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: i0=1, i1=i, i2=−1, i3=−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 full cycles vanishing with zero left over.
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 i1=i — the cycle restarting.
Remainder 2 — result is −1. Enter 82, 50, or 6. The right node highlights. This is the state behind the fundamental definitioni2=−1.
Remainder 3 — result is −i. Enter 323, 99, or 7. The bottom node activates. The chain i3=i2⋅i=−1⋅i=−i appears in the explanation panel — this is the tool's opening example.
Each remainder state produces its own unique cycle diagram highlighting, and each has a dedicated section below with the diagram frozen in that state.
Remainder 0: Full Cycles Vanish
When the exponent is a multiple of 4, nothing survives the reduction: i100=1, exactly as if no multiplication had happened at all.
i¹⁰⁰ — remainder 0, frozen
The left node glows: 100 ÷ 4 = 25 with nothing left over, so twenty-five full cycles vanish and i¹⁰⁰ = 1.
The frozen computation is 100÷4=25 remainder 0: twenty-five complete trips around the cycle and not a single step more. Each trip contributes a factor of i4=1, and 125 is still 1 — a hundred multiplications collapsing to nothing, as the step-by-step walkthrough shows line by line.
This state is the identity of the cycle, and it explains the other three: every power of i is "some full cycles, then a short remainder walk". Remainder 0 is the case where the walk has length zero. One step further takes you to remainder 1.
Remainder 1: The Cycle Restarts
An exponent one step past a multiple of 4 lands back on i itself: i17=i, seventeen multiplications with the effect of one.
i¹⁷ — remainder 1, frozen
The top node glows: 17 = 4×4 + 1, four silent cycles and one leftover quarter-step — seventeen multiplications with the effect of one.
The frozen computation reads 17=4×4+1: four vanishing cycles and a single leftover step. The top node of the cycle diagram lights up, which is the diagram's way of saying the sequence has begun again.
Remainder 1 is why the pattern is periodic rather than merely finite — the cycle does not stop at i4=1, it starts over from there. Every exponent congruent to 1(mod4), whether 5, 17, or 1001, is geometrically a single quarter-turn.
Remainder 2: The Definition Itself
Two steps into the cycle sits the identity everything else is built on: i82=i2=−1.
i⁸² — remainder 2, frozen
The right node glows: 82 mod 4 = 2 lands on −1, the defining value of the imaginary unit — half the cycle equals negation.
The frozen computation is 82=4×20+2: twenty silent cycles, then the two-step walk 1→i→−1. Whatever the exponent, remainder 2 always lands on the number whose existence defines i in the first place — i2=−1 is not derived from anything, it is the starting axiom, as the cycle explanation lays out.
Half the cycle equals negation: multiplying by i2 flips any number to its opposite. That is the algebraic shadow of a 180° rotation, and it is why even powers of i are always real (±1) while odd powers are always imaginary (±i).
Remainder 3: The Opening Example
The tool opens on i323 — a deliberately unwieldy exponent that the mod-4 shortcut dispatches in one line: remainder 3, so i323=−i.
i³²³ — remainder 3, frozen
The tool’s opening state: 323 = 4×80 + 3, and the bottom node glows on −i — three steps forward, which is also one step back.
The frozen computation is 323=4×80+3: eighty full cycles and a three-step walk 1→i→−1→−i, ending at the bottom node. Three hundred twenty-three multiplications, and the answer was decided entirely by the last two digits' remainder.
Remainder 3 has a second reading worth knowing: three steps forward is one step back, so i3=i−1 — the multiplicative inverse of i. That equivalence is exactly the negative exponent rule seen from the other side of the cycle.
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 — i1=i at the top, i2=−1 at the right, i3=−i at the bottom, and i4=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×q+r=k.
Step 2 — Rewrite: The power is decomposed as ik=i4q+r=i4q⋅ir, separating the full cycles from the leftover.
Step 3 — Apply i4=1: Since i4=1, raising 1 to any power still gives 1, so i4q=(i4)q=1q=1. The expression simplifies to 1⋅ir=ir.
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 ik=ir where r=kmod4.
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:
i0=1 — any number raised to the zero power equals 1.
i1=i — i to the first power is simply i.
i2=−1 — this is the defining property of the imaginary unit.
i3=−i — derived by multiplying: i3=i2⋅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 i0 through i100. Each row lists the power, its kmod4 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,… 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 i2=−1. Building up from there:
i0=1
i1=i
i2=−1
i3=i2⋅i=−i
i4=i3⋅i=(−i)(i)=−i2=−(−1)=1
At i4 the value returns to 1, which is where i0 started. From this point every subsequent multiplication by i just re-traces the same sequence: 1→i→−1→−i→1→…
Formally, the group generated by i under multiplication is the cyclic group of order 4: {1,i,−1,−i}. The remainder r=kmod4 identifies which element of this group ik 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 ((kmod4)+4)mod4 to ensure a non-negative result between 0 and 3.
For example, i−1: since i⋅i−1=1, we need the multiplicative inverse of i. Multiplying numerator and denominator by −i gives i−1=i1=−i2−i=1−i=−i. The calculator confirms this because −1mod4=3 and i3=−i.
Similarly, i−2=i21=−11=−1, matching remainder 2. And i−3=i31=−i1=i, matching remainder 1.
The zero exponent i0=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 formulaeiθ=cosθ+isinθ, each power of i maps to a quarter turn:
i0=ei⋅0=1 (angle 0)
i1=eiπ/2=i (angle 90°)
i2=eiπ=−1 (angle 180°)
i3=ei3π/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 z1=i and z2=i there, then mentally chaining the result.
That rotation behavior is a special case of De Moivre's Theorem. In polar form, i=eipi/2, so in=einpi/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 eipi/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 i2. 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°.