How this instrument works
The polar form of a complex number trades the pair (a, b) for a different pair, (r, θ), that describes the same point on the complex plane by distance and direction instead of by two perpendicular offsets. r is the modulus, |z| = √(a² + b²), the straight-line distance from the origin; θ is the argument, arg(z), the angle that line makes with the positive real axis, measured counterclockwise in radians. Jean-Robert Argand's 1806 essay is usually credited with fixing this picture — a complex number as an arrow on a plane rather than an abstract symbol — and the diagram still carries his name.
The reason engineers reach for this form is multiplication, not location. Multiplying two complex numbers in rectangular form means expanding four cross terms and collecting i² = −1; multiplying the same two numbers in polar form means multiplying two magnitudes and adding two angles, full stop. That shortcut is the entire reason AC circuit analysis tracks voltages and currents as phasors — magnitude and phase angle — since impedances combine by that same multiply-and-add rule once everything is expressed as re^{iθ} rather than a + bi.
One boundary case is worth knowing before it surprises you: at the origin itself, 0 + 0i, the modulus is zero but the argument is undefined — a point with no length has no direction to report, and a real instrument has to refuse that input rather than guess. Away from the origin the pair (r, θ) always exists and always recovers a and b exactly through a = r cos θ and b = r sin θ, the identity Euler's formula compresses into z = re^{iθ}.
- Enter the number's real part into the Real part field — the a in a + bi, positive, negative, or zero.
- Enter the imaginary part into the Imaginary part field — the b in a + bi, the coefficient attached to i.
- Read Magnitude for the modulus r = √(a² + b²), the number's distance from the origin on the complex plane.
- Read Angle for the argument θ in radians, measured counterclockwise from the positive real axis; multiply by 180⁄π to see degrees.
Worked example — the 3 + 4i phasor
Take the complex number 3 + 4i — three real units, four imaginary ones, the kind of pair an AC circuit might report as a voltage phasor. The modulus works out to r = √(3² + 4²) = √25 = 5.0 exactly, the familiar 3-4-5 right triangle turning up again. The argument is θ = atan2(4, 3) = 0.9272952180016122 radians, close to 53.13°, the same angle a 3-4-5 triangle's opposite corner always makes.
Running the pair back through a = r cos θ and b = r sin θ returns 5 cos(0.9272952180016122) + 5i sin(0.9272952180016122) = 3 + 4i on the nose — the companion a+bi Form calculator on this site performs exactly that return trip. The two sheets bracket the same conversion from opposite ends, which is a quick way to check either one: convert forward, then back, and the original pair should reappear untouched.
Questions
What do the modulus and argument of a complex number actually mean?
The modulus r = |z| is the complex number's straight-line distance from the origin on the complex plane, found by √(a² + b²) exactly as with any right triangle's hypotenuse. The argument θ = arg(z) is the angle that line makes with the positive real axis, measured counterclockwise in radians. Together, (r, θ) locate a + bi as precisely as (a, b) does, by length and direction rather than two perpendicular offsets.
Why does this use atan2 instead of a plain arctan(b/a)?
Because arctan alone cannot tell which quadrant a number sits in — tangent repeats every π radians, but a full turn around the complex plane is 2π, so 3 + 4i and −3 − 4i share the identical ratio b/a while pointing in opposite directions. atan2(b, a) reads the sign of a and b separately and places the argument in the correct quadrant automatically, catching the single most common mistake in hand-rolled polar conversions.
Why convert to polar form if rectangular form already works fine?
Because multiplying and dividing complex numbers collapses to almost nothing in polar form: magnitudes multiply and arguments add, instead of expanding four cross terms. That shortcut is why AC circuit work tracks voltages and currents as phasors, magnitude and phase angle, and only returns to a + bi when two phasors need adding, the one operation rectangular form still handles more directly.
Is this the same polar form as converting an (x, y) coordinate?
The arithmetic is identical — r = √(a² + b²) and θ = atan2(b, a) either way — but what it describes differs. A coordinate's polar form just relabels a location; a complex number's modulus and argument also describe an operation, because multiplying by re^{iθ} scales a plane by r and rotates it by θ, something a bare point has no equivalent for.
What angle comes back when the real part is zero?
A purely imaginary number, a = 0, sits straight up or down the imaginary axis, so θ = atan2(b, 0) resolves to +π/2 (90°) when b is positive and −π/2 (−90°) when b is negative, while the modulus simplifies to r = |b|. Only the origin, 0 + 0i, leaves the argument with nothing to measure.