $\theta$ :

R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

has characteristic polynomial $\lambda^2 - 2\cos\theta \cdot \lambda + 1$ .

The eigenvalues are $\lambda = \cos\theta \pm i\sin\theta = e^{\pm i\theta}$ .

For $\theta \neq 0, \pi$ , these are complex (non-real) eigenvalues!

Example 7.2: Complex Inner Product

For vectors $u, v \in \mathbb{C}^n$ , the standard inner product is:

\langle u, v \rangle = \sum_{k=1}^n \bar{u}_k v_k = \bar{u}^T v

Note the conjugation on the first argument! This ensures $\langle u, u \rangle \geq 0$ .

Remark 7.1: Why Conjugation in Inner Products?

Without conjugation, $\langle iu, iu \rangle = i^2 \|u\|^2 = -\|u\|^2 < 0$ .

With conjugation, $\langle iu, iu \rangle = \bar{i} \cdot i \|u\|^2 = |i|^2 \|u\|^2 = \|u\|^2 > 0$ . ✓

Example 7.3: Hermitian Matrices

A matrix $A$ is Hermitian if $A^* = A$ (conjugate transpose equals itself).

A = \begin{pmatrix} 2 & 3-i \\ 3+i & 5 \end{pmatrix}

Hermitian matrices have:

Real eigenvalues (even over $\mathbb{C}$ )
Orthogonal eigenvectors
Unitary diagonalization: $A = UDU^*$

Theorem 7.1: Complex Eigenvalues of Real Matrices

If $A$ is a real matrix and $\lambda \in \mathbb{C}$ is an eigenvalue with eigenvector $v$ , then $\bar{\lambda}$ is also an eigenvalue with eigenvector $\bar{v}$ .

Proof:

Since $Av = \lambda v$ and $A$ has real entries ( $\bar{A} = A$ ):

A\bar{v} = \bar{A}\bar{v} = \overline{Av} = \overline{\lambda v} = \bar{\lambda}\bar{v}

∎

8. Worked Examples

Example 8.1: Complex Division

Problem: Compute $\frac{3 + 4i}{1 + 2i}$ .

Solution: Multiply by conjugate:

\frac{3 + 4i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{(3 + 4i)(1 - 2i)}{1 + 4} = \frac{3 - 6i + 4i - 8i^2}{5}

= \frac{3 - 2i + 8}{5} = \frac{11 - 2i}{5} = \frac{11}{5} - \frac{2}{5}i

Example 8.2: Finding All Roots

Problem: Find all solutions to $z^4 = -16$ .

Solution: Write $-16 = 16e^{i\pi}$ .

z_k = 2e^{i(\pi + 2\pi k)/4} = 2e^{i\pi(1 + 2k)/4}, \quad k = 0, 1, 2, 3

$k=0$ : $z_0 = 2e^{i\pi/4} = \sqrt{2}(1 + i)$
$k=1$ : $z_1 = 2e^{i3\pi/4} = \sqrt{2}(-1 + i)$
$k=2$ : $z_2 = 2e^{i5\pi/4} = \sqrt{2}(-1 - i)$
$k=3$ : $z_3 = 2e^{i7\pi/4} = \sqrt{2}(1 - i)$

Example 8.3: Proving Trigonometric Identities

Problem: Prove $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ .

Solution: Use De Moivre:

\cos(2\theta) + i\sin(2\theta) = (\cos\theta + i\sin\theta)^2

= \cos^2\theta + 2i\cos\theta\sin\theta + i^2\sin^2\theta

= (\cos^2\theta - \sin^2\theta) + i(2\cos\theta\sin\theta)

Comparing real parts: $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ .

Example 8.4: Powers via Polar Form

Problem: Compute $(\sqrt{3} + i)^6$ .

Solution: Convert to polar form:

|\sqrt{3} + i| = \sqrt{3 + 1} = 2, \quad \arg(\sqrt{3} + i) = \arctan(1/\sqrt{3}) = \pi/6

So $\sqrt{3} + i = 2e^{i\pi/6}$ .

