i Calculator

Complex numbers consist of a real component and an imaginary component, typically written in the Cartesian form z = a + bi, where 'i' represents the imaginary unit defined by the fundamental property i² = -1. This calculator uses standard vector and trigonometric transformations to evaluate them. For instance, the magnitude of a complex number is calculated using the Pythagorean theorem as r = sqrt(a² + b²), while its phase angle is determined via the arctangent function, θ = atan2(b, a). Multiplication, division, powers, and logarithms are subsequently computed using polar coordinates and Euler's formula to ensure accurate results across all algebraic operations.

Let us walk through evaluating a complex number using the calculator. Suppose we have a first complex number z₁ = 3 + 4i, meaning our real part is 3 and our imaginary part is 4. First, we calculate its magnitude: magnitude = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5. Next, we determine its phase angle in radians: phase = atan2(4, 3) approximately equals 0.927 radians. If we add a second complex number, say z₂ = 1 + 2i, the real addition yields 3 + 1 = 4, and the imaginary addition yields 4 + 2 = 6, resulting in a sum of 4 + 6i. This step-by-step logic powers every calculation performed by the tool.

When working with complex numbers, always double-check whether your angles are set to degrees or radians, as trigonometric functions are sensitive to input units. Keep track of significant figures when rounding results to avoid precision loss in engineering applications. Finally, remember that raising a complex number to a power is often much simpler when converted into polar (phasor) form before applying exponent rules.