Title | 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