Complex Numbers¶

The Imaginary Unit¶

The equation $x^2 = -1$ has no solution in the real numbers. The imaginary unit $i$ is defined as a solution to this equation:

\[ i^2 = -1 \]

With $i$, all quadratic equations become solvable, including those with negative discriminant.

Representation and Basic Concepts¶

A complex number has the form $z = a + bi$ with $a, b \in \mathbb{R}$. Here $a = \operatorname{Re}(z)$ is the real part and $b = \operatorname{Im}(z)$ is the imaginary part.

The set of all complex numbers is denoted $\mathbb{C}$:

\[ \mathbb{C} = \{a + bi : a, b \in \mathbb{R}\} \]

Every real number is a complex number with $b = 0$, so $\mathbb{R} \subset \mathbb{C}$.

Conjugation and Absolute Value¶

The complex conjugate of $z = a + bi$ is:

\[ \bar{z} = a - bi \]

The absolute value (modulus) of $z$ is:

\[ |z| = \sqrt{a^2 + b^2} = \sqrt{z \cdot \bar{z}} \]

Example. For $z = 3 + 4i$: $\bar{z} = 3 - 4i$ and $|z| = \sqrt{9 + 16} = 5$.

Arithmetic Operations¶

For $z_1 = a + bi$ and $z_2 = c + di$:

Addition: $$ z_1 + z_2 = (a + c) + (b + d)i $$

Multiplication (using $i^2 = -1$): $$ z_1 \cdot z_2 = (ac - bd) + (ad + bc)i $$

Division (multiplying by the conjugate of the denominator): $$ \frac{z_1}{z_2} = \frac{z_1 \cdot \bar{z_2}}{|z_2|^2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} $$

Example. $(2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5 + i$.

Polar Form¶

Every complex number $z \neq 0$ can be written in polar form:

\[ z = r(\cos\varphi + i\sin\varphi) = r \cdot e^{i\varphi} \]

Here $r = |z|$ is the modulus and $\varphi = \arg(z)$ is the argument (angle to the positive real axis). The second equality uses Euler's formula:

\[ e^{i\varphi} = \cos\varphi + i\sin\varphi \]

"The formula $e^{i\pi} + 1 = 0$ connects the five most important constants in mathematics." — Eli Maor, e: The Story of a Number, Princeton University Press, 1994.

The polar form simplifies multiplication and exponentiation: moduli are multiplied, arguments are added.

Roots of Unity¶

The $n$-th roots of unity are the $n$ solutions of the equation $z^n = 1$:

\[ \zeta_k = e^{2\pi i k/n}, \quad k = 0, 1, \ldots, n-1 \]

The number $\zeta = e^{2\pi i/n}$ is called a primitive $n$-th root of unity. All $n$-th roots of unity are powers of $\zeta$: $\{\zeta^0, \zeta^1, \ldots, \zeta^{n-1}\}$.

Example. The cube roots of unity ($n = 3$) are:

\[ \zeta_0 = 1, \quad \zeta_1 = e^{2\pi i/3} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad \zeta_2 = e^{4\pi i/3} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \]

In number theory, roots of unity play a central role — for instance as the basis of cyclotomic rings $\mathbb{Z}[\zeta_p]$, which appear in Kummer's approach to proving Fermat's Last Theorem.

The Upper Half-Plane¶

The upper half-plane is the set of all complex numbers with positive imaginary part:

\[ \mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\} \]

The upper half-plane is the natural domain of modular forms — complex-valued functions with special symmetry properties.

Summary¶

Concept	Definition
Imaginary unit	$i^2 = -1$
Complex number	$z = a + bi$ with $a, b \in \mathbb{R}$
Conjugate	$\overline{a + bi} = a - bi$
Absolute value	$
Euler's formula	$e^{i\varphi} = \cos\varphi + i\sin\varphi$
$n$-th root of unity	$\zeta = e^{2\pi i/n}$
Upper half-plane	$\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}$

References¶

Needham, Tristan: Visual Complex Analysis. Oxford University Press, 1997. Chapters 1–4.
Ahlfors, Lars V.: Complex Analysis. McGraw-Hill, 3rd edition, 1979. Chapter 1.
Maor, Eli: e: The Story of a Number. Princeton University Press, 1994.

Concept	Definition
Imaginary unit	\(i^2 = -1\)
Complex number	\(z = a + bi\) with \(a, b \in \mathbb{R}\)
Conjugate	\(\overline{a + bi} = a - bi\)
Absolute value	$
Euler's formula	\(e^{i\varphi} = \cos\varphi + i\sin\varphi\)
\(n\)-th root of unity	\(\zeta = e^{2\pi i/n}\)
Upper half-plane	\(\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}\)