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
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
11 CalculatorsRelated Calculators
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