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Any integer exponent. Positive, negative, and zero all work. The result always falls in the 4-cycle: 1, i, -1, or -i.
i^ nPure imaginary
-i

Exact symbolic value of i raised to the entered power

Real part0
Imaginary part-1
Remainder (n mod 4)3
Cycle positionPosition 3 (i^3 = -i)
3 mod 4
i^0 = 1 (real)<0.5i^1 = i (imaginary)0.5-1.5i^2 = -1 (real)1.5-2.5i^3 = -i (imaginary)2.5+

i^7 = -i

  • The exponent 7 divided by 4 leaves remainder 3, which maps to -i in the cycle.
  • The 4-cycle of i is: i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, then it repeats.
  • Because the remainder is 3, i^7 is a pure imaginary number with real part 0.

Next stepTo evaluate i^n for any exponent, compute n mod 4 and look up the result in the cycle table. This works for positive and negative n alike.

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