Free · Arithmetic calculators ➗

Imaginary Unit Integer Power Cycle

Imaginary Unit Integer Power Cycle. Find the symbolic value of i raised to a positive, zero or negative integer exponent.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

i raised to the fifth power equals i. Integer powers repeat every four exponents.

i^n follows the cycle 1, i, −1, −i every four integer powers.

Worked example

Enter these known values and leave the other values blank.

Integer exponent
5
Result
i raised to 5 equals i.

Assumptions and limitations

  • Arithmetic uses 64 significant decimal digits, with separate rounding for displayed results. Very large integer arithmetic is not an exact arbitrary-length integer service.
  • The exponent must be an integer, positive, zero or negative. The result is displayed as a symbolic value in the explanation.
  • The internal remainder uses the absolute exponent; separate positive/negative cases preserve the four-value cycle.

Common questions

Can I enter the result to calculate a missing input?

These tools classify an entered number or exponent; they do not invert the classification.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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