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Complex Numbers

Introduction

Complex numbers extend the real number system so that equations such as $x^{2} + 1 = 0$ have solutions. A complex number has the form $z = a + b i$ , where $a$ and $b$ are real numbers and the imaginary unit $i$ satisfies $i^{2} = - 1$ . The real numbers are therefore contained inside the complex numbers: a real number $a$ can be written as $a + 0 i$ .

In this aiMOOC for Grades 11–13, you will connect algebra, geometry, trigonometry, and applications. You will learn to calculate with complex numbers, represent them on the complex plane, use conjugates, move between rectangular and polar forms, interpret multiplication as scaling and rotation, apply Euler's formula, and find roots using De Moivre's formula. You will also explore why complex numbers are important in polynomial equations, oscillations, electrical engineering, and signal analysis.

The diagram above shows how real, imaginary, and genuinely complex numbers can be located on a two-dimensional number plane.

The Khan Academy introduction above is useful for reviewing the transition from real to complex numbers before working with the more advanced representations in this course.

Learning Goals

By the end of the course, you should be able to explain the meaning of $a + b i$ , perform the four arithmetic operations, use conjugates and modulus, plot complex numbers, convert between rectangular and polar forms, calculate powers and roots, interpret geometric transformations caused by multiplication, and justify how complex numbers complete important parts of algebra.

Foundations of Complex Numbers

The Imaginary Unit and Rectangular Form

The defining relation $i^{2} = - 1$ gives a consistent way to work with square roots of negative real numbers. Powers of $i$ repeat in a cycle: $i^{0} = 1$ , $i^{1} = i$ , $i^{2} = - 1$ , $i^{3} = - i$ , and $i^{4} = 1$ . After that, the pattern repeats every four powers.

A complex number in rectangular form or Cartesian form is written $z = a + b i$ . The number $a$ is the real part, written $Re (z) = a$ , and $b$ is the imaginary part, written $Im (z) = b$ . Notice that the imaginary part is the real coefficient $b$ , not $b i$ .

Two complex numbers are equal exactly when their corresponding real and imaginary parts are equal. Thus $a + b i = c + d i$ implies $a = c$ and $b = d$ .

Addition and Subtraction

Addition and subtraction are performed componentwise: $(a + b i) + (c + d i) = (a + c) + (b + d) i$ , and $(a + b i) - (c + d i) = (a - c) + (b - d) i$ .

For example, $(3 + 2 i) + (- 5 + 4 i) = - 2 + 6 i$ . Geometrically, this works like adding two-dimensional vectors.

Multiplication

Use the distributive law and replace $i^{2}$ by $- 1$ : $(a + b i) (c + d i) = a c + a d i + b c i + b d i^{2} = (a c - b d) + (a d + b c) i$ .

For example, $(2 + 3 i) (1 - 4 i) = 2 - 8 i + 3 i - 12 i^{2} = 14 - 5 i$ .

Multiplication is more than an algebraic rule. In polar form, you will see that multiplying complex numbers multiplies their distances from the origin and adds their angles.

Conjugates, Modulus, and Division

Complex Conjugates

For $z = a + b i$ , the complex conjugate is $\overline{z} = a - b i$ . On the complex plane, conjugation reflects a point across the real axis.

A central identity is $z \overline{z} = (a + b i) (a - b i) = a^{2} + b^{2}$ , which is always a non-negative real number.

Modulus

The modulus or absolute value of $z = a + b i$ is $| z | = \sqrt{a^{2} + b^{2}}$ . Geometrically, this is the distance from the origin to the point $(a, b)$ in the complex plane.

For example, if $z = 3 + 4 i$ , then $| z | = \sqrt{3^{2} + 4^{2}} = 5$ .

The modulus satisfies useful properties such as $| z w | = | z | | w |$ and, for $w \neq 0$ , $| z / w | = | z | / | w |$ .

Division

To divide by a nonzero complex number, multiply numerator and denominator by the conjugate of the denominator: $\frac{a + b i}{c + d i} \cdot \frac{c - d i}{c - d i}$ . The denominator becomes the real number $c^{2} + d^{2}$ .

For example, $\frac{1 + 2 i}{3 - i} = \frac{(1 + 2 i) (3 + i)}{(3 - i) (3 + i)} = \frac{1 + 7 i}{10}$ .

This method works because a nonzero complex number has a reciprocal: $\frac{1}{z} = \frac{\overline{z}}{| z |^{2}}$ .

The Complex Plane

From Coordinates to Geometry

The Complex plane places the real axis horizontally and the imaginary axis vertically. The complex number $z = a + b i$ corresponds to the point $(a, b)$ and also to the vector from the origin to that point. This geometric interpretation makes addition, subtraction, modulus, argument, and multiplication easier to visualize.

