Complex Numbers: Algebra and Geometry

Why Real Numbers Are Not Enough

The equation $x^2 + 1 = 0$ has no solutions in real numbers. This is inconvenient—since many problems in physics and mathematics lead precisely to such equations. Mathematicians of the 16th century (Cardano, Bombelli) began to “pretend” that $\sqrt{-1}$ exists, and discovered that it works.

The imaginary unit $i$ is defined by the condition $i^2 = -1$. A complex number $z = a + bi$, where $a = \operatorname{Re}(z)$ is the real part, $b = \operatorname{Im}(z)$ is the imaginary part.

Algebraic Operations

Addition: $(a+bi) + (c+di) = (a+c) + (b+d)i$.

Multiplication: $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$ (using $i^2 = -1$).

Division: $\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{|c+di|^2} = \frac{(ac+bd) + (bc-ad)i}{c^2+d^2}$.

Conjugate: $\bar{z} = a - bi$. Properties: $z \cdot \bar{z} = a^2 + b^2 = |z|^2$ (a real number).

Geometry: The Complex Plane

A complex number $z = a+bi$ is represented as the point $(a, b)$ in the plane (the Gaussian plane). The modulus $|z| = \sqrt{a^2 + b^2}$ is the distance from the origin. The argument $\operatorname{arg}(z) = \arctan(b/a)$ is the angle with the real axis.

Trigonometric form: $z = |z|(\cos \varphi + i \sin \varphi)$, where $\varphi = \operatorname{arg}(z)$.

Euler’s Formula: $e^{i\varphi} = \cos \varphi + i \sin \varphi$.

This is one of the most beautiful formulas in mathematics. For $\varphi = \pi$: $e^{i\pi} = -1$, or $e^{i\pi} + 1 = 0$—“Euler’s formula”, uniting $e$, $i$, $\pi$, $1$, and $0$.

Exponential form: $z = |z| \cdot e^{i\varphi}$.

Multiplication in Geometric Form

$z_1 z_2 = |z_1||z_2| \cdot e^{i(\varphi_1+\varphi_2)}$: moduli are multiplied, arguments are added.

Multiplication by $e^{i\varphi}$ is a rotation by angle $\varphi$. Multiplication by $r$ is scaling by $r$.

De Moivre’s Formula: $(\cos \varphi + i \sin \varphi)^n = \cos(n\varphi) + i \sin(n\varphi)$.

Application: $\sin(3\varphi) = 3\cos^2\varphi \cdot \sin\varphi - \sin^3\varphi$ follows from $\operatorname{Re}[(\cos \varphi + i \sin \varphi)^3]$.

Roots of Complex Numbers

$n$th roots of $z = r \cdot e^{i\varphi}$: $z_k = r^{1/n} \cdot e^{i(\varphi + 2\pi k)/n}$, $k = 0, 1, ..., n-1$.

Roots of unity: $\omega_k = e^{2\pi i k/n}$—a regular $n$-gon on the unit circle. In the theory of DFT (Discrete Fourier Transform), roots of unity are main objects.

Fundamental Theorem of Algebra

Any polynomial of degree $n \geq 1$ with complex coefficients has exactly $n$ roots in $\mathbb{C}$ (counting multiplicities).

This is a theorem of existence—it does not give the roots explicitly. The proof uses topological arguments (fundamental group of the circle) or functional-analytic ones (Liouville’s theorem in complex analysis).