What topics does the complex numbers cheat sheet cover?

The cheat sheet covers the imaginary unit and powers of i, algebraic form and components, complex conjugate properties, arithmetic operations, modulus, geometric representation, trigonometric form, exponential form and Euler's formula, De Moivre's theorem and roots, equations and polynomials, and field properties of the complex numbers.

What is Euler's formula?

Euler's formula states that e^(iθ) = cos θ + i sin θ. It connects the exponential function to trigonometry and allows any complex number to be written in exponential form as z = re^(iθ), where r is the modulus and θ is the argument.

What is De Moivre's theorem?

De Moivre's theorem states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). It provides a formula for computing powers of complex numbers in trigonometric form: raise the modulus to the power and multiply the argument by n.

How many nth roots does a complex number have?

Every nonzero complex number has exactly n distinct nth roots. These roots all share the same modulus (r^(1/n)) and are equally spaced around a circle, separated by 360°/n, forming the vertices of a regular n-gon.

What are the three forms of a complex number?

A complex number can be written in algebraic form (a + bi), trigonometric form (r(cos θ + i sin θ)), or exponential form (re^(iθ)). All three are equivalent. Trigonometric and exponential forms are most efficient for multiplication, division, and powers.

Complex Numbers Cheat Sheet | Learn Math Class

Organized by category

The Imaginary Unit & Pure Imaginaries

Definition of

i

i^2 = -1

The imaginary unit was created to solve

x^2 + 1 = 0

, which has no solution in

\mathbb{R}

Square Roots of Negatives

\sqrt{-a} = i\sqrt{a}

for

a > 0

Trap:

\sqrt{-1} \cdot \sqrt{-1} \neq \sqrt{1}

. Always extract

i

first, then multiply:

i \cdot i = i^2 = -1

Powers of

i

— The 4-Cycle

i^1

i

i^2

-1

i^3

-i

i^4

1

(cycle restarts)

For

i^k

: divide

k

4

, use the remainder. E.g.

i^{323}

323 \div 4

remainder

3

→

i^{323} = i^3 = -i

Powers of

i

— Visualized

Multiplying by

i

= 90° counterclockwise rotation on the unit circle.

Pure Imaginary Numbers

Pure Imaginary: A number of the form $bi$ where $b \in \mathbb{R}$ and $b \neq 0$ . Equivalently, $z$ is pure imaginary when $Re(z) = 0$ and $z \neq 0$ .

Zero (

0 + 0i

) is both real and pure imaginary — the only number with dual membership.

Algebraic Form & Components

Standard (Algebraic) Form

z = a + bi

$a, b \in \mathbb{R}$ , and $i^2 = -1$
$a$ is the real part, $b$ is the imaginary part
The imaginary part $b$ is itself a real number — not $bi$

The Complex Plane

A complex number

z = a + bi

plotted as the point

(a, b)

Re(z)

and

Im(z)

Common mistake:

Im(5 - 3i) = -3

, not

3

. The sign belongs to the imaginary part.

Re(z)

a

— horizontal coordinate on the complex plane

Im(z)

b

— vertical coordinate (a real number, not

bi

)

Extracting Parts via Conjugate

Real part

Re(z) = \dfrac{z + \bar{z}}{2}

Imaginary part

Im(z) = \dfrac{z - \bar{z}}{2i}

Equality of Complex Numbers

a + bi = c + di \iff a = c

and

b = d

One complex equation yields two real equations. This is the key technique for solving complex equations by equating real and imaginary parts separately.

Classification of Complex Numbers

b = 0

z = a

is a real number

a = 0,\; b \neq 0

z = bi

is pure imaginary

a = b = 0

z = 0

(both real and pure imaginary)

a \neq 0,\; b \neq 0

Genuinely complex (off both axes)

Complex Conjugate

Definition

\bar{z} = a - bi

Not negation:

\overline{3+2i} = 3-2i

, but

-(3+2i) = -3-2i

Negate only the imaginary part. On the complex plane, this is reflection across the real axis. Also written

z^*

in physics.

Conjugate Reflection

The conjugate

\bar{z}

is the reflection of

z

across the real axis.

Conjugate Arithmetic Properties

$\overline{\bar{z}} = z$ (double conjugate)
$\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$
$\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}$
$\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}$
$\overline{z_1 / z_2} = \bar{z_1} / \bar{z_2}$
$\overline{z^n} = (\bar{z})^n$

Conjugation commutes with all arithmetic — conjugate each piece separately.

