Convert complex numbers between rectangular (a+bi), polar (r∠θ), Euler (re^iθ), and trigonometric forms with step-by-step conversion and a unit-circle diagram.
Every complex number has multiple equivalent representations. The rectangular form a + bi is convenient for addition and subtraction, while the polar form r∠θ simplifies multiplication, division, and exponentiation. Euler's formula, e^(iθ) = cos θ + i sin θ, provides an elegant bridge between the two through the exponential form re^(iθ).
This converter lets you move seamlessly between all four representations: rectangular, polar, Euler, and trigonometric (cis) form. Enter either the real and imaginary parts or the modulus and argument, and the calculator instantly shows every other form along with a step-by-step conversion table. A unit-circle diagram highlights the angle and direction of your complex number, and a quadrant reference table marks which quadrant the number falls in.
Understanding these conversions is vital in many disciplines. Electrical engineers use polar form for phasor analysis of AC circuits. Physicists work in Euler form for wave equations and quantum mechanics. Mathematicians rely on De Moivre's theorem (expressed most naturally in polar notation) to find roots of complex polynomials. Try the preset buttons for popular values like 1+i, 3+4i, or −1+√3i, and watch the diagram update in real time.
Converting between rectangular and polar form requires evaluating square roots and atan2 while tracking which quadrant the angle falls in — a frequent source of sign errors, especially for angles in Q II–IV. This converter shows every representation at once (rectangular, polar, Euler, trig/cis) plus the step-by-step algebra so you can verify each transformation. EE students checking phasor homework, physics students studying wave equations, and math students applying De Moivre's theorem all benefit from instant, visually annotated conversions.
r = √(a²+b²) | θ = atan2(b, a) | a = r cos θ | b = r sin θ | Euler: re^(iθ)
Result: 5 ∠ 53.13°
r = √(9+16) = 5, θ = atan2(4,3) ≈ 53.13°. Euler form: 5e^(0.93i). Trig form: 5(cos 53.13° + i sin 53.13°).
Rectangular form a + bi is ideal for addition and subtraction, because you simply add or subtract the real and imaginary parts separately. Polar form r∠θ, on the other hand, is far superior for multiplication, division, and exponentiation: multiply two complex numbers by multiplying their moduli and adding their arguments, divide by dividing moduli and subtracting arguments. De Moivre's theorem states (r cis θ)ⁿ = rⁿ cis(nθ), so raising to a power just scales the modulus and multiplies the angle. Euler form re^{iθ} is the most compact and connects complex arithmetic to calculus through the exponential function.
The most common mistake in hand conversion is using arctan(b/a) instead of atan2(b, a). Plain arctan only returns values in (−π/2, π/2), which covers Quadrants I and IV. If the point lies in Quadrant II or III, you need to add or subtract π — a step that is easy to forget. The atan2 function addresses this by considering the signs of both a and b, returning the correct angle in the range (−π, π]. This calculator uses atan2 internally and shows the resulting quadrant so you can confirm the angle is correct.
In electrical engineering, AC voltages and currents are represented as phasors in polar form: V = V_m ∠φ. Impedance is a complex number Z = R + jX, and multiplication in polar form gives voltage–current relationships directly. In physics, wave functions ψ = Ae^{i(kx − ωt)} use Euler form to encode amplitude and phase simultaneously. Computer graphics use complex-number multiplication to perform 2-D rotations: multiplying a point by e^{iθ} rotates it by θ radians about the origin. Signal processing relies on the Fourier transform, which decomposes signals into complex exponentials e^{iωt}. In every case, choosing the right form — rectangular for addition, polar for multiplication — makes the algebra dramatically simpler.
Polar form writes a complex number as r∠θ (or r cis θ), where r is the modulus (distance from the origin) and θ is the argument (angle from the positive real axis). Use this as a practical reminder before finalizing the result.
Compute r = √(a²+b²) and θ = atan2(b, a). This gives the unique polar representation.
Euler form writes the complex number as re^(iθ) using Euler's formula e^(iθ) = cos θ + i sin θ. Keep this note short and outcome-focused for reuse.
Polar form simplifies multiplication (multiply moduli, add arguments), division, powers (De Moivre), and root extraction. Apply this check where your workflow is most sensitive.
cis θ is shorthand for cos θ + i sin θ, so r cis θ = r(cos θ + i sin θ). Use this checkpoint when values look unexpected.
No. By convention r ≥ 0. A negative sign is absorbed into the argument by adding 180° (π radians).