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's Theorem Calculator

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

to

10

), the input field (

-20

to

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

. Each preset has a dedicated section below with the tool frozen on it: the squaring baseline, the outward spiral, escaping the window, the inward spiral, unit circle rotation, negative exponents, and the pure real base.

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 Squaring Baseline

The preset

(1+i)^2

is the smallest possible use of the theorem — one multiplication — and the cleanest place to check the machinery against plain algebra.

(1+i)², frozen

The smallest use of the theorem: the 45° base arc doubles to 90° and the result lands on 2i — one multiplication, checked against FOIL.

FOIL gives the answer in one line:

(1+i)^2 = 1 + 2i + i^2 = 2i

. De Moivre's route agrees from the other direction: modulus

(\sqrt{2})^2 = 2

, angle

2 \times 45° = 90°

, which is

2i

exactly. Two completely different computations, one point on the imaginary axis.

The frozen frame is deliberately spare — a single trail dot sitting under

z

itself, the base's

45°

arc doubled to the result's

90°

— because at

n = 2

the theorem barely differs from one complex multiplication. Every larger exponent on this page is this step iterated: the same base squared again gives the fourth power, and twice more, the eighth.

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.

(1+i)⁴, frozen

Three purple dots fan outward — 1+i, 2i, −2+2i — each 45° further around and √2 longer, before the teal vector lands on −4: a negative real reached with no negative signs.

The frozen frame catches the fourth power mid-flight: three purple dots at

1+i

,

2i

, and

-2+2i

, each

45°

further around and

\sqrt{2}

times further out, before the teal vector lands on

-4

. Both dashed circles are visible — the small navy one at

|z| = \sqrt{2}

and the teal one at

|z^4| = 4

— with the whole trail strung between them.

The landing point deserves a second look:

(1+i)^4

is a negative real number, reached without a single negative sign in the base. Angle accumulation did it — four lots of

45°

make the half-turn. Push on to the eighth power and the spiral escapes the window entirely.

Escaping the Window

The preset

(1+i)^8

pushes the outward spiral past the edge of the visible plane: the result,

+16

, is real, positive — and outside the

\pm 10

window.

(1+i)⁸, frozen

The spiral escapes: seven dots trace two full revolutions, the last at 8 − 8i, and a dashed ray aims at the off-screen +16 on the positive real axis.

The tool answers with its off-screen convention: a dashed teal ray aimed along the positive real axis, arrowhead at the boundary, label pointing at the unreachable

z^8

. The direction is exact even though the endpoint is not drawn — and the seven trail dots, the last of them at

8 - 8i

, still trace the two full revolutions that got there.

The arithmetic behind the direction:

8 \times 45° = 360°

, a whole number of turns, so the eighth power comes home to the positive real axis with modulus

(\sqrt{2})^8 = 16

. Squaring the fourth power's

-4

confirms it:

(-4)^2 = 16

.

No teal

n\theta

arc appears in this frame — the normalized result angle is

0°

, so there is no angle left to draw. Growth without net rotation: the opposite corner of the theorem from the unit-circle state.

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.

(0.5+0.5i)⁶, frozen

Five dots coil into the origin as the modulus decays by √2/2 per step; the zoom inset magnifies the final −0.125i that full scale cannot show.

The frozen frame shows both of the tool's smallness devices at once: five purple dots coiling into the origin, and the zoom inset in the corner magnifying the final

-0.125i

that would otherwise be invisible at full scale.

Note the symmetry with the outward spiral: this base is exactly half of

1+i

, same

45°

angle, and the two spirals are mirror processes — one compounds, one decays. The boundary between them is the unit circle, explored in the rotation state, where powers neither grow nor shrink.

Unit Circle Rotation — When |z| = 1

Set

z = i

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

|z| = 1

. Now slide

n

from

1

to

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.

i³, frozen

Pure rotation: the navy and teal dashed circles coincide at radius 1, two trail dots wait at i and −1, and the result is −i — a quarter-turn walk with no growth.

The frozen

i^3

frame is the smallest complete example: two trail dots at

i

and

-1

, the result at

-i

, and the navy and teal dashed circles perfectly superimposed at radius

1

— the visual signature of a modulus that refuses to change.

The teal

n\theta

arc sweeps clockwise to

-90°

rather than counterclockwise to

270°

: the tool normalizes result angles into

(-180°, 180°]

, and the two descriptions name the same point. This unit-circle behavior is also where roots of unity live — points whose powers walk in place around the circle and come home.

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.

(3+4i)⁻¹, frozen

The reciprocal caught in the zoom inset: modulus 5 inverts to 0.2, the angle mirrors to −53.1°, and the moduli multiply back to exactly 1.

The frozen frame pairs the extremes: the navy

z = 3+4i

reaching out to modulus

5

, and its reciprocal caught in the zoom inset at modulus

0.2

— the two moduli multiplying to exactly

1

, as reciprocals must. The teal

n\theta

arc dips below the axis to

-53.1°

, the mirror of the base's angle above it.

Reciprocal-taking is the one power that ties directly back to division:

z^{-1} = \bar{z}/|z|^2

, so this preset's answer

0.12 - 0.16i

is just

\frac{3-4i}{25}

— the conjugate shrunk by the squared modulus. The five-step panel reaches the same place through polar arithmetic alone.

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 — though only

2

,

4

, and

8

remain inside the visible window. There is no rotation because

n \times 0° = 0°

for any

n

.

Now try changing

z

to

-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°

is

0°

for even

n

and

180°

for odd

n

.

2¹⁰, frozen

De Moivre with the rotation switched off: no arcs, three doubling dots at 2, 4 and 8, and a dashed ray carrying the invisible 1024 off to the right.

The frozen frame is De Moivre's theorem with the rotation half switched off: no arcs anywhere, three doubling dots on the axis, and the dashed ray carrying the invisible

1024

off to the right. Ordinary real exponentiation is the

\theta = 0

slice of the complex theory.

That reading runs both ways: real powers were never a different subject, just the special case where angle multiplication has nothing to multiply. Compare the fully angular opposite — the unit-circle state, where only the angle moves — and the general spirals are every mixture in between.

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 only makes sense in polar form — the whole point is that exponentiation becomes trivial when you write

z = re^{i\theta}

. The tools below build up everything this theorem depends on.

The essential prerequisite is the Polar & Rectangular Converter. De Moivre's theorem lives entirely in polar coordinates —

r

gets raised to the

n

th power,

\theta

gets multiplied by

n

. If you want to understand why the step-by-step panel here converts to polar first, that tool shows the conversion geometry in full detail.

Right behind it is Euler's Formula Explorer. The exponential notation

re^{i\theta}

is not just shorthand — it is why De Moivre's theorem works at all.

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

follows directly from the exponent rules once you accept Euler's formula.

The Multiplication Visualizer shows the single-step version of what this tool does repeatedly. Each intermediate power in the purple trail is one more multiplication — modulus scales by

|z|

, angle shifts by

\theta

. The trail you see here is that operation applied

n

times in sequence.

Finally, for the unit circle behavior — where

|z| = 1

and powers only rotate — the Powers of i Calculator is the cleanest special case. Setting

r = 1

and

\theta = 90°

here reproduces exactly the

i, -1, -i, 1

cycle that tool is built around.

De Moivre Law Visual Calculator

De Moivre's Theorem

Getting Started — Set z and Choose n

The Squaring Baseline

The Outward Spiral — When |z| > 1

Escaping the Window

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