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Powers of i Calculator.

Evaluate iⁿ using the repeating cycle.

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01

Set your values

Results update as you type.

Value: −i

Value

−i
Cycle index: 3

Cycle index

0.000000

Results update automatically as you type.

Use Cases

Simplify complex number expressions

When working with complex numbers, you often need to simplify expressions involving powers of i. This calculator quickly evaluates i^n for any integer exponent, saving time and reducing errors.

Example: Simplify i^27: 27 mod 4 = 3, so i^27 = -i.

Check your algebra homework

Students learning about complex numbers can use this calculator to verify their manual calculations of powers of i, ensuring they understand the pattern correctly.

Example: Check i^(-5) = 1 / i^5 = 1 / i = -i.

Frequently Asked Questions

What is the imaginary unit i?
The imaginary unit i is defined as the square root of -1. It is a fundamental concept in complex numbers, used to represent numbers that are not real. Powers of i follow a repeating cycle: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then the pattern repeats.
How do I calculate powers of i?
To calculate i^n, divide the exponent n by 4 and use the remainder. If the remainder is 0, the result is 1; if 1, the result is i; if 2, the result is -1; if 3, the result is -i. For negative exponents, use the property i^(-n) = 1 / i^n, and then simplify using the same cycle.
What is the cycle of powers of i?
The powers of i repeat every 4 exponents: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, and so on. This cycle allows you to reduce any integer exponent modulo 4 to find the result quickly.

Tips & Common Mistakes

Tips

  • Remember the cycle: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. Any exponent can be reduced by dividing by 4 and using the remainder.
  • For negative exponents, use the rule i^(-n) = 1 / i^n. Then simplify the denominator using the cycle.
  • If the exponent is large, don't try to multiply i by itself many times. Use the modulo 4 method for instant results.
  • This calculator works for any integer exponent, including zero: i^0 = 1.

Common Mistakes to Avoid

  • Forgetting that i^0 = 1, not 0 or undefined.
  • Incorrectly handling negative exponents: i^(-1) is -i, not i. Remember to take the reciprocal first.
  • Mixing up the cycle order: i^3 is -i, not i^3 = i. Double-check the remainder.

Last updated: August 13, 2026