Key Conjugate Identities

z + \bar{z}

2\,Re(z)

— always real

z - \bar{z}

2i\,Im(z)

— always pure imaginary

z \cdot \bar{z}

|z|^2 = a^2 + b^2

— always real and

\geq 0

Classification Theorems

z

is Real

z = \bar{z}

(fixed by reflection)

z

is Pure Imaginary

\bar{z} = -z

(negated by reflection)

Conjugate Pairs in Real Polynomials

Consequence: real polynomials of odd degree always have at least one real root.

p(z)

has all real coefficients and

p(z_0) = 0

, then

p(\bar{z_0}) = 0

. Non-real roots of real polynomials always come in conjugate pairs.

Arithmetic Operations

Addition & Subtraction

(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i

Combine real parts, combine imaginary parts independently. Geometrically: vector addition (parallelogram rule).

Vector Addition — Parallelogram Rule

Adding

z_1 + z_2

follows the parallelogram (tip-to-tail) rule.

Multiplication

(a+bi)(c+di) = (ac - bd) + (ad + bc)i

Expand using FOIL / distribution

Replace every

i^2

with

-1

Collect real and imaginary parts

Multiplication as Rotation & Scaling

Multiply moduli, add arguments:

|z_1 z_2| = |z_1||z_2|

\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)

Division

\dfrac{z_1}{z_2} = \dfrac{z_1 \cdot \bar{z_2}}{|z_2|^2}

Multiply numerator and denominator by

\bar{z_2}

Denominator becomes

z_2 \cdot \bar{z_2} = |z_2|^2

(real)

Expand numerator, write result as

a + bi

The conjugate eliminates

i

from the denominator.

Multiplicative Inverse

z^{-1} = \dfrac{\bar{z}}{|z|^2} = \dfrac{a - bi}{a^2 + b^2}

Exists for every

z \neq 0

. Geometrically: reflect across real axis and rescale modulus to

1/|z|

Common Pitfalls

Forgetting $i^2 = -1$ in multiplication — the $bdi^2$ term must become $-bd$
Sign errors in subtraction — distribute the minus across both parts: $(5+3i)-(2-4i) = 3+7i$
Leaving complex denominators — always multiply by conjugate to get real denominator
Confusing conjugate with negation — conjugate flips only the imaginary sign

Modulus (Absolute Value)

Definition

|z| = |a + bi| = \sqrt{a^2 + b^2}

Distance from the origin to

(a, b)

in the complex plane. Generalizes real absolute value to two dimensions.

Modulus — Right Triangle

The modulus

|z|

is the hypotenuse of the right triangle with legs

a

and

b

Special Cases

Real

z = a

|a| = |a|

(ordinary absolute value)

Pure imaginary

z = bi

|bi| = |b|

z = 0

|0| = 0

(the only

z

with

|z| = 0

)

Modulus–Conjugate Identity

z \cdot \bar{z} = |z|^2

The product

z \cdot \bar{z}

is always real and non-negative. This identity powers division and connects modulus to the conjugate.

Algebraic Properties of Modulus

$|z| \geq 0$ , with $|z| = 0 \iff z = 0$
$|\bar{z}| = |z|$
$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$
$|z_1 / z_2| = |z_1| / |z_2|$
$|z^n| = |z|^n$

Triangle Inequalities

Triangle

|z_1 + z_2| \leq |z_1| + |z_2|

Reverse

\big||z_1| - |z_2|\big| \leq |z_1 - z_2|

Equality in the triangle inequality holds iff

z_1

and

z_2

point in the same direction (one is a non-negative real multiple of the other).

Distance Between Two Points

d(z_1, z_2) = |z_1 - z_2|

The Euclidean distance in the complex plane. The set

|z - z_0| = r

is a circle centered at

z_0

with radius

r

Geometric Representation

The Complex Plane (Argand Diagram)

$z = a + bi \longleftrightarrow (a, b)$: Every complex number maps to a unique point. Horizontal axis = real numbers, vertical axis = pure imaginaries. Also called the Argand diagram.

Quadrants of the Complex Plane

(+,+)

, Q2

(-,+)

, Q3

(-,-)

, Q4

(+,-)

Axes & Quadrants

Real axis

Horizontal — all numbers with

b = 0

Imaginary axis

Vertical — all numbers with

a = 0

Origin

0 + 0i

— the only point on both axes

Complex Numbers as Vectors

$z = a + bi$ is an arrow from origin to $(a,b)$
Addition follows the parallelogram rule (tip-to-tail)
$z_1 - z_2$ is the vector from $z_2$ to $z_1$
$|z|$ = length of the vector (modulus)

Geometric Meaning of Operations

Addition

Vector (parallelogram) addition

Conjugate

Reflection across real axis

Multiply by

r

Scale by

r

(no rotation)

Multiply by

i

Rotate

90°

counterclockwise

Multiply by

-1

Rotate

180°

General product

Scale by

|z_2|

and rotate by

\arg(z_2)

No Ordering in

\mathbb{C}

Complex numbers cannot be ordered. The plane extends in all directions — no consistent "

<

" exists. Only moduli (distances) can be compared.

