Lecture 3: Complex Numbers and Polynomials - Study Guide

Course Overview and Introduction

• This lecture constitutes the third session of the Mathematics I module, focused on complex numbers.

• Delivered on April 29, 2026, by Stephan Block from the AG Bionanogrenzflächen, Institute for Physical and Theoretical Chemistry, Department of Biology, Chemistry, Pharmacy (BCP), Freie Universität Berlin.

• The motivation for introducing complex numbers arises from equations that cannot be solved using only real numbers, such as solving for $x$ in the quadratic equation: $x^2 + 2x + 3 = 0$

Learning Objectives and Structure

• The lecture is divided into four primary sections:   - 1. Complex Numbers: Definition and arithmetic rules (Rechenregeln).   - 2. Methods of representing complex numbers.   - 3. Polynomials and the Fundamental Theorem of Algebra (Nullstellensatz).   - 4. The Horner Scheme (Horner-Schema).

• By the end of the lecture, students are expected to:   - Know the fundamental properties of complex numbers.   - Recognize the advantages offered by the various forms of representing complex numbers.   - Be able to represent polynomials as products of linear factors.

Definition of Complex Numbers

• The set of complex numbers $\mathbb{C}$ is defined as follows: $\mathbb{C} \coloneq {z = x + i \cdot y : x, y \in \mathbb{R}}$

• The symbol $i$ represents the imaginary unit, defined as $i = \sqrt{-1}$.

• Complex numbers are composed of two real numbers, which leads to two structural consequences:   - They cannot be represented in a one-dimensional space.   - They cannot be ordered (sorted) like real numbers.

• The two-dimensional representation of complex numbers is known as the Gaussian number plane (Gausssche Zahlenebene), where the horizontal axis is the Real part (Re) and the vertical axis is the Imaginary part (Im).

Properties and Forms of Complex Numbers I: Cartesian Form

• The representation $z = x + i \cdot y$ is called the Cartesian form (kartesische Form).

• The components are defined as:   - $x = \text{Re}(z)$ (the Real part).   - $y = \text{Im}(z)$ (the Imaginary part).

• The absolute value or modulus of $z$, denoted as $|z|$ or $r$, is calculated using the Pythagorean theorem: $r = |z| = \sqrt{x^2 + y^2}$

• The angle $\phi$ enclosed between $z$ and the real axis is called the argument of $z$: $\phi = \text{arg}(z) = \arctan(y/x)$

• For any complex number $z = x + i \cdot y$, the complex conjugate (komplex konjugierte Zahl), denoted as $z^<em>$, is defined as: $z^</em> = x - i \cdot y$

• A key property is that the modulus of a number and its conjugate are identical: $|z| = |z^*|$

Properties and Forms of Complex Numbers II: Arithmetic Operations

• Addition and subtraction are performed by applying the operation to the real and imaginary parts separately: $(a + i \cdot b) \pm (c + i \cdot d) = (a \pm c) + i \cdot (b \pm d)$

• Multiplication involves mixing the real and imaginary parts via the distributive property: $(a + i \cdot b) \cdot (c + i \cdot d) = ac + i \cdot (ad + bc) + i^2 \cdot bd$

• Since $i^2 = -1$, the multiplication formula simplifies to: $(a + i \cdot b) \cdot (c + i \cdot d) = (ac - bd) + i \cdot (ad + bc)$

• Division utilizes the property that $z \cdot z^* = |z|^2$, where $|z|^2 = x^2 + y^2$. Thus, the reciprocal of $z$ is: $\frac{1}{z} = \frac{z^*}{|z|^2} = \frac{x}{x^2 + y^2} - i \cdot \frac{y}{x^2 + y^2}$

• To divide two complex numbers, multiply the fraction by the complex conjugate of the denominator: $\frac{a + i \cdot b}{c + i \cdot d} = \frac{a + i \cdot b}{c + i \cdot d} \cdot \frac{c - i \cdot d}{c - i \cdot d} = \frac{(a + i \cdot b) \cdot (c - i \cdot d)}{c^2 + d^2}$

• Example of division calculation: $\frac{1 + i \cdot 2}{3 + i \cdot 4} = \frac{1 + i \cdot 2}{3 + i \cdot 4} \cdot \frac{3 - i \cdot 4}{3 - i \cdot 4} = \frac{3 + 8 + i \cdot (6 - 4)}{9 + 16} = \frac{11 + i \cdot 2}{25} = \frac{11}{25} + i \cdot \frac{2}{25}$

• Note: The transcript contains a manual calculation correction where $\frac{1 + i \cdot 2}{3 + i \cdot 4} \cdot \frac{3 - i \cdot 4}{3 - i \cdot 4}$ leads to a result designated as $\frac{15 + i \cdot 2}{25}$ in the provided graphic, though standard algebra yields $\frac{11 + i \cdot 2}{25}$.

