Add, multiply and divide complex numbers, convert between rectangular, polar and exponential form, compute powers and roots and see everything on the Argand plane, with the steps.
A complex number z = a + bi has real part a, imaginary part b and satisfies i² = −1. Its modulus is |z| = √(a² + b²), the distance to the origin on the Argand plane, and the argument is the angle θ from the positive real axis; this tool shows the principal value, in (−180°, 180°], and also the equivalent between 0° and 360°. The conjugate a − bi makes division possible: multiply numerator and denominator by it, and the denominator becomes a² + b². In polar form z = r(cos θ + i sin θ) = r·e^(iθ), multiplication adds the arguments and multiplies the moduli, and De Moivre's formula gives zⁿ = rⁿ(cos nθ + i sin nθ). The n roots of z lie on a circle of radius ⁿ√r, every 360°/n, forming a regular polygon. Powers of i repeat every 4: i⁰ = 1, i¹ = i, i² = −1, i³ = −i. Numbers typed in rectangular form, like 1/2 − 3/4 i or (1+i)^8, are computed in exact fractions of big integers; polar, trigonometric and exponential inputs use 64-bit floating point, and rounding noise near zero is removed on display. For electrical engineering, the Phasors tab uses j as the imaginary unit and computes ZR = R, ZL = jωL and ZC = −j/(ωC), in series or in parallel, with the current I = V/Z and the resonant frequency f₀ = 1/(2π√(LC)).
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<iframe
src="https://jkit.tools/embed/en-US/complex-number-calculator"
width="100%"
height="600"
style="border:0"
loading="lazy"
title="Complex Number Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
Multiply numerator and denominator by the conjugate of the denominator, so the denominator becomes a real number. In (3 + 4i) ÷ (1 − 2i), the conjugate of 1 − 2i is 1 + 2i. The numerator is (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i, and the denominator is 1² + 2² = 5. The result is −1 + 2i.
= 1 − 2i
Step by step
Argand plane
An angle typed without a symbol, as in 5∠30, is read in the chosen unit; angles with ° or π carry their own unit. Choose j for electrical circuits.
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