Math July 13, 2026 · 8 Min Read

Imaginary Number Calculator – Guide & Formulas

Solve powers of the imaginary unit i (i^n) instantly. Simples imaginary square roots and equations with complete explanations of the cyclical i-pattern.

Simplify any expression containing imaginary terms. This tool resolves powers of i (e.g. i¹⁵), extracts imaginary parts from square roots of negative numbers, and explains the 4-step power cycle.

Key Takeaway

Use the free Imaginary Number Calculator to solve powers of the imaginary unit i (i^n) instantly. simples imaginary square roots and equations with complete explanations of the cyclical i-pattern. Get instant results with step-by-step explanations.

How to Use the Imaginary Number Calculator

  1. Enter an integer power (n) to calculate the exact value of i^n.
  2. Or enter a negative radicand to simplify its imaginary square root (e.g., √-16).
  3. Review the cycling rule explanation showing how i^n is derived.

The Formula

i^n is cyclic: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1

Variable Definitions

  • i: The fundamental imaginary unit √-1
  • n: The exponent or power of the expression

Simplifying i^45 and √-36

A precalculus student needs to simplify the power expression i^45 and extract the square root of -36.

  1. Step 1: For i^45, divide the exponent 45 by 4. The remainder is 1.
  2. Step 2: Therefore, i^45 = i^1 = i.
  3. Step 3: For √-36, rewrite as √36 * √-1.
  4. Step 4: Evaluate as 6i.

Frequently Asked Questions

What is the cyclic pattern of imaginary powers?

Powers of i repeat in a continuous 4-step cycle: i, -1, -i, 1. To find the value for any positive integer exponent, simply divide the exponent by 4 and use the remainder.

Is zero an imaginary number?

Zero is both a purely real number and a purely imaginary number because 0 = 0 + 0i.

How do you multiply two imaginary numbers?

Multiply the coefficients and apply i² = -1. For example, 3i * 4i = 12i² = -12.