For engineers, complex numbers aren't abstract — they're how alternating-current circuits are solved. Master a handful of complex number arithmetic moves and impedance problems become simple algebra. Here's the practical core.
On this page
j, not i
Electrical engineers write the imaginary unit as j because i is already current. It's the same number (j² = −1). When using the calculator, just enter complex values with i.
Phasors
A phasor represents a sinusoid by amplitude and phase. The signal V(t) = Vm·cos(ωt + φ) becomes the phasor Vm∠φ. Differentiation and integration of sinusoids turn into multiplication by jω — which is exactly why complex numbers tame AC analysis.
Impedance & adding in rectangular
Impedance is complex: Z = R + jX, where R is resistance and X is reactance (positive for inductors, negative for capacitors). Series impedances add, and addition is cleanest in rectangular form.
Z₁ = 3 + 4i Ω (inductive), Z₂ = 1 − 2i Ω (capacitive).
Z = Z₁ + Z₂ = 4 + 2i Ω. In polar that's |Z| = √(4²+2²) = √20 ≈ 4.472 Ω ∠ 26.57°.
Ohm's law & multiplying in polar
Ohm's law for AC is V = I·Z — a multiplication, which is cleanest in polar form (multiply magnitudes, add angles).
Drive the Z above with current I = 2∠0° A.
V = I·Z = (2∠0°)(4.472∠26.57°) = 8.944 ∠ 26.57° V. Equivalently, in rectangular form, 2·(4 + 2i) = 8 + 4i V — the same phasor either way.
Add and subtract in rectangular form; multiply and divide in polar form. The calculator's ►r∠θ and ►a+bi let you switch between them instantly mid-problem.
On the calculator
Work in CMPLX mode. Enter impedances as a+bi to add them, then press ►r∠θ to read magnitude and phase. For Ohm's law, enter currents and impedances and let the calculator multiply complex values directly. The notation behind r∠θ is covered in modulus and argument explained, and the conversion steps in converting a+bi to polar form. The full feature set is in the complex number calculator pillar guide.
Frequently asked questions
Why do engineers use j instead of i?
Because i is current; j is the same imaginary unit. Enter i on the calculator.
When do I use rectangular vs polar?
Add/subtract in rectangular; multiply/divide in polar (ideal for V = IZ).
What is a phasor?
A complex number capturing a sinusoid's amplitude and phase: Vm·cos(ωt+φ) ↔ Vm∠φ.
Solve AC problems faster
Use CMPLX mode to add impedances and apply Ohm's law without phasor bookkeeping.
Open the complex number calculator →