Powers of i Calculator: Simplify Imaginary Unit Exponents Instantly

Calculate i to the Power of n

Enter any integer exponent (positive, negative, or zero).
Discover the fascinating cyclic pattern of the imaginary unit 'i' with our free Powers of i Calculator. Instantly compute i raised to any positive or negative integer exponent, simplifying complex number problems. Perfect for students, engineers, and anyone working with complex numbers.

Formula:

The imaginary unit `i` is fundamentally defined as the square root of -1 (i.e., `i = √-1`). When `i` is raised to an integer power `n` (represented as `in`), its value exhibits a distinct cyclic pattern that repeats every four powers:

  • `i0 = 1` (By definition, any non-zero number to the power of zero is 1)
  • `i1 = i`
  • `i2 = -1` (Since `i * i = √-1 * √-1 = -1`)
  • `i3 = -i` (Since `i3 = i2 * i = -1 * i = -i`)
  • `i4 = 1` (Since `i4 = i2 * i2 = -1 * -1 = 1`)
This cycle of `1, i, -1, -i` is crucial. To calculate `in` for any integer `n`, we can determine its position in this cycle by finding the remainder when `n` is divided by 4.
Let `r = n mod 4`.
Then, `in` will have the same value as `ir`.
For negative exponents, `i-n = 1 / in`, which also follows the same cyclic pattern.

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