Complex Numbers and the Complex Plane

The field of real numbers $R$ is insufficient for algebra because many simple polynomial equations, such as $x^{2} + 1 = 0$ , have no real solutions. To achieve Algebraic Closure, we extend the real line into a two-dimensional space by introducing the imaginary unit $i$ , defined by the property $i^{2} = - 1$ . The resulting field, the Complex Numbers ( $C$ ), is the cornerstone of modern analysis, quantum mechanics, and engineering.

Algebraic Structure of C

A complex number $z \in C$ is an expression of the form $z = a + bi$ , where $a, b \in R$ .

Real Part: $Re (z) = a$ .
Imaginary Part: $Im (z) = b$ .
Conjugate: $\overset{z}{ˉ} = a - bi$ . The product $z \overset{z}{ˉ} = a^{2} + b^{2}$ is always a non-negative real number.

Field Properties

The complex numbers form a field under the following operations:

Addition: $(a + bi) + (c + d i) = (a + c) + (b + d) i$ .
Multiplication: $(a + bi) (c + d i) = (a c - b d) + (a d + b c) i$ .
Division: $\frac{z}{w} = \frac{z w ˉ}{w w ˉ}$ , where $w \neq = 0$ .

Unlike $R$ , $C$ is not an ordered field. There is no consistent way to define $z < w$ such that it respects the field operations.

Geometric Interpretation: The Argand Diagram

We visualize $C$ as a 2D plane where the x-axis represents the real part and the y-axis represents the imaginary part. Every $z = a + bi$ corresponds to a vector $(a, b)$ .

Modulus (Magnitude): $∣ z ∣ = a^{2} + b^{2}$ , represents the distance from the origin.
Argument (Phase): $θ = ar g (z)$ is the angle the vector makes with the positive real axis.

Polar and Exponential Forms

Using trigonometry, we can express $z$ as: $z = r (cos θ + i sin θ),$ where $r = ∣ z ∣$ . By Euler’s Formula, $e^{i θ} = cos θ + i sin θ$ , we arrive at the most concise representation: $z = r e^{i θ} .$

In this form, multiplication and division become trivial:

$z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$ (Magnitudes multiply, angles add).
$z_{1} / z_{2} = (r_{1} / r_{2}) e^{i (θ_{1} - θ_{2})}$ (Magnitudes divide, angles subtract).

De Moivre’s Theorem and Roots of Unity

De Moivre’s Theorem states that $(r e^{i θ})^{n} = r^{n} e^{in θ}$ . This allows us to find the $n$ -th roots of any complex number. The solutions to $z^{n} = 1$ are called the $n$ -th Roots of Unity: $ω_{k} = e^{i \frac{2 πk}{n}}, k = 0, 1, \dots, n - 1$ These points form a regular polygon in the complex plane.

The Fundamental Theorem of Algebra

Proven by Gauss, this theorem states that every non-constant polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots (counting multiplicity). This means $C$ is Algebraically Closed. In contrast, $R$ is not (as seen with $x^{2} + 1$ ).

Analytic Properties: Holomorphic Functions

When we extend calculus to complex-valued functions of a complex variable, $f (z)$ , we enter the field of Complex Analysis. A function is Holomorphic if its complex derivative exists. Such functions are incredibly “rigid”—if a function is differentiable once, it is infinitely differentiable and equal to its Taylor series (analytic). This leads to the Cauchy-Riemann Equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$

Applications in Science and Engineering

Complex numbers allow us to represent oscillatory phenomena as static vectors (phasors):

Electromagnetism: Impedance in AC circuits is $Z = R + j X$ .
Quantum Mechanics: The state of a particle is a complex-valued wave function $Ψ$ , and probabilities are derived from $∣Ψ ∣^{2}$ .
Digital Signal Processing: The Discrete Fourier Transform (DFT) maps signals from the time domain to the complex frequency domain.

Computational Modeling

Most high-level languages provide a complex type. In languages like Typescript, we implement them as custom structures.

class Complex {
    constructor(public re: number, public im: number) {}

    get modulus(): number {
        return Math.sqrt(this.re ** 2 + this.im ** 2);
    }

    get argument(): number {
        return Math.atan2(this.im, this.re);
    }

    multiply(other: Complex): Complex {
        // (a + bi)(c + di) = (ac - bd) + (ad + bc)i
        return new Complex(
            this.re * other.re - this.im * other.im,
            this.re * other.im + this.im * other.re
        );
    }

    static fromPolar(r: number, theta: number): Complex {
        return new Complex(r * Math.cos(theta), r * Math.sin(theta));
    }
}

By transitioning from the 1D real line to the 2D complex plane, mathematics gains the power to describe rotation, wave propagation, and the fundamental roots of all polynomial systems.

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