Complex Numbers - Theory & Concepts

Complex Numbers

Complex numbers extend the concept of the one-dimensional number line to a two-dimensional complex plane. They are essential in engineering for representing alternating currents, oscillations, and roots of polynomials that do not intersect the x-axis.

The Imaginary Unit

The foundation of complex numbers is the imaginary unit, denoted as $i$ (or $j$ in electrical engineering), defined as the square root of negative one.

Imaginary Unit (i)

i = \sqrt{-1} \quad \text{and} \quad i^2 = -1

Powers of i

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

Note

The powers of $i$ cycle every four powers: $i, -1, -i, 1$ . To find $i^n$ , divide $n$ by $4$ and look at the remainder.

Standard Form

A complex number $z$ is written in standard rectangular form as $z = a + bi$ .

Complex Number (Standard Form)

Real Part: $\text{Re}(z) = a$
Imaginary Part: $\text{Im}(z) = b$
Complex Conjugate: $\bar{z} = a - bi$ . Multiplying a complex number by its conjugate always yields a real number: $(a+bi)(a-bi) = a^2 + b^2$ .

Use the interactive vector laboratory below to explore complex number arithmetic. Adjust the real and imaginary components of $z_1$ and $z_2$ to visualize addition, subtraction, multiplication, and division geometrically as vector operations.

Complex Vector Arithmetic Plane

Arithmetic Operation

Vector

z_1 = a + bi

Real part (a)2

Imaginary part (b)3i

Vector

z_2 = c + di

Real part (c)1

Imaginary part (d)-2i

Algebraic Result Evaluation

z_1 + z_2 = (2 + 1) + (3 + -2)i = 3 + 1i

Interactive Complex Plane (Argand Diagram)

Complex numbers can be plotted on a 2D plane where the x-axis is real and the y-axis is imaginary. Explore the polar representation (magnitude and angle) of a complex number using the simulation below.

Polar Form and Euler's Formula

Complex numbers can also be expressed using a magnitude ( $r$ ) and an angle ( $\theta$ ), which is extremely useful for multiplication and division.

Polar and Exponential Forms

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

Where $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ .

De Moivre's Theorem

This theorem makes it easy to compute powers and roots of complex numbers.

De Moivre's Theorem

A formula useful for finding powers and roots of complex numbers.

[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))

Variables

Symbol	Description	Unit
$r$	The modulus (magnitude) of the complex number	-
$\theta$	The argument (angle) of the complex number in radians	-
$n$	The integer power to which the complex number is raised	-
$i$	The imaginary unit, satisfying i² = -1	-

Finding n-th Roots

Extending De Moivre's Theorem to find all n distinct roots of a complex number.

\begin{aligned} z_k = r^{1/n} \left[ \cos\left(\frac{\theta + 360^{\\circ} k}{n}\right) + i \sin\left(\frac{\theta + 360^{\\circ} k}{n}\right) \right] \end{aligned}

Variables

Symbol	Description	Unit
$z_{k}$	The k-th root of the complex number	-
$n$	The degree of the root (e.g., n=3 for cube roots)	-
$k$	An integer counter from 0 to n-1	-
$360^{\circ} \text{ or } 2\pi$	A full rotation ensuring all roots are found within one cycle	-

Explore De Moivre's Theorem visually by raising a complex number to a given power $n$ . Observe how the magnitude and argument scale geometrically on the polar complex plane.

De Moivre's Theorem Explorer

Complex Number z

z = 1.20 \left(\cos 45^\circ + i \sin 45^\circ\right)

z = 1.20 e^{i 0.785}

Magnitude r (Radius)1.20

Angle θ (Degrees)45°

Power nn = 3

Theorem Application

Result:

z^{n}

z^{3} = 1.20^{3} \left(\cos (3 \cdot 45^\circ) + i \sin (3 \cdot 45^\circ)\right)

z^{3} = 1.728 \left(\cos 135^\circ + i \sin 135^\circ\right)

Notice that the magnitude raises geometrically to 1.728, while the angle multiplies linearly to 135° (or 135° coterminal).

z (Original)

z^{n}

(Result)

Intermediates

Key Takeaways

The i Cycle: Powers of $i$ repeat every four cycles ( $i, -1, -i, 1$ ).
Division: To divide complex numbers, multiply the top and bottom by the complex conjugate of the denominator.
Geometry: Multiplying by $i$ geometrically represents a $90^\\circ$ counterclockwise rotation in the complex plane.
Polar Power: Polar form ( $re^{i\theta}$ ) makes multiplying and finding powers of complex numbers significantly easier via De Moivre's Theorem.

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