Last updated: July 31, 2026
i Calculator (Imaginary Unit)
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
i^n repeats every 4 powers: 1, i, -1, -i
Where:
- n=Exponent applied to i
- i=Imaginary unit where i² = -1
Worked Examples
Zero exponent
Base case of the cycle.
- 1Normalize 0 modulo 4.
- 2Normalized exponent is 0.
- 3Cycle value at 0 is 1.
Positive exponent 5
A full cycle plus one extra step.
- 1Compute 5 mod 4 = 1.
- 2Use cycle position 1.
- 3Return i with imaginary part 1.
Negative exponent -1
Negative inputs wrap correctly using normalized modulo arithmetic.
- 1Normalize with ((n % 4) + 4) % 4.
- 2For n = -1, normalized exponent is 3.
- 3Cycle value at 3 is -i.
Large exponent 100
Large powers still reduce immediately.
- 1Compute 100 mod 4 = 0.
- 2Use the cycle position 0.
- 3Return the scalar value 1.
Introduction
Powers of the imaginary unit i are simple because they repeat in a fixed cycle of four values. This calculator reduces any exponent to its modulo-4 equivalent and reports both the symbolic answer and its real/imaginary components.
The Period-4 Cycle
Repeated multiplication by i moves through four states and then repeats.
- 1
i⁰ = 1
- 2
i¹ = i
- 3
i² = -1
- 4
i³ = -i
- 5
i⁴ returns to 1
How Normalization Works
Any exponent can be reduced to one of four cases using modulo arithmetic.
Take n mod 4.
Adjust negative results back into 0..3.
Use ((n % 4) + 4) % 4.
Map 0,1,2,3 to the cycle.
Ignore larger magnitude once normalized.
Real and Imaginary Parts
Each cycle value has a simple coordinate on the complex plane.
1 has real part 1.
i has imaginary part 1.
-1 has real part -1.
-i has imaginary part -1.
The missing component is always 0.
Negative Exponents in This Tool
The implementation normalizes negative integers into the same four-value cycle.
-1 normalizes to 3.
-2 normalizes to 2.
-3 normalizes to 1.
-4 normalizes to 0.
This keeps the answer consistent with algebraic repetition.
Input Rules
The calculator accepts any finite numeric input and truncates it to an integer.
Finite values are valid.
Infinity is rejected.
NaN is rejected.
Math.trunc removes fractional parts.
Invalid values return safe defaults with an error string.
How to Use This Calculator
Enter an exponent and read the normalized cycle result.
Input exponent n.
Normalize modulo 4.
Read the symbolic value.
Check realPart and imaginaryPart.
Use normalizedExponent for manual verification.
Why This Cycle Matters
Understanding powers of i is foundational in complex numbers and signal math.
Complex-number simplification
Polynomial factorization
Electrical engineering notation
Fourier-analysis intuition
Classroom practice with imaginary numbers
FAQs
Why do powers of i repeat every four steps?
Because multiplying by i cycles through 1, i, -1, and -i, and the next multiplication returns to 1 again.
What is i²?
i² = -1 by definition of the imaginary unit.
What is i⁵?
i⁵ = i because 5 mod 4 = 1.
How does the calculator handle negative exponents?
It uses normalized modulo arithmetic so negative exponents still map to one of the four cycle positions.
What happens to decimal exponents?
The calculation truncates the input with Math.trunc before reducing it modulo 4.
Why does the output include realPart and imaginaryPart?
Those fields show the complex-number coordinates of the simplified symbolic result.
Can the result ever be something other than 1, i, -1, or -i?
No. Every integer power of i reduces to one of those four values.
What if the exponent is not finite?
The wrapper returns zero-ish defaults and an error string instead of attempting the computation.