Trigonometric Form

The Trigonometric Representation

z = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta

r = |z|

Modulus — distance from origin

\theta = \arg(z)

Argument — angle from positive real axis

Trigonometric Form — Visualized

The modulus

r

is the length,

\theta

is the angle from the positive real axis.

Argument & Principal Argument

$\arg(z)$ is determined only up to multiples of $2\pi$
$\text{Arg}(z) \in (-\pi, \pi]$ is the principal argument (unique)
$\arg(0)$ is undefined

Finding the Argument by Quadrant

Q1 (

a>0, b>0

)

\theta = \arctan(b/a)

Q2 (

a<0, b>0

)

\theta = \arctan(b/a) + \pi

Q3 (

a<0, b<0

)

\theta = \arctan(b/a) - \pi

Q4 (

a>0, b<0

)

\theta = \arctan(b/a)

Or use

\text{atan2}(b, a)

which handles all quadrants automatically.

Special Angles on the Axes

Positive real (

a > 0

)

\theta = 0

Positive imaginary (

b > 0

)

\theta = \pi/2

Negative real (

a < 0

)

\theta = \pi

Negative imaginary (

b < 0

)

\theta = -\pi/2

Unit Circle — Key Angles

Memorize these standard angles — they appear constantly in problems.

Conversions

Algebraic → Trig:

r = \sqrt{a^2+b^2}

\theta = \text{atan2}(b, a)

Trig → Algebraic:

a = r\cos\theta

b = r\sin\theta

Multiplication & Division

Product

z_1 z_2 = r_1 r_2\,\text{cis}(\theta_1 + \theta_2)

Quotient

z_1 / z_2 = (r_1/r_2)\,\text{cis}(\theta_1 - \theta_2)

Multiply moduli & add arguments. Divide moduli & subtract arguments. This is why trigonometric form excels at multiplication.

Exponential Form & Euler's Formula

Euler's Formula

e^{i\theta} = \cos\theta + i\sin\theta

Proved via Taylor series. Connects exponential functions to trigonometry. The point

e^{i\theta}

traces the unit circle as

\theta

varies.

e^{i\theta}

on the Unit Circle

\theta

varies,

e^{i\theta}

traces the unit circle through

1, i, -1, -i

Euler's Identity

e^{i\pi} + 1 = 0

Unites five fundamental constants:

e

i

\pi

1

0

. Often called the most beautiful formula in mathematics.

Exponential Form

z = re^{i\theta}

Equivalent to

r\,\text{cis}\,\theta

via Euler. The three representations:

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

Operations in Exponential Form

Product

z_1 z_2 = r_1 r_2\, e^{i(\theta_1 + \theta_2)}

Quotient

z_1 / z_2 = (r_1/r_2)\, e^{i(\theta_1 - \theta_2)}

Power

z^n = r^n e^{in\theta}

n

-th roots

z_k = R^{1/n} e^{i(\phi + 2\pi k)/n}

Standard exponent rules apply directly — this is why exponential form is the most efficient for computation.

Conjugate in Exponential Form

\overline{re^{i\theta}} = re^{-i\theta}

Conjugation negates the argument. Reflection across the real axis = flipping the angle sign.

De Moivre's Theorem & Roots

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

For any complex number:

z^n = r^n\,\text{cis}(n\theta)

. Raise the modulus to the power, multiply the argument by

n

Computing Powers — Procedure

Convert to trigonometric form: find

r

and

\theta

Apply De Moivre:

z^n = r^n\,\text{cis}(n\theta)

Reduce angle modulo

360°

if needed

Convert back to algebraic form

Example:

(1+i)^{10} = (\sqrt{2})^{10}\text{cis}(450°) = 32\text{cis}(90°) = 32i

n

-th Root Formula

z_k = R^{1/n}\,\text{cis}\!\left(\dfrac{\phi + 360°k}{n}\right)

\;k = 0, 1, \ldots, n{-}1

Every nonzero complex number has exactly $n$ distinct $n$-th roots
All roots share modulus $R^{1/n}$ — they lie on a circle
Roots are spaced $360°/n$ apart — vertices of a regular $n$ -gon

Roots of Unity (

z^n = 1

)

z_k = \text{cis}\!\left(\dfrac{2\pi k}{n}\right) = \omega^k

, where

\omega = \text{cis}\!\left(\dfrac{2\pi}{n}\right)

Sum of all roots

1 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0

4th roots of unity

1,\; i,\; -1,\; -i

(square on unit circle)

6th roots of unity

Regular hexagon on unit circle

6th Roots of Unity — Visualized

Six equally spaced roots forming a regular hexagon on the unit circle.

