What are the powers of i?

The imaginary unit i cycles through four values when raised to successive integer powers: i⁰ = 1, i¹ = i, i² = −1, i³ = −i, and then i⁴ = 1 again. This four-step pattern repeats indefinitely for all integer exponents.

How do you simplify any power of i?

Divide the exponent by 4 and use the remainder. If the remainder is 0 the answer is 1, if 1 the answer is i, if 2 the answer is −1, and if 3 the answer is −i. This works because i⁴ = 1, so full groups of 4 in the exponent contribute nothing.

Why do powers of i repeat every 4?

Because i² = −1 by definition, and multiplying twice more gives i⁴ = (i²)² = (−1)² = 1. Once the value returns to 1, every subsequent power re-traces the same cycle of 1, i, −1, −i.

What is i to a negative power?

Negative exponents follow the same cycle. Compute the remainder of the exponent modulo 4 (adjusted to be non-negative). For example, i⁻¹ = −i because −1 mod 4 = 3, and i³ = −i.

How do powers of i relate to the complex plane?

The four powers of i correspond to quarter-turn rotations on the unit circle: 1 is at 0°, i is at 90°, −1 is at 180°, and −i is at 270°. Multiplying by i rotates a complex number 90° counterclockwise.

Powers of i Calculator/Visualizer

Powers of i

i⁴ = 1, so powers cycle every 4. Just find k mod 4.

r = 0

1

i⁰ = 1

r = 1

i

i¹ = i

r = 2

−1

i² = −1

r = 3

−i

i³ = −i

1Divide by 4:323 ÷ 4 = 80 remainder 3 (4×80+3=323 ✓)

2Rewrite:i³²³ = i^4×80+3 = i^4×80 · i³

3Apply i⁴=1:(i⁴)⁸⁰ · i³ = 1 · i³ = i³

4Lookup r=3:i³ = −i

i³²³ = −iiᵏ = i^r where r = k mod 4

Explanation

i⁰ = 1Any number to power 0 equals 1

i¹ = ii to the first power is i

i² = −1By definition: i² = −1

i³ = −ii³ = i² · i = −1 · i = −i

Getting Started — Enter Any Exponent

The Four Remainder States

Reading the Cycle Diagram

Step-by-Step Calculation Walkthrough

Explanation Panel and Special Cases

Quick Reference Table

Why Powers of i Cycle Every 4

Negative and Zero Exponents

Connection to Complex Numbers and Euler's Formula

Related Concepts and Tools

Getting Started — Enter Any Exponent

Type any integer into the input field next to the large

i

symbol. The calculator immediately shows the result, a four-step solution, and highlights which of the four possible outcomes applies.

Six preset buttons — 17, 100, 323, 1000, 45, and 82 — let you jump to specific examples. Click Random to generate a value between 0 and 1000, or Clear to reset the input and hide the calculation panel.

The four-case strip at the top always stays visible, displaying the four values in the cycle:

i^0 = 1

i^1 = i

i^2 = -1

i^3 = -i

. Whichever remainder matches the current exponent gets highlighted with a blue background, so you can see at a glance where the input lands in the cycle.

The Four Remainder States

Every power of

i

reduces to one of exactly four values, determined by the remainder when the exponent is divided by 4. Each state produces a distinct visual configuration in the calculator.

Remainder 0 — result is

1

. Try entering 100, 44, or any multiple of 4. The left node on the cycle diagram lights up and the case cell for

r = 0

highlights. The four-step breakdown shows the division leaving zero remainder.

Remainder 1 — result is

i

. Enter 17, 45, or 1001. The top node activates. The calculation confirms that the exponent equals

4q + 1

, so the final lookup gives

i^1 = i

.

Remainder 2 — result is

-1

. Enter 82, 50, or 6. The right node highlights. This is the state behind the fundamental definition

i^2 = -1

.

Remainder 3 — result is

-i

. Enter 323, 99, or 7. The bottom node activates. The chain

i^3 = i^2 \cdot i = -1 \cdot i = -i

appears in the explanation panel.

Each remainder state produces its own unique cycle diagram highlighting, making four distinct visual snapshots.

Reading the Cycle Diagram

The SVG cycle diagram sits between the case strip and the calculation steps. It shows four nodes arranged in a circle —

i^1 = i

at the top,

i^2 = -1

at the right,

i^3 = -i

at the bottom, and

i^4 = 1

at the left — connected by curved arrows indicating the clockwise progression through the cycle.

The active node glows with a ring effect and full opacity while inactive nodes appear faded. Inside each node, the power label appears above a divider line and the resulting value below it.

To the right of the circle, a Shortcut box lists all four remainder-to-value mappings with the active row highlighted in blue. At the bottom, an example bar shows the current computation in compact form: the exponent, its mod 4 result, and the final value.

The center of the circle reads "cycle of 4," reinforcing the key insight that powers of

i

repeat every four steps. This diagram is the visual anchor of the entire tool — each of the four remainder states produces a different highlighted configuration.

Step-by-Step Calculation Walkthrough

When an exponent is entered, the calculator displays a four-step breakdown on the left panel.

Step 1 — Divide by 4: The exponent

k

is divided by 4, showing the quotient

q

and remainder

r

with a verification check:

4 \times q + r = k

.

Step 2 — Rewrite: The power is decomposed as

i^k = i^{4q + r} = i^{4q} \cdot i^r

, separating the full cycles from the leftover.

Step 3 — Apply $i^4 = 1$: Since

i^4 = 1

, raising 1 to any power still gives 1, so

i^{4q} = (i^4)^q = 1^q = 1

. The expression simplifies to

1 \cdot i^r = i^r

.