(\sqrt{3} + i)^6 = 2^6 e^{i \cdot 6 \cdot \pi/6} = 64 e^{i\pi} = 64(-1) = -64

Example 8.5: Solving Quadratics

Problem: Solve $z^2 + 2z + 5 = 0$ .

Solution: Use the quadratic formula:

z = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i

Example 8.6: Verifying Euler's Formula

Problem: Verify $e^{i\pi} + 1 = 0$ .

Solution:

e^{i\pi} = \cos\pi + i\sin\pi = -1 + i \cdot 0 = -1

Therefore $e^{i\pi} + 1 = -1 + 1 = 0$ . ✓

Example 8.7: Product of Complex Numbers

Problem: Compute $(1 + i)(2 - 3i)(1 + 2i)$ .

Solution: First multiply $(1 + i)(2 - 3i)$ :

(1 + i)(2 - 3i) = 2 - 3i + 2i - 3i^2 = 2 - i + 3 = 5 - i

Then multiply by $(1 + 2i)$ :

(5 - i)(1 + 2i) = 5 + 10i - i - 2i^2 = 5 + 9i + 2 = 7 + 9i

Example 8.8: Complex Exponential

Problem: Simplify $e^{2 + i\pi/4}$ .

Solution:

e^{2 + i\pi/4} = e^2 \cdot e^{i\pi/4} = e^2(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}) = e^2 \cdot \frac{\sqrt{2}}{2}(1 + i)

= \frac{e^2\sqrt{2}}{2}(1 + i) \approx 5.22(1 + i)

Example 8.9: Using De Moivre for Trig

Problem: Express $\cos(3\theta)$ in terms of $\cos\theta$ .

Solution: Let $c = \cos\theta$ and $s = \sin\theta$ .

\cos(3\theta) + i\sin(3\theta) = (c + is)^3

= c^3 + 3c^2(is) + 3c(is)^2 + (is)^3 = c^3 + 3c^2 is - 3cs^2 - is^3

= (c^3 - 3cs^2) + i(3c^2s - s^3)

Taking real parts: $\cos(3\theta) = c^3 - 3c(1-c^2) = 4c^3 - 3c = 4\cos^3\theta - 3\cos\theta$ .

Example 8.10: Absolute Value Inequality

Problem: Show that $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)$ .

Solution: Expand using $|w|^2 = w\bar{w}$ :

|z_1 + z_2|^2 = (z_1 + z_2)(\bar{z}_1 + \bar{z}_2) = |z_1|^2 + z_1\bar{z}_2 + \bar{z}_1 z_2 + |z_2|^2

|z_1 - z_2|^2 = (z_1 - z_2)(\bar{z}_1 - \bar{z}_2) = |z_1|^2 - z_1\bar{z}_2 - \bar{z}_1 z_2 + |z_2|^2

Adding: $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$ . ✓

(This is the parallelogram law.)

9. Common Mistakes

Mistake 1: Forgetting conjugation in modulus

$|z|^2 = z \cdot \bar{z}$ , NOT $z \cdot z = z^2$ . For example, $|i|^2 = i \cdot (-i) = 1$ , but $i^2 = -1$ .

Mistake 2: Wrong argument quadrant

$\arctan(b/a)$ doesn't account for the quadrant. For $-1 + i$ , $\arctan(-1) = -\pi/4$ , but the correct argument is $3\pi/4$ .

Mistake 3: Missing roots

When finding $n$ th roots, there are always $n$ distinct roots. Don't forget to find all of them using $e^{2\pi ik/n}$ for $k = 0, 1, \ldots, n-1$ .

Mistake 4: Treating i as a variable

$i$ is a constant with $i^2 = -1$ . Don't treat it like an unknown. Simplify powers of $i$ : $i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$ , then repeat.

Mistake 5: Trying to order complex numbers

You cannot say $1 + i > 2$ or $i > 0$ . Complex numbers have no compatible ordering. Only compare magnitudes: $|1 + i| = \sqrt{2} < 2$ .

