Powers of i Calculator

The imaginary unit i is defined as the square root of -1. It is not a real number but is essential in complex number mathematics, electrical engineering, and quantum physics.

Because i² = -1, squaring i twice gives i⁴ = (-1)² = 1. Multiplying by i repeatedly generates the cycle: i, -1, -i, 1, then repeats.

Any non-zero number raised to the power of 0 equals 1. Since i⁴ = 1 and 0 mod 4 = 0, we have i^0 = 1.

Yes, negative powers of i follow the same cyclic pattern. For example, i^(-1) = -i because -1 mod 4 = 3, and i^3 = -i.

The imaginary unit is used in electrical engineering (AC circuit analysis), signal processing, quantum mechanics, and control systems theory.

i^1 = i. Any number raised to the first power equals itself.

i^2 = -1. This is the fundamental definition of the imaginary unit: i is the square root of -1.

i^3 = i^2 × i = -1 × i = -i. Multiplying i^2 by i gives negative i.

i^4 = i^2 × i^2 = (-1) × (-1) = 1. This completes the cycle, and the pattern repeats from here.

100 ÷ 4 = 25 remainder 0, so i^100 = i^0 = 1. Any exponent divisible by 4 gives 1.

-3 mod 4 = 1 (since -3 + 4 = 1), so i^(-3) = i^1 = i. Alternatively, i^(-3) = 1/i^3 = 1/(-i) = i.

Complex numbers are of the form a + bi, where a is the real part and b is the imaginary part. Powers of i are used to simplify complex number expressions.

Yes, fractional powers of i are complex numbers. For example, i^(1/2) = ±(1+i)/√2. This calculator focuses on integer powers.

When multiplying complex numbers, combine like terms and simplify powers of i. For example, (2 + 3i)(1 - i) = 2 - 2i + 3i - 3i² = 2 + i + 3 = 5 + i.

The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate gives a² + b², which is always a real number.

The modulus of a + bi is √(a² + b²). It represents the distance from the origin in the complex plane.

Euler's formula states e^(iθ) = cos θ + i sin θ. It connects exponential functions with trigonometric functions through the imaginary unit.

In electrical engineering, j is used instead of i (to avoid confusion with current). It represents the imaginary unit in AC circuit analysis and impedance calculations.

The complex plane is a 2D coordinate system where the x-axis represents real numbers and the y-axis represents imaginary numbers. Complex numbers are plotted as points on this plane.

Yes, but you must simplify each power first, then combine like terms. For example, i^3 + i^5 = -i + i = 0.

For odd n: if n mod 4 = 1, i^n = i; if n mod 4 = 3, i^n = -i. Odd powers alternate between i and -i.

For even n: if n mod 4 = 0, i^n = 1; if n mod 4 = 2, i^n = -1. Even powers alternate between 1 and -1.

Divide the exponent by 4 and find the remainder. The remainder tells you which position in the cycle the power corresponds to.

The imaginary unit was introduced by Gerolamo Cardano in the 16th century to solve cubic equations. The term "imaginary" was coined by René Descartes as a derogatory term.

Despite its name, the imaginary unit is essential for solving equations that have no real solutions, modeling oscillations and waves, and describing quantum mechanics.

The Powers of i Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Powers of i Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.