Complex numbers

Arithmetic

Real and imaginary parts are collected separately. Multiplication uses $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$. Division multiplies top and bottom by the conjugate of the denominator.

Polar form

The modulus is $r = |z| = \sqrt{a^2 + b^2}$, and the argument is $\theta = \arg z$.

Then $z = r(\cos\theta + i\sin\theta) = re^{i\theta}.$

Powers and roots

De Moivre gives $z^n = r^n(\cos n\theta + i\sin n\theta)$. A principal $n$th root is $z_0 = |z|^{1/n}e^{i\theta/n}$. With $\omega = e^{2\pi i/n}$, the roots are $\{1, \omega, \ldots, \omega^{n-1}\}$ on the unit circle, or $z_0$ times those values for a general $z$.

Argand diagram

The horizontal axis is the real part and the vertical axis is the imaginary part. Inputs are shown in a neutral colour, results and roots in the accent colour. A dashed circle marks the modulus when it is relevant.

Export

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