Calculation Result
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Calculator Workspace
Argand Diagram
The diagram plots inputs, outputs, vectors, and argument angles.
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Formula Used
A complex number combines real and imaginary components. Its rectangular form is written as a plus bi. The symbol i satisfies the rule i squared equals negative one.
Polar form uses a magnitude and an angle. Exponential form writes the same value using Euler’s identity. These forms simplify multiplication, division, powers, and root calculations greatly.
How to Use
- Select an arithmetic, expression, equation, or list mode.
- Enter values using rectangular, polar, or exponential form.
- Choose angle units, precision, and output notation settings.
- Press the main button to calculate the result.
- Review exact forms, steps, plots, and exported records.
Example Calculations
| Task | Input | Expected result |
|---|---|---|
| Addition | (3+4i)+(2-5i) | 5-i |
| Multiplication | (3+4i)(2-5i) | 26-7i |
| Division | (3+4i)/(2-5i) | -14/29+23/29i |
| Magnitude | |3+4i| | 5 |
| Polar conversion | 3+4i | 5∠53.130102° |
| Square roots | √(-9) | 3i and -3i |
| Cube roots | ∛1 | 1 and two complex roots |
Frequently Asked Questions
What is a complex number?
A complex number contains real and imaginary parts. It usually appears as a plus bi. Both parts may be zero or nonzero values.
Can I enter j instead of i?
Yes, the expression parser accepts i or j. Engineers often use j for imaginary values. The output uses i for consistent mathematical notation.
How is the argument calculated?
The calculator uses the two-argument arctangent function. This method identifies the correct complex quadrant. Results appear in degrees or radians as selected.
Does the calculator show all roots?
Yes, nth-root calculations return every distinct root. Their angles differ by equal circular intervals. Each root appears in rectangular and polar forms.
Can it solve complex quadratic equations?
Yes, coefficients may contain real or imaginary parts. The solver applies the complex quadratic formula. Both roots include detailed intermediate calculation steps.
How are polynomial roots found?
The page uses an iterative Durand-Kerner method. It supports real and complex polynomial coefficients. Very difficult polynomials may require additional numerical precision.
What does the Argand diagram show?
The horizontal axis represents real values. The vertical axis represents imaginary values. Vectors show magnitude, direction, and plotted result positions.
Can I export calculations?
Yes, results can be copied or exported. CSV files include formatted values and steps. Printing supports saving the result as a PDF.
Why is division by zero rejected?
Zero has no reciprocal in complex arithmetic. Dividing by zero creates an undefined result. The calculator displays a clear validation message instead.