You can interpret $z + w$ as vector addition. The difference $z - w$ represents the displacement from $w$ to $z$ , so the distance between the points represented by $z$ and $w$ is $| z - w |$ .

Loci in the Complex Plane

Equations and inequalities involving modulus often describe geometric sets. For example, $| z | = 3$ is a circle of radius 3 centered at the origin, while $| z - 2 | = 1$ is a circle of radius 1 centered at the point represented by the real number 2.

The condition $| z - a | = | z - b |$ describes all points that are equally distant from the points $a$ and $b$ . Geometrically, this is the perpendicular bisector of the segment joining them.

Polar and Exponential Forms

Argument and Polar Coordinates

For a nonzero complex number $z = a + b i$ , let $r = | z |$ and let $θ$ be an angle from the positive real axis to the vector representing $z$ . Then $a = r \cos θ$ and $b = r \sin θ$ , so $z = r (\cos θ + i \sin θ)$ .

The angle $θ$ is an argument of $z$ . Because adding a full turn does not change the point, all angles $θ + 2 π k$ , where $k$ is an integer, represent the same direction. A common principal-value convention is $- π < Arg (z) \leq π$ . The number zero has no defined argument.

The polar form is especially efficient for multiplication, division, powers, and roots.

Multiplication as Scaling and Rotation

Suppose $z_{1} = r_{1} (\cos θ_{1} + i \sin θ_{1})$ and $z_{2} = r_{2} (\cos θ_{2} + i \sin θ_{2})$ . Their product is $z_{1} z_{2} = r_{1} r_{2} (\cos (θ_{1} + θ_{2}) + i \sin (θ_{1} + θ_{2}))$ .

Therefore multiplication by a nonzero complex number has a clear geometric effect: distances from the origin are scaled by its modulus and directions are rotated by its argument. Division reverses these operations by dividing moduli and subtracting arguments.

A particularly useful example is multiplication by $i$ . Since $i = \cos (π / 2) + i \sin (π / 2)$ , multiplying by $i$ rotates a point by 90 degrees counterclockwise without changing its distance from the origin.

Euler's Formula

Euler's formula connects the exponential function with trigonometry: $e^{i θ} = \cos θ + i \sin θ$ .

This lets you write every nonzero complex number in exponential form: $z = r e^{i θ}$ .

Setting $θ = π$ gives $e^{i π} = - 1$ , or equivalently $e^{i π} + 1 = 0$ . This identity links five fundamental mathematical constants in one equation.

The MIT OpenCourseWare video above develops complex numbers and Euler's formula in a context that connects naturally to differential equations and oscillations.

Powers and Roots

De Moivre's Formula

For an integer $n$ , $(\cos θ + i \sin θ)^{n} = \cos (n θ) + i \sin (n θ)$ . More generally, if $z = r (\cos θ + i \sin θ)$ , then $z^{n} = r^{n} (\cos (n θ) + i \sin (n θ))$ .

This is De Moivre's formula. It turns repeated complex multiplication into two simpler operations: raise the modulus to a power and multiply the argument by the exponent.

For example, if $z = 2 (\cos 3 0^{\circ} + i \sin 3 0^{\circ})$ , then $z^{3} = 8 (\cos 9 0^{\circ} + i \sin 9 0^{\circ}) = 8 i$ .

Complex Roots

If $z = r (\cos θ + i \sin θ)$ and you want all solutions of $w^{n} = z$ , the roots are $w_{k} = r^{1 / n} (\cos \frac{θ + 2 π k}{n} + i \sin \frac{θ + 2 π k}{n})$ for $k = 0, 1, \dots, n - 1$ .

These $n$ roots have equal modulus and equally spaced arguments, so they form the vertices of a regular $n$ -gon centered at the origin.

Roots of Unity

The nth roots of unity are the solutions of $z^{n} = 1$ . They are $e^{2 π i k / n}$ for $k = 0, 1, \dots, n - 1$ .

For example, the fifth roots of unity lie at five equally spaced points on the unit circle.

Roots of unity appear in polynomial factorization, discrete Fourier transforms, symmetry, and many areas of higher mathematics.

Polynomial Equations and Algebraic Closure

A quadratic equation can have no real solutions but still have complex solutions. For example, $x^{2} + 4 = 0$ has solutions $x = \pm 2 i$ .

More generally, the Fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has at least one complex root. Equivalently, a degree $n$ polynomial has exactly $n$ complex roots when multiplicities are counted. This is why the complex number system is described as algebraically closed.

