Complex Number Calculator
This complex number calculator works with numbers of the form a + bi and shows the result, polar form, Euler form and modulus. Example: (3 + 4i) · (1 − 2i) = 11 − 2i; the modulus of 3 + 4i is 5.
Example: What do you want to calculate: Arithmetic with two numbers (+ − · ÷) · z₁: real part a 3 · z₁: imaginary part b 4 · Operation: Multiplication z₁ · z₂ · z₂: real part c 1 · z₂: imaginary part d -2 → Result (rectangular): z = 11 − 2i. Source: Wolfram MathWorld – Complex number. Updated: .
How it is calculated
A complex number has the form z = a + bi with real part a, imaginary part b and the imaginary unit i, for which i² = −1. The calculator carries out the basic operations, powers and roots, and gives every number in polar and Euler form too.
The rules
- Addition and subtraction: real and imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.
- Multiplication: expand and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
- Division: multiply by the complex conjugate: (a + bi) / (c + di) = (a + bi)(c − di) / (c² + d²).
- Modulus: |z| = √(a² + b²), the length of the arrow in the complex plane.
Polar form and Euler form
With r = |z| and the angle φ (argument, −180° to 180°), z = r · (cos φ + i · sin φ) = r · eiφ. In this form multiplying means multiplying the moduli and adding the angles; powers and roots follow from de Moivre's formula: zⁿ = rⁿ · (cos nφ + i · sin nφ).
Example
(3 + 4i) · (1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i. Check with the moduli: 5 · √5 = √125 = 11.18 = |11 − 2i|. The cube roots of 1 are 1 and −0.5 ± 0.866i: three points evenly spaced on the unit circle.
What are complex numbers used for?
In electrical engineering for alternating current (impedance, phase), in signal processing (Fourier transform), in control engineering and quantum physics – and in mathematics, so that every polynomial equation has solutions.
Frequently asked questions
How do you multiply complex numbers?
Expand the brackets and replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. In polar form you multiply the moduli and add the angles.
How do you divide complex numbers?
Multiply numerator and denominator by the complex conjugate of the denominator. The denominator becomes real: c² + d². Division by 0 + 0i is not possible.
How do you find the modulus and the angle?
Modulus r = √(a² + b²). The angle φ comes from atan2(b, a); the calculator gives the principal value between −180° and 180° (also in radians).
How do you take the root of a complex number?
Use polar form: the nth root has modulus ⁿ√r and angles (φ + 360° · k) / n for k = 0 to n − 1. There are always exactly n solutions, and the calculator lists them all.
What is the complex conjugate?
For z = a + bi it is z̄ = a − bi. The product z · z̄ = a² + b² is real and equals the square of the modulus.
Sources and legal basis
- Wolfram MathWorld – Complex number
- DLMF §1.9: NIST Digital Library of Mathematical Functions – complex numbers, de Moivre
- Wikipedia – De Moivre’s formula
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