Complex Number Converter Form
Use rectangular inputs for a + bi, or switch to modulus-angle inputs for polar, trigonometric, or exponential entry.
Example Data Table
| Input | Rectangular Form | Polar Form | Exponential Form |
|---|---|---|---|
| 3 + 4i | 3 + 4i | 5 ∠ 53.1301° | 5e^(i0.9273 rad) |
| -2 + 2i | -2 + 2i | 2.8284 ∠ 135.0000° | 2.8284e^(i2.3562 rad) |
| 5 ∠ -30° | 4.3301 - 2.5000i | 5 ∠ -30.0000° | 5e^(i-0.5236 rad) |
| 2e^(iπ/2) | 0 + 2i | 2 ∠ 90.0000° | 2e^(i1.5708 rad) |
Formula Used
Rectangular form: z = a + bi
Modulus: |z| = √(a² + b²)
Argument: arg(z) = atan2(b, a)
Polar form: z = r(cosθ + i sinθ)
Exponential form: z = reiθ
Conjugate: z̄ = a - bi
Reciprocal: 1/z = (a - bi)/(a² + b²), when z ≠ 0
Power using De Moivre: zn = rn(cos(nθ) + i sin(nθ))
n-th roots: zk = r1/n[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]
How to Use This Calculator
- Select the input mode that matches your known values.
- Enter either the rectangular parts or the modulus-angle pair.
- Choose degrees or radians for the angle unit.
- Set decimal precision for cleaner output formatting.
- Enter an exponent to compute zn.
- Enter the root order to list all n-th roots.
- Press the convert button to show results above the form.
- Use the CSV and PDF buttons to export the summary.
Frequently Asked Questions
1. What does this calculator convert?
It converts a complex number into rectangular, polar, trigonometric, and exponential forms. It also returns modulus, principal argument, conjugate, reciprocal, powers, and n-th roots.
2. What is the principal argument?
The principal argument is the standard angle chosen for the complex number. This page normalizes it to the interval from -π to π, or the equivalent degree range.
3. Why do polar and exponential forms look similar?
Both forms use the same modulus and angle. Polar form shows cosine and sine geometry, while exponential form uses Euler notation, which is often shorter and easier in advanced algebra.
4. What happens if I enter a negative modulus?
The tool adjusts the angle by 180 degrees or π radians and uses the positive modulus. That keeps the represented complex number mathematically equivalent.
5. Can this calculator find powers of complex numbers?
Yes. Enter an integer exponent and the page uses De Moivre’s theorem to compute the new modulus, angle, and rectangular coordinates for z raised to that power.
6. Can it list all n-th roots?
Yes. It returns every distinct n-th root, indexed by k. Their angles differ by equal spacing around the Argand plane, which the graph helps visualize clearly.
7. Why is the reciprocal sometimes undefined?
The reciprocal of 0 + 0i is undefined because division by zero is not allowed. Every nonzero complex number has a valid reciprocal.
8. What does the plot show?
The chart places the original number, its conjugate, and the computed roots on the Argand plane. This makes real and imaginary components easier to interpret visually.