How do you raise a complex number to a power?

Convert to polar form z = r·e^(iθ), then apply De Moivre's theorem: z^n = r^n · e^(inθ). The modulus is raised to the n-th power and the angle is multiplied by n. Convert back to rectangular form using cos(nθ) and sin(nθ).

What do negative exponents mean for complex numbers?

A negative exponent gives the reciprocal: z^(−n) = 1/z^n = r^(−n) · e^(−inθ). The modulus becomes the reciprocal power and the angle reverses direction. For example, (3+4i)^(−1) has modulus 1/5 and angle −53.1°.

Why do the powers of a complex number form a spiral?

Each multiplication by z scales by |z| and rotates by θ. When |z| > 1, the distance from the origin grows exponentially, creating an outward spiral. When |z| < 1, the distance shrinks, creating an inward spiral. When |z| = 1, there is no spiral — only rotation around the unit circle.

How does De Moivre's theorem relate to roots of unity?

The n-th roots of unity are the solutions to z^n = 1. Using De Moivre's theorem in reverse, they are z_k = e^(i·2πk/n) for k = 0, 1, ..., n−1 — n equally spaced points around the unit circle.

De Moivre Law Visual Calculator

De Moivre's Theorem

(r · e^iθ)ⁿ = rⁿ · e^inθ

Re:Im:n =

Try these

Input

z1 + i

|z|1.41

θ45° = π/4 rad

n3

z³ — Result

-2 + 2i

|z³| = 2.83, θ = 135° = 3π/4 rad

Step-by-Step (De Moivre's Theorem)

1Convert to polar form:

z = 1.41 · ei·π/4

2Apply De Moivre's Theorem: (r · e^iθ)ⁿ = rⁿ · e^inθ

3Raise the modulus to the n-th power:

|z|³ = 1.41³ = 2.83

4Multiply the angle by n:

nθ = 3 × 45° = 135°

5Convert back to rectangular:

z³ = 2.83(cos(135°) + i sin(135°))

= 2.83(-0.71 + 0.71i)

= -2 + 2i

Key Ideas

De Moivre's Theorem says (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). In exponential form: (r·e^iθ)ⁿ = rⁿ·e^inθ. Raising to a power means raising the modulus and multiplying the angle.

The purple trail shows intermediate powers z¹, z², z³, … Each step multiplies the modulus by |z| and adds θ to the angle. If |z| > 1 the points spiral outward; if |z| < 1 they spiral inward toward zero.

Negative exponents give reciprocals: z⁻ⁿ = 1/zⁿ. The modulus shrinks (r⁻ⁿ) and the angle reverses (−nθ). Try "(3+4i)⁻¹" to see how the result is a tiny vector in the opposite angular direction.

On the unit circle (|z|=1), powers only rotate — the point stays on the circle. This is why powers of i cycle: i¹, i², i³, i⁴ = i, −1, −i, 1 are four 90° rotations.

Getting Started — Set z and Choose n

The Outward Spiral — When |z| > 1

The Inward Spiral — When |z| < 1

Unit Circle Rotation — When |z| = 1

Negative Exponents — Reciprocals and Reversal

Pure Real Base — No Spiral, Just Scaling

Reading the Purple Trail and Modulus Circles

The Five-Step Calculation

What is De Moivre's Theorem?

Connection to Roots of Unity

Related Concepts and Tools

Getting Started — Set z and Choose n

Drag the navy point

z

anywhere on the complex plane, or type values into the Re and Im input fields (range

\pm 10

). Then set the exponent

n

using the slider (

-10

10

), the input field (

-20

20

), or one of seven presets:

(1+i)^2

(1+i)^4

i^3

(1+i)^8

(3+4i)^{-1}

2^{10}

, and

(0.5+0.5i)^6

.

The green result vector

z^n

updates instantly along with the step-by-step panel, polar values, and the purple intermediate power trail. Click Random to generate a random base and exponent.

The input summary on the right shows

z

in rectangular form, its modulus

|z|

, its argument

\theta

in both degrees and radians, and the current exponent

n

. All values update in real time as you drag or type.

The Outward Spiral — When |z| > 1

Click

(1+i)^4

to see the classic outward spiral. Since

|1+i| = \sqrt{2} \approx 1.414 > 1

, each successive power has a larger modulus:

|z^k| = |z|^k

grows with

k

. The purple trail dots

z^1, z^2, z^3

fan outward from the origin, connected by a dashed polyline.

