Complex Numbers
Complex number calculator: addition, multiplication, division, modulus and argument, polar and exponential forms, De Moivre powers and roots.
The complex number calculator adds, subtracts, multiplies and divides complex numbers, finds the modulus, argument, polar and exponential forms and the conjugate, raises to a power by De Moivre's formula and finds n-th roots. Operations in algebraic form are exact, with fractions.
Operations in algebraic form z = a + bi
| Operation | Formula |
|---|---|
| Sum | (a + bi) + (c + di) = (a + c) + (b + d)i |
| Product | (a + bi)(c + di) = (ac − bd) + (ad + bc)i, since i² = −1 |
| Quotient | (a + bi)/(c + di) = (a + bi)(c − di) / (c² + d²) |
| Conjugate | z̄ = a − bi, z · z̄ = a² + b² |
Modulus, argument and forms
|z| = √(a² + b²), φ = arg z, z = |z|(cos φ + i sin φ) = |z|eiφ
Example. z = 1 + i: |z| = √2, φ = π/4, z = √2(cos π/4 + i sin π/4).
De Moivre's formula and roots
zⁿ = |z|ⁿ(cos nφ + i sin nφ), wk = ⁿ√|z| · (cos((φ + 2πk)/n) + i sin((φ + 2πk)/n)), k = 0, …, n − 1
Example. (1 + i)⁸ = (√2)⁸(cos 2π + i sin 2π) = 16. The cube roots of 8: 2, −1 + √3·i, −1 − √3·i.
Every non-zero number has exactly n distinct n-th roots; in the complex plane they form a regular n-gon.
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