Imaginary Number (i) Calculator — Powers & Complex Operations
Compute powers of the imaginary unit i, raise any complex number a + bi to an integer power n via De Moivre's theorem, and explore the modulus, argument, conjugate and square — with a live Argand-plane diagram.
Distance from the origin in the complex (Argand) plane
- 1
Real part squared
a² = 3² = 9 - 2
Imaginary part squared
b² = 4² = 16 - 3
Sum of squares
a² + b² = 9 + 16 = 25 - 4
Modulus |z| = √(a² + b²)
√25 = 5Distance from the origin to z in the Argand plane
How does this calculator work?
Powers of i cycle with period 4 (i, i², i³, i⁴ = 1, −1, −i, 1). For any z = a + bi and integer n, z^n = r^n(cos nθ + i sin nθ) by De Moivre's theorem, where r = √(a²+b²) and θ = atan2(b,a). Reduce large i^n exponents by taking n mod 4.
Formula
How this is calculated
The imaginary unit i satisfies i² = −1. Successive powers cycle with period 4: i⁰ = 1, i¹ = i, i² = −1, i³ = −i, then i⁴ = 1 again. For any integer n, the calculator reduces n modulo 4 to immediately return i^n — useful for simplifying large exponents like i^{2025}.
For a general complex number z = a + bi, the calculator applies De Moivre's theorem: z^n = r^n × (cos(nθ) + i sin(nθ)), where the modulus r = √(a² + b²) is the vector length and the argument θ = atan2(b, a) is the angle from the positive real axis in radians. The result is computed numerically; for large n and non-unit modulus, z^n can grow rapidly or collapse toward zero.
The Argand-plane diagram shows z and i^n as arrows from the origin, helping visualise why multiplication by i rotates a vector 90° anticlockwise — so i^4 brings it back to start. The conjugate z̄ = a − bi reflects across the real axis, and |z̄| = |z|.
Frequently asked questions
By definition, i is the square root of −1, so i² = −1. This extends the real numbers to allow solutions to equations like x² + 1 = 0, which has no real solution.
47 mod 4 = 3, so i^47 = i³ = −i. The powers of i repeat in a 4-cycle: 1, i, −1, −i. Enter any exponent and the calculator reduces it automatically.
The modulus |z| = √(a² + b²) is the distance from the origin to the point (a, b) in the Argand plane — analogous to the absolute value for real numbers. The argument θ = atan2(b, a) is the angle this vector makes with the positive real axis.
TG we-Calculate Editorial Team. (2026). Imaginary Number (i) Calculator — Powers & Complex Operations [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/i-number-calculator
TG we-Calculate Editorial Team. "Imaginary Number (i) Calculator — Powers & Complex Operations." TG we-Calculate. 2026. https://we-calculate.com/calculator/i-number-calculator.
TG we-Calculate Editorial Team, "Imaginary Number (i) Calculator — Powers & Complex Operations," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/i-number-calculator
@misc{wecalculate_i_number_calculator, title = {Imaginary Number (i) Calculator — Powers & Complex Operations}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/i-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?