Calculator Inputs
Choose an operation, enter both complex numbers, then calculate. Power and root inputs are only used when their matching operation is selected.
Argand Plane Plot
The graph places the real axis horizontally and the imaginary axis vertically. Submitted values and computed outputs appear as labeled points.
Example Data Table
| Case | Operation | z1 | z2 | Setting | Output |
|---|---|---|---|---|---|
| 1 | Addition | 3 + 4i | 1 - 2i | — | 4 + 2i |
| 2 | Multiplication | 2 + 3i | 4 + i | — | 5 + 14i |
| 3 | Division | 5 + 2i | 1 - i | — | 1.5 + 3.5i |
| 4 | Power | 1 + i | 0 + 0i | n = 3 | -2 + 2i |
| 5 | Nth Roots | 8 + 0i | 0 + 0i | n = 3 | 2, -1 + 1.732051i, -1 - 1.732051i |
Formula Used
(a + bi) ± (c + di) = (a ± c) + (b ± d)i
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
(a + bi) / (c + di) = [(ac + bd) / (c² + d²)] + [(bc - ad) / (c² + d²)]i
conj(a + bi) = a - bi
1 / (a + bi) = (a - bi) / (a² + b²)
|z| = √(a² + b²)
arg(z) = atan2(b, a)
z = r(cos θ + i sin θ)
zⁿ = rⁿ(cos nθ + i sin nθ)
zk = r1/n[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], for k = 0, 1, ..., n-1
These equations let the page solve rectangular form operations directly and also switch into polar form for powers, roots, magnitude, and angle interpretation.
How to Use This Calculator
- Enter the real and imaginary parts for the first complex number z1.
- Enter the real and imaginary parts for the second complex number z2.
- Select the operation you want to perform from the dropdown list.
- For powers, enter the exponent n. For roots, enter the root order n.
- Click the calculate button to place the result above the form.
- Review the rectangular result, polar form, and plotted points on the Argand plane.
- Use the download buttons to save the output as CSV or PDF.
FAQs
1. What does this calculator handle?
It handles addition, subtraction, multiplication, division, conjugates, reciprocals, modulus, argument, integer powers, and nth roots for complex numbers.
2. What form should I enter numbers in?
Enter each complex number using separate real and imaginary fields. For example, 3 + 4i becomes real 3 and imaginary 4.
3. When is the exponent field used?
The exponent field is used only for the power operations. It is ignored for addition, subtraction, multiplication, division, conjugates, and modulus tasks.
4. Why can division fail?
Division fails when the second complex number is zero. A zero denominator makes the quotient undefined, so the calculator displays an error message.
5. Why are there multiple roots?
An nth root of a complex number usually produces n distinct solutions. They are evenly spaced around a circle on the Argand plane.
6. What does the Argand graph show?
It shows the real part on the horizontal axis and the imaginary part on the vertical axis. Inputs and results appear as labeled points.
7. Can I export my result?
Yes. After a successful calculation, you can download the output as a CSV file or create a simple PDF summary directly from the page.
8. Does the calculator use polar form internally?
Yes for powers and roots. Polar form makes angle scaling and root spacing straightforward, while rectangular form remains convenient for basic arithmetic.