Calculator Inputs
The page remains single column, while the input area uses a responsive 3, 2, and 1 column grid.
Example Data Table
| z₁ | z₂ | Operation | Expected Output |
|---|---|---|---|
| 2 + 3i | 1 - 4i | Addition | 3 - 1i |
| 4 + 2i | 3 + 1i | Multiplication | 10 + 10i |
| 1 + 1i | — | Power (n = 3) | -2 + 2i |
| 8 + 0i | — | 3rd Roots | 2, -1 + 1.7321i, -1 - 1.7321i |
Formula Used
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division: (a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c² + d²)
Conjugate: conj(a + bi) = a - bi
Modulus: |a + bi| = √(a² + b²)
Argument: arg(a + bi) = atan2(b, a)
Power: zⁿ = rⁿ [cos(nθ) + i sin(nθ)]
n-th Roots: z1/n = r1/n [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
How to Use This Calculator
Enter the real and imaginary parts for z₁. Enter z₂ when you need a two-number operation like addition, subtraction, multiplication, or division.
Select the operation from the dropdown list. For power mode, supply the exponent. For root mode, choose the root degree.
Pick the decimal precision and preferred angle unit. Click the calculate button to place the result above the form.
Review the summary table, the Argand plane graph, and the export buttons. Download a CSV for spreadsheets or a PDF for reports.
FAQs
1. What is a complex number?
A complex number has a real part and an imaginary part. It is written as a + bi, where i² equals -1. This format helps represent rotation, oscillation, and two-dimensional relationships clearly.
2. Why is polar form useful?
Polar form makes multiplication, division, powers, and roots easier. Instead of handling long rectangular expressions, you work with magnitude and angle, which often reveals structure more quickly.
3. What happens if I divide by zero?
Division by zero is undefined. When z₂ equals 0 + 0i, the calculator stops and shows an error message rather than returning a misleading value.
4. Does the calculator support negative angles?
Yes. Arguments can be positive or negative depending on the quadrant of the point. The calculator uses atan2, so the angle follows the correct sign and position.
5. Why are there multiple n-th roots?
Complex numbers usually have n distinct n-th roots. They are spaced evenly around a circle in the complex plane, each separated by the same angular step.
6. Why do powers rotate points on the graph?
In polar form, raising a complex number to a power multiplies its angle and scales its magnitude. That changes both direction and distance from the origin.
7. What does the plotly graph display?
The graph shows z₁, z₂ when relevant, the result point, and all roots in root mode. This helps you see location, symmetry, distance, and rotation on the Argand plane.
8. Which precision setting should I choose?
Use 2 to 4 decimals for quick classroom work. Choose 5 to 8 decimals when you need more detailed comparisons, exported records, or closer numerical verification.