Complex Numbers

The complex numbers $\mathbb{C}$ extend the real numbers $\mathbb{R}$ by adjoining a square root of $-1$ . This algebraic completion enables powerful geometric and analytic techniques unavailable in real analysis.

Definition and Basic Properties

Definition1.1Complex number

A complex number is an expression of the form $z = x + iy$ where $x, y \in \mathbb{R}$ and $i$ satisfies $i^2 = -1$ . The set of all complex numbers is

$\mathbb{C} = \{x + iy \mid x, y \in \mathbb{R}\}.$

The real part of $z$ is $\text{Re}(z) = x$ and the imaginary part is $\text{Im}(z) = y$ .

RemarkAlgebraic structure

$\mathbb{C}$ forms a field under the operations $(x + iy) + (u + iv) = (x+u) + i(y+v)$ and $(x+iy)(u+iv) = (xu - yv) + i(xv + yu)$ . The multiplicative identity is $1 = 1 + 0i$ and the additive identity is $0 = 0 + 0i$ .

ExampleBasic arithmetic

$(2 + 3i) + (1 - 4i) = 3 - i$
$(1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2$
$(2 + i)(3 + 2i) = 6 + 4i + 3i + 2i^2 = 6 + 7i - 2 = 4 + 7i$
$i^3 = i^2 \cdot i = -i$ , $i^4 = (i^2)^2 = 1$

Geometric Representation

Definition1.2Complex plane

The complex plane (or Argand diagram) identifies $\mathbb{C}$ with $\mathbb{R}^2$ via $z = x + iy \leftrightarrow (x, y)$ . The $x$ -axis is the real axis and the $y$ -axis is the imaginary axis.

RemarkPolar coordinates

Every nonzero $z \in \mathbb{C}$ can be written in polar form:

$z = r e^{i\theta} = r(\cos\theta + i\sin\theta)$

where $r = |z| = \sqrt{x^2 + y^2}$ is the modulus (or absolute value) and $\theta = \arg(z)$ is the argument (angle from the positive real axis). The argument is determined up to integer multiples of $2\pi$ .

ExamplePolar representation

$z = 1 + i$ has $|z| = \sqrt{2}$ and $\arg(z) = \pi/4$ , so $z = \sqrt{2} e^{i\pi/4}$ .
$z = -1$ has $|z| = 1$ and $\arg(z) = \pi$ , so $z = e^{i\pi}$ , confirming Euler's identity $e^{i\pi} + 1 = 0$ .
$z = i$ has $|z| = 1$ and $\arg(z) = \pi/2$ , so $z = e^{i\pi/2}$ .

Conjugation and Modulus

Definition1.3Complex conjugate

The complex conjugate of $z = x + iy$ is $\bar{z} = x - iy$ . Geometrically, $\bar{z}$ is the reflection of $z$ across the real axis.

RemarkProperties of conjugation

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

$\overline{z + w} = \bar{z} + \bar{w}$
$\overline{zw} = \bar{z}\,\bar{w}$
$\overline{(z/w)} = \bar{z}/\bar{w}$ (for $w \neq 0$ )
$z + \bar{z} = 2\text{Re}(z)$ and $z - \bar{z} = 2i\text{Im}(z)$
$z\bar{z} = |z|^2 = x^2 + y^2$
$\overline{\bar{z}} = z$

ExampleReciprocal via conjugation

To compute $1/(3 + 4i)$ , multiply numerator and denominator by the conjugate:

$\frac{1}{3 + 4i} = \frac{1}{3+4i} \cdot \frac{3-4i}{3-4i} = \frac{3 - 4i}{9 + 16} = \frac{3 - 4i}{25} = \frac{3}{25} - \frac{4}{25}i.$

Triangle Inequality and Algebraic Identities

Theorem1.1Triangle 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 (i.e., $w = tz$ for some $t \geq 0$ ).

RemarkReverse triangle inequality

From the triangle inequality, we also have

$\big||z| - |w|\big| \leq |z - w|.$

This is useful for proving continuity of the modulus function.

ExampleNumerical verification

Let $z = 3 + 4i$ and $w = 1 - 2i$ . Then $|z| = 5$ , $|w| = \sqrt{5}$ , and $z + w = 4 + 2i$ so $|z+w| = \sqrt{20} = 2\sqrt{5} \approx 4.47$ . Indeed, $2\sqrt{5} < 5 + \sqrt{5} \approx 7.24$ , confirming the triangle inequality.

Multiplication as Rotation and Scaling

RemarkGeometric interpretation of multiplication

If $z = r e^{i\alpha}$ and $w = s e^{i\beta}$ , then

$zw = (rs) e^{i(\alpha + \beta)}.$

Thus multiplication by $w$ corresponds to:

Scaling by $|w|$
Rotation by $\arg(w)$

This geometric interpretation is fundamental in complex analysis and explains many phenomena (e.g., the behavior of analytic functions as conformal maps).

ExampleMultiplication by i

Multiplication by $i = e^{i\pi/2}$ rotates a complex number by $90^\circ$ counterclockwise:

$i \cdot 1 = i$
$i \cdot i = -1$
$i \cdot (3 + 4i) = 3i + 4i^2 = -4 + 3i$

Geometrically, $(3, 4)$ rotates to $(-4, 3)$ .

Roots of Unity and De Moivre's Formula

Theorem1.2De Moivre's formula

For any integer $n$ and real $\theta$ :

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

In exponential form: $(e^{i\theta})^n = e^{in\theta}$ .

Definition1.4Roots of unity

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

$\omega_k = e^{2\pi i k/n}, \quad k = 0, 1, \ldots, n-1.$

These points are equally spaced around the unit circle, forming the vertices of a regular $n$ -gon.

ExampleCube roots of unity

The solutions to $z^3 = 1$ are:

$1, \quad e^{2\pi i/3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}, \quad e^{4\pi i/3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}.$

These form an equilateral triangle centered at the origin.

ExampleGeneral nth roots

To solve $z^n = w$ where $w = r e^{i\theta}$ , write

$z = \sqrt[n]{r}\, e^{i(\theta + 2\pi k)/n}, \quad k = 0, 1, \ldots, n-1.$

For instance, the square roots of $i = e^{i\pi/2}$ are $e^{i\pi/4}$ and $e^{i5\pi/4}$ :

$\sqrt{i} = \frac{1}{\sqrt{2}}(1 + i) \quad \text{and} \quad -\frac{1}{\sqrt{2}}(1 + i).$

Summary

RemarkKey takeaways

$\mathbb{C}$ is a field extending $\mathbb{R}$ .
The geometric viewpoint (complex plane, polar form) is essential.
Conjugation and modulus satisfy algebraic identities that mirror real analysis.
Multiplication combines scaling and rotation, providing a rich geometric structure.
Roots of unity and De Moivre's formula unlock polynomial equations and periodicity.

These foundational properties underpin all of complex analysis, from differentiability (analytic functions) to integration (Cauchy's theorem).