If a polynomial has real coefficients and a non-real root $a + b i$ , then its conjugate $a - b i$ is also a root. This conjugate-pair property explains why non-real roots of real polynomials occur in pairs.

Applications and Connections

Oscillations and Waves

Complex exponentials provide a compact language for sinusoidal behavior. By Euler's formula, a rotating complex quantity $e^{i ω t}$ contains both $\cos (ω t)$ and $\sin (ω t)$ . This makes calculations involving waves, harmonic motion, and differential equations more systematic.

Electrical Engineering

In alternating-current circuit analysis, sinusoidal voltages and currents can be represented by complex quantities called phasors. Resistance, capacitance, and inductance combine through complex impedance, allowing phase shifts and amplitudes to be handled algebraically.

Signal Processing

Fourier analysis represents signals using combinations of sinusoidal components. Complex exponentials are central because they encode amplitude and phase efficiently. The roots of unity also appear in the discrete Fourier transform, which is widely used in digital signal processing.

Two-Dimensional Rotations

If a point in the plane is represented by a complex number $z$ , multiplication by $e^{i θ}$ rotates it through angle $θ$ around the origin. This provides a concise alternative to a two-dimensional rotation matrix in suitable problems.

Worked Examples

Example: Solving a Quadratic

Solve $x^{2} - 6 x + 13 = 0$ . The quadratic formula gives $x = \frac{6 \pm \sqrt{36 - 52}}{2} = \frac{6 \pm \sqrt{- 16}}{2} = 3 \pm 2 i$ . The two solutions are conjugates because the polynomial has real coefficients.

Example: Converting to Polar Form

For $z = - 1 + \sqrt{3} i$ , $| z | = \sqrt{(- 1)^{2} + (\sqrt{3})^{2}} = 2$ . The point lies in quadrant II with reference angle 60 degrees, so a principal argument is 120 degrees, or $2 π / 3$ radians. Therefore $z = 2 (\cos (2 π / 3) + i \sin (2 π / 3)) = 2 e^{2 π i / 3}$ .

Example: Geometric Multiplication

Let $z = 2 e^{i π / 6}$ and $w = 3 e^{i π / 4}$ . Then $z w = 6 e^{i (5 π / 12)}$ . The modulus becomes $2 \cdot 3 = 6$ , while the arguments add: $π / 6 + π / 4 = 5 π / 12$ .

Interactive Tasks

Quiz: Test Your Knowledge

What value defines the square of the imaginary unit i? (Negative one) (!One) (!Zero) (!Two)

What is the real part of three minus four i? (Three) (!Negative four) (!Four) (!Negative three)

What is the modulus of three plus four i? (Five) (!Seven) (!One) (!Twenty five)

What is the conjugate of two minus five i? (Two plus five i) (!Negative two plus five i) (!Two minus five i) (!Five plus two i)

What is i multiplied by two plus three i? (Negative three plus two i) (!Three plus two i) (!Negative two plus three i) (!Two plus three i)

What is the reciprocal of i? (Negative i) (!i) (!One) (!Negative one)

What happens to arguments when two nonzero complex numbers are multiplied? (Their arguments are added) (!Their arguments are multiplied) (!Their arguments are always equal) (!Their arguments become zero)

What does Euler's formula connect with the complex exponential? (Sine and cosine) (!Prime numbers and factors) (!Matrices and determinants) (!Logarithms and inequalities)

Which equation is satisfied by every fourth root of unity? (z to the fourth power equals one) (!z squared equals two) (!z cubed equals negative one) (!z to the fifth power equals zero)

How many complex roots does a degree six polynomial have when multiplicity is counted? (Six) (!Three) (!Five) (!Twelve)

Memory Game

Imaginary unit	Number whose square is negative one
Complex conjugate	Reflection of a complex number across the real axis
Modulus	Distance of a complex number from the origin
Argument	Angle from the positive real axis to a nonzero complex number
Polar form	Representation using a magnitude and an angle
Root of unity	Solution obtained from an equation in which a power equals one

Drag and Drop

Match the correct terms.	Topic
Real part	Horizontal coordinate on the complex plane
Imaginary part	Vertical coordinate on the complex plane
Conjugation	Reflection across the real axis
Multiplication	Scaling of modulus and addition of arguments
Division	Division of moduli and subtraction of arguments

...

Crossword Puzzle

Modulus	What word names the distance of a complex number from the origin?
Argument	What word names the direction angle of a nonzero complex number?
Conjugate	What is the reflected partner of a complex number across the real axis called?
Imaginary	Which word describes the part whose coefficient multiplies i?
Euler	Which mathematician is associated with the formula connecting complex exponentials and trigonometry?
Rotation	What geometric motion is produced by multiplication with a unit complex number?