Quick Root Examples

\sqrt{i}

\text{cis}(45°)

and

\text{cis}(225°)

\sqrt[3]{-8}

2\text{cis}(60°),\; -2,\; 2\text{cis}(300°)

\sqrt[4]{-16}

\sqrt{2}(1+i),\; \sqrt{2}(-1+i),\; \sqrt{2}(-1-i),\; \sqrt{2}(1-i)

Cube Roots of

-8

Three roots forming an equilateral triangle on a circle of radius 2.

Fourth Roots of

-16

Four roots forming a square on a circle of radius 2, rotated 45° from axes.

Equations & Polynomials

Fundamental Theorem of Algebra

Every polynomial of degree

n \geq 1

has exactly

n

roots in

\mathbb{C}

(counted with multiplicity).

\mathbb{C}

is algebraically closed — every polynomial factors completely into linear terms:

p(z) = a_n(z - z_1)(z - z_2)\cdots(z - z_n)

Vieta's Formulas (Quadratic)

z^2 + bz + c = 0

with roots

z_1, z_2

Sum of roots

z_1 + z_2 = -b

Product of roots

z_1 \cdot z_2 = c

Extends to higher degrees: the

k

-th symmetric sum of roots

= (-1)^k \times

coefficient of

z^{n-k}

. See Vieta's formulas.

Real Polynomials — Conjugate Pair Theorem

Non-real roots come in conjugate pairs: if $z_0$ is a root, so is $\bar{z_0}$
Odd-degree real polynomials always have at least one real root
Each conjugate pair gives a real quadratic factor: $(z - z_0)(z - \bar{z_0}) = z^2 - 2\,Re(z_0)\,z + |z_0|^2$

Solving

z^n = \bar{z}

n + 2

solutions total

$z = 0$ (from $r = 0$ )
$n + 1$ points on the unit circle: $z_k = \text{cis}\!\left(\dfrac{360°k}{n+1}\right)$ for $k = 0, \ldots, n$

Write

z = re^{i\theta}

, match moduli (

r^n = r

) and arguments (

(n+1)\theta = 360°k

Conjugate Equations as Geometric Loci

z + \bar{z} = k

Vertical line

Re(z) = k/2

z - \bar{z} = ki

Horizontal line

Im(z) = k/2

z \cdot \bar{z} = k

Circle

|z| = \sqrt{k}

centered at origin

Combine conditions to find intersections. See equations & polynomials.

Field Properties & Structure

The 11 Field Axioms of

\mathbb{C}

Closure: $z_1 + z_2 \in \mathbb{C}$ ; $z_1 \cdot z_2 \in \mathbb{C}$
Commutativity: $z_1 + z_2 = z_2 + z_1$ ; $z_1 z_2 = z_2 z_1$
Associativity: $(z_1+z_2)+z_3 = z_1+(z_2+z_3)$ ; $(z_1 z_2)z_3 = z_1(z_2 z_3)$
Additive identity: $z + 0 = z$
Multiplicative identity: $z \cdot 1 = z$
Additive inverse: $z + (-z) = 0$
Multiplicative inverse: $z \cdot z^{-1} = 1$ for $z \neq 0$
Distributivity: $z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3$

\mathbb{C}

is a field — all techniques from real algebra transfer directly.

The 19 Core Properties

Complete Property Set: 11 field axioms + 7 conjugate properties (including double conjugate) + 7 modulus properties + 5 argument properties − shared overlaps = 19 distinct rules governing complex arithmetic.

These properties are catalogued on the properties page.

Argument Properties (mod

2\pi

)

\arg(z_1 z_2)

\arg(z_1) + \arg(z_2)

\arg(z_1/z_2)

\arg(z_1) - \arg(z_2)

\arg(z^n)

n \cdot \arg(z)

\arg(\bar{z})

-\arg(z)

\arg(-z)

\arg(z) + \pi

Why

\mathbb{C}

Cannot Be Ordered

Suppose

i > 0

: then

i^2 > 0

, so

-1 > 0

— contradiction. Suppose

i < 0

: then

-i > 0

, so

(-i)^2 > 0

, meaning

-1 > 0

— same contradiction. No consistent ordering exists.

Algebraic Closure

\mathbb{C}

is algebraically closed: every non-constant polynomial has at least one root. No further number system extension is needed. See the Fundamental Theorem of Algebra.

Complex Numbers Cheat Sheets