Calculator
Formula Used
- Binomial expansion: (a+bi)n = Σ C(n,k) · an−k · (bi)k
- Imaginary cycle: ik repeats every four powers: 1, i, −1, −i
- Polar form: r = √(a²+b²), θ = atan2(b,a)
- De Moivre: (r(cosθ + i sinθ))n = rn(cos(nθ)+i sin(nθ))
How to Use This Calculator
- Enter the real part a and imaginary part b for a+bi.
- Choose an exponent n. Use small n to view every term easily.
- Select output mode: binomial, polar, or both.
- Press Submit to display results above the form.
- Use Download CSV or Download PDF to export the same table.
Example Data Table
| a | b | n | Expected rectangular output |
|---|---|---|---|
| 2 | 3 | 2 | -5 + 12i |
| 1 | -1 | 3 | -2 - 2i |
| 0 | 4 | 4 | 256 + 0i |
These examples help you verify sign handling and ik cycling.
FAQs
1) What does this calculator expand?
It expands powers of a complex number in the form (a + bi)^n. You can view the summed rectangular result, and optionally the polar result computed with De Moivre’s relationship.
2) Why do the terms switch between real and imaginary parts?
Because i^k cycles through 1, i, −1, and −i. That cycle controls whether each term contributes to the real sum, the imaginary sum, or changes sign.
3) Is the answer exact?
For integer inputs and small n, results are usually clean. Internally it uses floating-point arithmetic, so very large n or large values may introduce rounding. Increase precision to inspect differences.
4) What exponent values are supported?
This page is designed for non‑negative integers. The interface limits n to 60 for readability and export size, but the math concept extends further with careful numeric handling.
5) What is the polar output showing?
It reports r and θ from the original number, then applies (a+bi)^n = r^n(cos(nθ) + i sin(nθ)). The shown polar rectangular value should match the binomial sum closely.
6) How should I interpret the term table?
Each row represents k in the binomial sum. It lists C(n,k), a^(n−k), b^k, i^k, and the term’s real and imaginary contributions. The footer totals them into the final result.
7) What’s included in the CSV and PDF exports?
Both exports include your inputs, the final rectangular output, polar parameters, and the full term table. Run Submit first so the download matches your latest calculation.
8) Why does n=0 return 1?
Any nonzero value raised to the power 0 equals 1. For complex numbers, the same rule applies, so (a+bi)^0 is treated as 1 + 0i.