• Complex numbers satisfy commutative, associative, and distributive laws, meaning they form a field (Körper). All basic arithmetic operations (+, –, $\cdot$, :) are performable except for division by zero.

Properties and Forms of Complex Numbers III: Polar and Trigonometric Forms

• Complex numbers can be represented in trigonometric form: $z = r \cdot \cos(\phi) + i \cdot r \cdot \sin(\phi)$

• They can also be represented in Polar form (also called Euler's form / Eulersche Formel): $z = r \cdot e^{i \cdot \phi} = r \cdot \exp(i \cdot \phi)$

• In these forms, $e$ is Euler's number ($\approx 2.71828$) and $\phi$ must be provided in radians (rad). The conversion from degrees ($\phi^\circ$) to radians ($\phi_{\text{rad}}$) is: $\phi_{\text{rad}} = \phi^\circ \times \frac{2\pi}{360^\circ}$

• Key logarithmic/exponential identities include: $|e^{i \cdot \phi}| = 1$ $e^{i \cdot \phi} \cdot e^{i \cdot \psi} = e^{i \cdot (\phi + \psi)}$

• Multiplication and division are significantly easier in polar form:   - Multiplication: $r \cdot e^{i \cdot \phi} \cdot s \cdot e^{i \cdot \psi} = (r \cdot s) \cdot e^{i \cdot (\phi + \psi)}$   - Division: $\frac{r \cdot e^{i \cdot \phi}}{s \cdot e^{i \cdot \psi}} = \frac{r}{s} \cdot e^{i \cdot (\phi - \psi)}$

• Practical Rule: Addition/Subtraction is best performed in Cartesian form; Multiplication/Division is best performed in Polar form.

Roots of Complex Numbers

• Calculating the $n$-th root of a complex number $z$ is best done in polar representation.

• Because of periodicity, $z = r \cdot e^{i \cdot \phi}$ is equivalent to $z = r \cdot e^{i \cdot (\phi + 2\pi k)}$ for any integer $k \in \mathbb{Z}$.

• The $n$-th roots are given by the formula: $z_k = \sqrt[n]{r} \cdot e^{i \cdot \frac{\phi + 2\pi k}{n}}$

• There are exactly $n$ distinct $n$-th roots, found by evaluating for $k = 0, 1, \dots, n - 1$.

Polynomials and the Fundamental Theorem of Algebra

• A polynomial of degree $n$ is an expression of the form: $a(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

• While in school these typically involve real variables ($x$) and real coefficients ($a_i$), we can define complex-valued polynomials where the variable ($z$) and the coefficients ($a_i$) are elements of $\mathbb{C}$.

• The Fundamental Theorem of Algebra states that a complex-valued polynomial of degree $n$ has exactly $n$ complex roots (Nullstellen), counting multiplicity (where a $k$-fold root is counted $k$ times).

• If $z_1, \dots, z_n$ are the roots, the polynomial can be factored as a product of linear factors: $a(z) = (z - z_1) \cdot (z - z_2) \cdot \dots \cdot (z - z_n)$

• Difference between Real and Complex Polynomials: Real polynomials may not have real roots. For example, $z^2 + 2 = 0$ has two roots in $\mathbb{C}$, but zero roots in $\mathbb{R}$.

Dividing Polynomials and the Horner Scheme

• For polynomials of degree 3 or higher, there is no general formula for finding roots (unlike the p-q formula for degree 2).

• If one root of a polynomial is known, the complexity can be reduced by "dividing out" (Abdividieren) the corresponding linear factor, thereby reducing the degree of the polynomial.

• The Horner Scheme is a visual tool to perform this division, applicable to both real and complex-valued polynomials.

• Example Process for $x^3 + 2x^2 - 5x - 6$:   - 1. Identify a root: By observation, $-1$ is a root because $(-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0$.   - 2. The polynomial can be rewritten as $a(x) = (x + 1) \cdot (x - z_2) \cdot (x - z_3)$.   - 3. Set up the scheme: Write the coefficients of the polynomial in the top row: $1$, $2$, $-5$, $-6$.   - 4. Procedure: Bring the first coefficient ($1$) down. Multiply the root ($-1$) by the lower number and add it to the next coefficient in the top row.     - $1 \times (-1) = -1$; Add to next coefficient ($2$): $2 + (-1) = 1$.     - $1 \times (-1) = -1$; Add to next coefficient ($-5$): $-5 + (-1) = -6$.     - $-6 \times (-1) = 6$; Add to next coefficient ($-6$): $-6 + 6 = 0$.   - 5. Result: The last number is the remainder (0 confirms it is a root). The remaining numbers ($1$, $1$, $-6$) are the coefficients of the resulting lower-degree polynomial: $\frac{x^3 + 2x^2 - 5x - 6}{x + 1} = x^2 + x - 6$