Calculator
Example Data Table
| Case | Input Form | Input Value | Rectangular Output | Polar Output |
|---|---|---|---|---|
| 1 | Rectangular | 3 + 4i | 3 + 4i | 5 ∠ 53.1301° |
| 2 | Polar | 2 ∠ 225° | -1.4142 - 1.4142i | 2 ∠ 225° |
| 3 | Cis | 6 cis 30° | 5.1962 + 3i | 6 ∠ 30° |
| 4 | Exponential | 4e^(i60°) | 2 + 3.4641i | 4 ∠ 60° |
Formula Used
Rectangular to polar: If z = a + bi, then modulus r = √(a² + b²) and argument θ = atan2(b, a).
Polar to rectangular: If z = r∠θ, then a = r cos θ and b = r sin θ.
Exponential form: z = re^(iθ), which matches the same magnitude and angle used in polar notation.
Trigonometric form: z = r(cos θ + i sin θ).
Cis form: z = r cis θ, where cis θ means cos θ + i sin θ.
Conjugate: For z = a + bi, the conjugate is a - bi.
Reciprocal: 1 / z = (a - bi) / (a² + b²), valid when z ≠ 0.
n-th roots: zk = r1/n[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], for k = 0, 1, 2, ..., n - 1.
How to Use This Calculator
- Choose the input form you already have.
- Enter either rectangular parts or magnitude and angle.
- Select degrees or radians for angle interpretation.
- Pick the principal argument range and desired decimal precision.
- Set the root order if you want roots listed.
- Press Convert Complex Number to view results above the form.
- Review the table, graph, conjugate, reciprocal, and n-th roots.
- Use the CSV or PDF buttons to export the output.
FAQs
1. What forms does this calculator support?
It accepts rectangular input and polar-style input labeled as polar, exponential, trigonometric, or cis. It then rebuilds the same number in all major forms automatically.
2. What is the modulus of a complex number?
The modulus is the distance from the origin to the point on the Argand plane. For z = a + bi, it equals √(a² + b²).
3. Why can the argument change between ranges?
A complex number can be represented by many coterminal angles. The selected range simply controls how the principal argument is normalized for display.
4. What happens when the input is zero?
The origin has modulus zero. Its argument is undefined mathematically, so the calculator displays zero by convention and marks the note in the result area.
5. What is the difference between polar and exponential form?
They describe the same complex number. Polar writes r∠θ, while exponential writes re^(iθ). Both store magnitude and angle information.
6. Why is the reciprocal undefined for zero?
The reciprocal formula divides by a² + b². At zero, that denominator becomes zero, so no finite reciprocal exists.
7. How are the n-th roots calculated?
The calculator takes the n-th root of the modulus and spreads the angles evenly around the plane using (θ + 2πk) / n for each root.
8. What does the graph show?
The chart plots the original complex number, its conjugate, and all selected n-th roots. This makes geometric relationships easier to inspect.