Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
Complex Number Calculator
Enter complex numbers and choose an operation to see the result.
This calculator is also known as Complex Plane Plotter.
Read the complete guideWhat Is the Argand Diagram?
The Argand diagram is a coordinate system where the x-axis represents real numbers and the y-axis represents imaginary numbers. Each complex number z = a + bi corresponds to the point (a, b). Operations like addition become vector addition, multiplication becomes rotation and scaling, and conjugation becomes reflection across the real axis.
Examples
Plotting 3 + 4i
Visualize the complex number 3 + 4i on the Argand diagram.
The point (3, 4) is plotted. Its distance from the origin is |z| = 5 (the modulus), and the angle from the positive real axis is about 53.13°.
Key takeaway: The modulus is the distance from the origin, and the argument is the angle — these are the polar coordinates of the point.
Visualizing Complex Operations
Use geometric intuition to understand operations:
- Addition: translate one vector by the other (parallelogram rule)
- Multiplication: scale by the modulus and rotate by the argument
- Conjugation: reflect the point across the real axis
Frequently Asked Questions about Complex Plane Plotter
Why is it called the Argand diagram?
It is named after Jean-Robert Argand, who published the geometric interpretation of complex numbers in 1806, though Caspar Wessel described it earlier in 1799.
How does multiplication look on the Argand diagram?
Multiplying by z = r∠θ scales every point by r and rotates it by θ counterclockwise. For example, multiplying by i rotates 90° counterclockwise.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
Operation
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