LearningApps

Cloze Text

A complex number can be written in rectangular form using the imaginary unit

. The horizontal coordinate on the complex plane is the

. The distance from a complex number to the origin is its

. Reflecting a point across the real axis produces its

. A direction angle of a nonzero complex number is called an

. In polar form, multiplication causes the moduli to be multiplied while the angles are

. Euler's formula links the complex exponential with sine and

. De Moivre's formula is especially useful for calculating powers and

. The solutions of a power equation equal to one are called roots of

. The fundamental theorem of algebra guarantees complex roots for every nonconstant

.

Open-Ended Tasks

Easy

Complex plane poster: Create a labeled poster showing the real axis, imaginary axis, four complex numbers, their conjugates, and the modulus of one selected number.
Imaginary unit pattern: Build a visual or digital pattern showing at least twelve consecutive powers of i, then explain why the values repeat every four powers.
Complex number story: Write a short explanation for a younger learner that shows why extending the real numbers makes an equation such as x squared plus one equals zero solvable.
Geometry photo analogy: Find or create an image that illustrates reflection and rotation, then annotate it to compare those motions with conjugation and multiplication by a unit complex number.

Standard

Complex arithmetic tutorial: Produce a one-page tutorial or short video that teaches addition, multiplication, conjugation, and division with one fully explained example of each.
Polar form investigation: Choose six nonzero complex numbers in different quadrants, calculate their moduli and principal arguments, and compare rectangular, trigonometric, and exponential forms.
STEM interview: Interview a mathematics, physics, computing, or engineering teacher or professional about where complex numbers appear in their work, then summarize the mathematical ideas in the interview.
Roots of unity model: Draw or construct a digital model of the sixth roots of unity, explain the symmetry, and verify algebraically that each point satisfies the defining equation.

Advanced

Complex transformation experiment: Use dynamic geometry or graphing software to multiply many points by one fixed complex number and infer the scaling-and-rotation rule from your observations.
Polynomial roots project: Investigate several real polynomials with non-real roots, test the conjugate-pair pattern, and explain how your evidence relates to the fundamental theorem of algebra.
Phasor application study: Research a simple alternating-current circuit example, represent a voltage or current with a complex phasor, and explain what modulus and argument mean in that context.
Fourier connection project: Create a report, animation, or presentation showing how complex exponentials and roots of unity contribute to a discrete Fourier transform and why this is useful for signal analysis.

Learning Assessment

Representation and reasoning: Given a complex number in rectangular form, represent it geometrically and in polar form, then justify every step and explain what each representation makes easier to see.
Error analysis: Analyze a worked solution in which a student incorrectly divides two complex numbers, identify the exact mathematical error, correct it, and explain why the conjugate method works.
Geometric transfer: Predict the image of a set of points after multiplication by a fixed nonzero complex number, then verify your prediction algebraically and describe the transformation in terms of scaling and rotation.
Polynomial connection: Solve a polynomial problem with non-real roots and explain how conjugate pairs and the fundamental theorem of algebra help you check whether your answer is complete.
Method comparison: Calculate a high power of a complex number using both rectangular multiplication and polar or exponential form, then compare the efficiency and reliability of the two approaches.
Application interpretation: Choose a context such as oscillations, alternating-current circuits, or signal processing and explain how modulus and argument can represent measurable features such as amplitude and phase.

Evidence of Learning

Knowledge: You can define the imaginary unit, real and imaginary parts, conjugate, modulus, argument, rectangular form, polar form, exponential form, and roots of unity, and you can state the central ideas of Euler's formula, de Moivre's formula, and the fundamental theorem of algebra.

Skills: You can perform arithmetic with complex numbers, convert between representations, locate numbers on the complex plane, determine modulus and argument, calculate powers and roots, and interpret multiplication geometrically.

Products: Your diagrams, written explanations, calculations, digital models, investigations, interviews, videos, or presentations communicate correct mathematics and make your reasoning visible.

Reasoning: You can choose an efficient representation for a problem, justify why a method works, check results using geometric and algebraic relationships, and explain patterns such as conjugate root pairs and equally spaced roots of unity.

Transfer: You can apply complex-number ideas to unfamiliar polynomial, geometric, oscillation, circuit, or signal problems and explain how the mathematics models the new context.

OERs on the Topic

The English Wikipedia article on complex numbers provides a broad reference for definitions, history, representations, operations, and applications.

Linked Learning Areas

Complex Numbers

Complex numbers connect strongly with Algebra, Geometry, Trigonometry, Precalculus, polynomials, differential equations, Electrical engineering, and Signal processing. These connections make the topic a bridge between upper-secondary mathematics and university-level mathematics, physics, engineering, and computing.