At each step the angle also increases by

\theta = 45°

, so the trail rotates

45°

per step while expanding. The result

(1+i)^4 = -4

lands on the negative real axis at angle

4 \times 45° = 180°

with modulus

(\sqrt{2})^4 = 4

.

Now click

(1+i)^8

. The spiral extends further — the modulus reaches

(\sqrt{2})^8 = 16

— but the angle wraps back to

8 \times 45° = 360° = 0°

, placing the result at

+16

on the positive real axis. The trail makes two full revolutions.

Try increasing

n

with the slider while watching the spiral grow. Each click of

n

adds one more purple dot and pushes the green result further outward.

The Inward Spiral — When |z| < 1

Click

(0.5 + 0.5i)^6

to see the inward spiral. Since

|0.5 + 0.5i| = \frac{\sqrt{2}}{2} \approx 0.707 < 1

, each power has a smaller modulus:

|z^k| = |z|^k

shrinks toward zero. The purple trail dots spiral inward, converging on the origin.

The result modulus is

(0.707)^6 \approx 0.125

— very small. The zoom inset appears in the upper-right corner of the plane, magnifying the area near the origin so the green result point is visible and labeled.

This behavior is the opposite of the outward spiral: if

|z| < 1

, repeated multiplication pulls the point closer to zero. The angle still accumulates (

6 \times 45° = 270°

), but the distance from the origin shrinks exponentially.

Drag

z

to any point inside the unit circle and increase

n

to see the inward spiral get tighter. The higher the exponent, the closer the result is to zero.

Unit Circle Rotation — When |z| = 1

Set

z = i

(type Re = 0, Im = 1) so that

|z| = 1

. Now slide

n

from

1

4

. Because

|z| = 1

, the modulus never changes:

|z^n| = 1^n = 1

. Every power stays on the unit circle — the trail dots form a perfect arc with no spiraling.

At

n = 1

i^1 = i

(angle

90°

). At

n = 2

i^2 = -1

(angle

180°

). At

n = 3

i^3 = -i

(angle

270°

). At

n = 4

i^4 = 1

(angle

360° = 0°

). The dashed modulus circles for

z

and the result overlap because both have radius

1

.

Click the

i^3

preset to see three purple trail dots evenly spaced

90°

apart on the unit circle, ending at

-i

.

This is why powers of i cycle every four steps. On the unit circle, De Moivre's theorem is pure rotation: the modulus is fixed and only the angle changes. Any point on the unit circle produces this behavior — try

z = \cos(60°) + i\sin(60°)

and watch the dots walk around the circle in

60°

steps.

Negative Exponents — Reciprocals and Reversal

Click

(3+4i)^{-1}

to see a negative exponent. De Moivre's theorem handles negative powers naturally:

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

. The modulus shrinks (reciprocal) and the angle reverses (negative direction).

Here

|3+4i| = 5

, so

|z^{-1}| = 1/5 = 0.2

. The angle is

\theta \approx 53.1°

, so the result angle is

-53.1°

. The green vector is short and points below the real axis — the reciprocal is a tiny vector in the opposite angular direction.

Try sliding

n

into negative values with any base. The trail reverses: instead of spiraling outward (for

|z| > 1

), it spirals inward. Instead of accumulating angle, it subtracts angle.

At

n = 0

, the result is always

z^0 = 1

regardless of

z

— the green point sits at

(1, 0)

on the positive real axis. This is the boundary between positive and negative exponents.

Pure Real Base — No Spiral, Just Scaling

Click

2^{10}

to see a purely real base. Since

z = 2

has angle

\theta = 0°

, every power also has angle

0°

: the result sits on the positive real axis. The modulus is

2^{10} = 1024

— far off-screen — so the green result appears as a dashed ray pointing right with an arrow.

The purple trail dots

z^1, z^2, \dots, z^9

march along the positive real axis, each one doubling the previous:

2, 4, 8, 16, \dots, 512

. There is no rotation because

n \times 0° = 0°

for any

n

.

Now try changing

z

-2

(angle

180°

). The powers alternate between the positive and negative real axes:

(-2)^1 = -2

(-2)^2 = 4

(-2)^3 = -8

, and so on. The trail zigzags left and right along the real axis, growing in magnitude. The angle alternates between

0°

and

180°

because

n \times 180°

0°

for even

n

and

180°

for odd

n

Reading the Purple Trail and Modulus Circles

The purple trail shows every intermediate power from

z^1

up to

z^{n-1}

(for positive

n

) or from

z^{-1}

down to

z^{-(|n|-1)}

(for negative

n

). Each dot is labeled with its power. A dashed polyline connects them in order, tracing the spiral path from

z

to the final result.

