Calculate iⁿ for any integer n, see the repeating 4-cycle (1, i, −1, −i), and extend to complex number powers. Visual cycle indicator, sequence table, and unit circle position.
The imaginary unit i, defined as √(−1), produces a perfectly repeating cycle when raised to successive integer powers: i⁰ = 1, i¹ = i, i² = −1, i³ = −i, and then i⁴ = 1 again. This four-step cycle — 1, i, −1, −i — repeats forever in both directions, so any power of i can be determined simply by finding n mod 4.
This elegant pattern is one of the first things students encounter in complex number algebra, and it underpins everything from electrical engineering (where j = i is used for phasors) to quantum mechanics (where complex amplitudes describe probability). The cycle also maps naturally to the unit circle in the complex plane: 1 sits at 0°, i at 90°, −1 at 180°, and −i at 270°.
This calculator lets you enter any integer exponent and instantly see the result, the cycle position, and the real and imaginary parts. It also extends to general complex expressions (a + bi)ⁿ using De Moivre's theorem, showing the result, magnitude, and angle. The cycle visualization and sequence table make the repeating pattern crystal clear, and presets let you quickly explore classic examples from homework and exams.
Powers of i Calculator helps you solve powers of i problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Exponent (n), Real part (a), Imaginary part (b) once and immediately inspect n mod 4, Real Part, Imaginary Part to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
iⁿ = i^(n mod 4). Cycle: i⁰=1, i¹=i, i²=−1, i³=−i. For (a+bi)ⁿ: use polar form r·e^(iθ), result = rⁿ · e^(inθ).
Result: n mod 4 shown by the calculator
Using the preset "i⁰", the calculator evaluates the powers of i setup, applies the selected algebra rules, and reports n mod 4 with supporting checks so you can verify each transformation.
This calculator takes Exponent (n), Real part (a), Imaginary part (b) and applies the relevant powers of i relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use n mod 4, Real Part, Imaginary Part, Magnitude to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
i is the imaginary unit, defined as the square root of −1. It extends the real numbers into the complex number system.
Because i² = −1, so i⁴ = (i²)² = (−1)² = 1. Once you reach 1 again, the cycle repeats.
Use the same cycle: i⁻¹ = 1/i = −i (since i·(−i) = 1). More generally, add 4 to the exponent until it is non-negative.
Each power of i corresponds to a 90° rotation on the unit circle in the complex plane: 1→i→−1→−i. Use this as a practical reminder before finalizing the result.
It states that (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). It generalizes powers of complex numbers using polar form.
This calculator works with integer exponents. Non-integer complex powers involve branch cuts and are typically handled with the principal value of the complex logarithm.