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Reduce an integer power of the imaginary unit i using its four-step cycle and return its real and imaginary parts.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: i⁴ = 1, so iⁿ = iʳ where r is n modulo 4; the cycle is 1, i, −1, −i for remainders 0, 1, 2, 3.
Multiplying by i rotates through four values before returning to 1. Reducing the exponent modulo 4 works for positive and negative integer exponents because i is nonzero and the same cycle extends in both directions.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Integer exponent n — minimum -1000000000; maximum 1000000000
Worked example
| Input | Value |
|---|---|
| Integer exponent n | 35 |
35 mod 4 = 3, so i³ = −i
Assumptions and limits
- The exponent is an integer.
- i denotes the principal imaginary unit satisfying i² = −1.
- The output is an exact one-term complex result rather than a decimal approximation.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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