Two dashed modulus circles appear when relevant. The inner navy circle has radius

|z|

— the modulus of the base. The outer green circle has radius

|z^n|

— the modulus of the result. When

|z| > 1

, the result circle is larger; when

|z| < 1

, it is smaller. When

|z| = 1

, both circles coincide.

Two angle arcs appear near the origin. The navy arc labeled

\theta

shows the argument of

z

. The green arc labeled

n\theta

shows the argument of

z^n

. The green arc is always

n

times as wide as the navy arc — this is the "multiply the angle" rule of De Moivre's theorem made visible.

The Five-Step Calculation

The Step-by-Step panel on the right walks through De Moivre's theorem in five numbered stages.

Step 1 — convert

z

to polar form:

z = r \cdot e^{i\theta}

, showing the computed modulus and argument.

Step 2 — state the theorem:

(r \cdot e^{i\theta})^n = r^n \cdot e^{in\theta}

.

Step 3 — raise the modulus to the

n

-th power:

|z|^n = r^n

.

Step 4 — multiply the angle by

n

n\theta

. If the result exceeds

\pm 180°

, a normalized angle also appears.

Step 5 — convert back to rectangular form using

\cos(n\theta)

and

\sin(n\theta)

, showing the evaluation and the final complex number.

Every value in every step updates dynamically with each drag, keystroke, or slider change. This makes the panel a live worked example for any base-exponent combination you choose.

What is De Moivre's Theorem?

De Moivre's theorem states that for any integer

n

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

In exponential notation using Euler's formula:

(re^{i\theta})^n = r^n \cdot e^{in\theta}

Raising a complex number to a power means raising the modulus to that power and multiplying the angle by

n

. This converts exponentiation — normally a difficult operation in rectangular form — into a simple combination of real exponentiation and angle multiplication.

The theorem is a direct consequence of Euler's formula and the laws of exponents. Since

e^{i\theta}

is a complex exponential,

(e^{i\theta})^n = e^{in\theta}

follows from the rule

a^{mn} = (a^m)^n

.

De Moivre's theorem works for all integers

n

— positive, negative, and zero. It is also the foundation for finding roots of unity and deriving trigonometric identities for

\cos(n\theta)

and

\sin(n\theta)

in terms of powers of

\cos\theta

and

\sin\theta

Connection to Roots of Unity

De Moivre's theorem in reverse gives the

n

-th roots of any complex number. The

n

-th roots of unity — the solutions to

z^n = 1

— are:

z_k = e^{i \cdot 2\pi k / n} = \cos\frac{2\pi k}{n} + i\sin\frac{2\pi k}{n}, \quad k = 0, 1, \dots, n-1

These

n

points are equally spaced around the unit circle, separated by angles of

\frac{360°}{n}

.

In the visualizer, set

z

to any

n

-th root of unity and raise it to the

n

-th power — the result will always be

1

. For example, set

z = \cos(72°) + i\sin(72°) \approx 0.31 + 0.95i

and

n = 5

. The trail traces five equally spaced dots around the unit circle, and the result lands at

1

.

Roots of unity appear throughout mathematics: in Fourier transforms, polynomial factoring, and group theory. De Moivre's theorem is the computational tool that makes finding and verifying these roots straightforward.

Related Concepts and Tools

De Moivre's theorem ties together many areas of complex number theory. Explore these related pages.

Euler's Formula Explorer — the identity

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

is the foundation of De Moivre's theorem. The explorer visualizes the unit circle trace and the right triangle connection.

Complex Multiplication Visualizer — multiplication is the single-step version of De Moivre: multiply moduli, add angles. Repeated multiplication gives exponentiation.

Polar-Rectangular Converter — De Moivre's theorem requires polar form. This converter handles the conversion between

a + bi

and

re^{i\theta}

.

Powers of i Calculator — the

i^n

cycle is a special case of De Moivre with

r = 1

and

\theta = 90°

. The mod 4 pattern emerges because

4 \times 90° = 360°

.

Complex Division Visualizer — division subtracts angles and divides moduli, the inverse of the power operation.

Complex Numbers — foundational theory covering the imaginary unit, algebraic operations, and the modulus-argument representation.