10. Key Takeaways

Construction

$\mathbb{C} = \mathbb{R}^2$ with multiplication $(a,b)(c,d) = (ac-bd, ad+bc)$ . This makes $i^2 = -1$ .

Polar Form

$z = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$ . Multiplication scales and rotates.

Euler's Formula

$e^{i\theta} = \cos\theta + i\sin\theta$ . The bridge between exponentials and trigonometry.

Algebraically Closed

Every polynomial over $\mathbb{C}$ factors completely. Every degree- $n$ polynomial has exactly $n$ roots.

Complex Numbers Practice

Questions

Correct

Accuracy

What is the modulus of

z = 3 + 4i

Easy

Not attempted

What is

i^{2023}

Easy

Not attempted

Express

e^{i\pi/3}

in rectangular form.

Medium

Not attempted

What is

\overline{(2 + 3i)(1 - i)}

Medium

Not attempted

How many cube roots does

-8

have in

\mathbb{C}

Medium

Not attempted

|z| = 2

and

\arg(z) = \pi/4

, what is

z^4

Hard

Not attempted

Which statement about

\mathbb{C}

is FALSE?

Hard

Not attempted

What is

\sum_{k=0}^{n-1} e^{2\pi i k/n}

for

n \geq 2

Hard

Not attempted

What is

\text{Re}(e^{i\theta})

Easy

Not attempted

z = 2e^{i\pi/6}

, what is

\bar{z}

Medium

Not attempted

The 4th roots of unity are:

Easy

Not attempted

What is the principal argument of

z = -1 - i

Hard

Not attempted

Frequently Asked Questions

Why do we need complex numbers in linear algebra?

Complex numbers are essential for spectral theory. Over ℝ, not all matrices have eigenvalues (e.g., rotation matrices). Over ℂ, the Fundamental Theorem of Algebra guarantees that every polynomial factors completely, so every matrix has eigenvalues.

What's the geometric meaning of complex multiplication?

Multiplying by z = re^{iθ} scales by r and rotates by θ. This makes complex numbers perfect for describing rotations and scaling in 2D, which generalizes to understanding linear operators geometrically.

Is there a 3D analog of complex numbers?

No in a strict sense—you can't extend ℂ to 3D while preserving all field properties. However, quaternions (4D) extend complex numbers and are useful for 3D rotations. Octonions extend further but lose associativity.

What does 'algebraically closed' mean precisely?

A field F is algebraically closed if every non-constant polynomial with coefficients in F has a root in F. Equivalently, every polynomial of degree n over F has exactly n roots (counting multiplicity) in F.

How do complex conjugates help with division?

To divide by z = a + bi, multiply numerator and denominator by z̄ = a - bi. This gives z·z̄ = |z|² in the denominator, which is a real number, making the division straightforward.

What is the triangle inequality for complex numbers?

For complex numbers z and w: |z + w| ≤ |z| + |w|. Geometrically, this says the length of one side of a triangle is at most the sum of the other two sides. Equality holds iff z and w point in the same direction.

Why is e^{iπ} = -1 called 'the most beautiful equation'?

Euler's identity e^{iπ} + 1 = 0 connects five fundamental constants: e, i, π, 1, and 0, using three basic operations: addition, multiplication, and exponentiation. It links analysis, algebra, and geometry in one elegant formula.

Can complex numbers be compared with < or >?

No! ℂ cannot be ordered compatibly with its field structure. If we tried to say i > 0, then i² = -1 > 0, which contradicts -1 < 0. Complex numbers have no natural ordering, only magnitude comparisons via |z|.

What's the difference between argument and principal argument?

The argument of z is any angle θ such that z = |z|e^{iθ}. Since e^{iθ} = e^{i(θ+2πk)}, the argument is only defined up to multiples of 2π. The principal argument restricts θ to a specific range, usually (-π, π] or [0, 2π).

How are complex numbers used in physics?

Complex numbers appear throughout physics: AC circuit analysis uses impedance (complex resistance), quantum mechanics uses complex wave functions, and signal processing uses Fourier transforms with complex exponentials.