Step 4 — Lookup: The remainder

r

maps directly to one of the four known values:

1

i

-1

, or

-i

.

Below the steps, the answer bar shows the final result in large text alongside the general formula

i^k = i^r

where

r = k \bmod 4

Explanation Panel and Special Cases

The right-side explanation panel lists all four base cases with their derivation chains. The row matching the current remainder is highlighted with a blue background and a left border accent. This makes it easy to see both the active result and the reasoning behind it.

The four explanations are:

i^0 = 1

— any number raised to the zero power equals 1.

i^1 = i

—

i

to the first power is simply

i

i^2 = -1

— this is the defining property of the imaginary unit.

i^3 = -i

— derived by multiplying:

i^3 = i^2 \cdot i = (-1)(i) = -i

.

Try entering small exponents like 0, 1, 2, and 3 to confirm each base case directly. Then try a large number like 1000 — the same four-step process applies regardless of magnitude, because only the remainder after dividing by 4 matters.

Quick Reference Table

Click the Quick Reference toggle at the bottom to expand a scrollable table showing

i^0

through

i^{100}

. Each row lists the power, its

k \bmod 4

value, and the result.

Every fourth row is separated by a thicker border, visually reinforcing the length-4 cycle. Scanning down the result column, the repeating pattern

1, i, -1, -i, 1, i, -1, -i, \dots

becomes immediately obvious.

This table serves as a verification tool. If you enter 47 in the calculator and get

-i

, you can scroll to row 47 in the reference table and confirm the result independently. It also helps students who learn by pattern recognition — seeing dozens of repetitions of the same four-value cycle builds intuition faster than any single example.

The table is compact by default (collapsed) so it does not overwhelm the main interface. Open it when you need to verify or explore, collapse it when you are done.

Why Powers of i Cycle Every 4

The repeating pattern comes from the definition

i^2 = -1

. Building up from there:

i^0 = 1

i^1 = i

i^2 = -1

i^3 = i^2 \cdot i = -i

i^4 = i^3 \cdot i = (-i)(i) = -i^2 = -(-1) = 1

i^4

the value returns to

1

, which is where

i^0

started. From this point every subsequent multiplication by

i

just re-traces the same sequence:

1 \to i \to -1 \to -i \to 1 \to \dots

Formally, the group generated by

i

under multiplication is the cyclic group of order 4:

\{1, i, -1, -i\}

. The remainder

r = k \bmod 4

identifies which element of this group

i^k

equals. This is why the mod 4 shortcut works for any integer exponent, no matter how large.

Negative and Zero Exponents

The calculator handles negative exponents using the same mod 4 logic. For any negative integer

k

, the remainder is computed as

((k \bmod 4) + 4) \bmod 4

to ensure a non-negative result between 0 and 3.

For example,

i^{-1}

: since

i \cdot i^{-1} = 1

, we need the multiplicative inverse of

i

. Multiplying numerator and denominator by

-i

gives

i^{-1} = \frac{1}{i} = \frac{-i}{-i^2} = \frac{-i}{1} = -i

. The calculator confirms this because

-1 \bmod 4 = 3

and

i^3 = -i

.

Similarly,

i^{-2} = \frac{1}{i^2} = \frac{1}{-1} = -1

, matching remainder 2. And

i^{-3} = \frac{1}{i^3} = \frac{1}{-i} = i

, matching remainder 1.

The zero exponent

i^0 = 1

follows the standard convention that any nonzero number raised to the power 0 equals 1.

Connection to Complex Numbers and Euler's Formula

The four powers of

i

correspond to four special points on the unit circle in the complex plane:

1

sits on the positive real axis,

i

on the positive imaginary axis,

-1

on the negative real axis, and

-i

on the negative imaginary axis.

Using Euler's formula

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

, each power of

i

maps to a quarter turn:

i^0 = e^{i \cdot 0} = 1

(angle

0

)

i^1 = e^{i\pi/2} = i

(angle

90°

)

i^2 = e^{i\pi} = -1

(angle

180°

)

i^3 = e^{i3\pi/2} = -i

(angle

270°

)

Multiplying by

i

is equivalent to rotating a point 90° counterclockwise on the complex plane. Four such rotations return to the starting position — this is the geometric reason behind the length-4 cycle.

Related Concepts and Tools

The mod 4 cycle here is algebraically simple, but its geometric meaning becomes clear once you see it on the complex plane.

The direct geometric explanation lives in the Multiplication Visualizer. Each multiplication by

i

is a 90° rotation — modulus stays at 1, angle increases by 90°. Four rotations return to the start. The cycle

i, -1, -i, 1

is just that rotation applied repeatedly, and you can reproduce it exactly by setting

z_1 = i

and

z_2 = i

there, then mentally chaining the result.

That rotation behavior is a special case of De Moivre's Theorem. In polar form,

i = e^{ipi/2}

, so

i^n = e^{inpi/2}

— which means the angle just increments by

pi/2

each time and wraps around at

2pi

. Set

z = i

in that tool and drag

n

through 1, 2, 3, 4 to watch it happen visually.

If the polar notation

e^{ipi/2}

is unfamiliar, Euler's Formula Explorer is the place to start.

heta = pi/2

places you exactly at

i

on the unit circle — and

heta = pi

lands at

-1

, which is

i^2

. The four powers of

i

are the four cardinal points of the unit circle, and that tool shows why.

The Polar & Rectangular Converter ties it together practically — convert

i

-1

-i

, and

1

to polar form and you'll see that all four have

r = 1

and angles that are exact multiples of 90°.