How Exponential Form of a Complex Number Works
Every complex number z = a + bi (real part a, imaginary part b) can be rewritten in exponential form as z = r·e^(iθ), where r is the modulus (distance from the origin on the complex plane) and θ is the argument (angle from the positive real axis), measured in radians. This form follows directly from Euler's formula, e^(iθ) = cos θ + i sin θ, which connects the exponential function to the trigonometric functions on the complex plane.
Formula and derivation
Given a + bi, first find the modulus with the Pythagorean theorem: r = √(a² + b²). Then find the argument with the two-argument arctangent, θ = atan2(b, a), which places θ in the correct quadrant for any signs of a and b — unlike a plain arctan(b/a), which only spans a 180° range and cannot distinguish, for example, the first and third quadrants. Since a = r cos θ and b = r sin θ, substituting gives z = r(cos θ + i sin θ) = r·e^(iθ) by Euler's formula. The angle θ inside the exponential must be in radians, because Euler's formula is derived from the Taylor series of e^x, cos x, and sin x, which are only equal to those series when x is in radians.
Working with the angle unit
Degrees are often more intuitive for reporting the argument (for example, θ = 53.13°), so this calculator lets you display θ in degrees or radians. Internally it always converts to radians before building the exponential form r·e^(iθ), because e^(iθ) is only mathematically correct with a radian-measured angle — writing "e^(i53.13°)" is a shorthand seen in some engineering texts, but it is not the correct numeric value unless 53.13° is first converted to about 0.9273 radians (θ_rad = θ_deg × π/180).
Common sources of error
- Using degrees directly in e^(iθ): the exponent must be in radians; plugging in a degree value gives a wrong numeric result even though the notation looks similar.
- Using arctan instead of atan2: arctan(b/a) alone cannot tell quadrant II from quadrant IV — always use the two-argument arctangent (atan2) to get the correct principal argument.
- Forgetting the a = b = 0 case: the origin has modulus 0 but an undefined argument, since every angle points to the same location when r = 0.
Applications
Exponential (phasor) form is the standard way to represent AC voltages, currents, and impedances in electrical engineering, because multiplying and dividing complex numbers becomes simple multiplication and subtraction of moduli and angles. It is also used in signal processing (Fourier and Laplace transforms), control theory, and anywhere rotations or oscillations are modeled on the complex plane.