Complex Number Form Converter
Convert a complex number between rectangular, polar, trigonometric, and exponential forms. Includes exact angles, exact modulus when possible, steps, copy buttons, and an Argand-plane view.
Argand plane
Step-by-step conversion
- Algebraic form: 1 + i
- Modulus: √2
- Argument: π/4 + 2π·k, k ∈ Z ≈ 0.7854 rad
- Forms using modulus and argument:
- Polar: √2 ∠ π/4
- Trigonometric: √2 · (cos π/4 + i·sin π/4)
- Exponential: √2 · e^(i · π/4)
Representation summary for z
| Form | Value |
|---|---|
| Algebraic | 1 + i |
| Polar | √2 ∠ π/4 |
| Trigonometric | √2 · (cos π/4 + i·sin π/4) |
| Exponential | √2 · e^(i · π/4) |
Exact angle reference
| Radians | Degrees | Cosine | Sine | Exponential form |
|---|---|---|---|---|
| 0 | 0° | 1 | 0 | 1 |
| π/6 | 30° | √3/2 | 1/2 | √3/2 + i/2 |
| π/4 | 45° | √2/2 | √2/2 | √2/2·(1+i) |
| π/3 | 60° | 1/2 | √3/2 | 1/2 + (√3/2)·i |
| π/2 | 90° | 0 | 1 | i |
| π | 180° | −1 | 0 | −1 |
| 3π/2 | 270° | 0 | −1 | −i |
Convert between the common complex forms
Search results for this topic mix calculator intent and learning intent. This converter keeps both: it accepts a complex number, shows the equivalent forms, and gives enough steps to verify the conversion.
Rectangular form uses real and imaginary coordinates.
Polar form uses modulus and argument.
Trigonometric and exponential forms use the same r and theta.
When each form is useful
| Form | Best use |
|---|---|
| Rectangular | Addition, subtraction, and coordinate checks |
| Polar | Magnitude-angle input and quick engineering notation |
| Trigonometric | Classroom work, powers, roots, and identities |
| Exponential | Compact notation for multiplication, division, and Euler form |
Frequently Asked Questions
Sources and References
- Complex Number Forms ReviewKhan Academy
- Complex Number Conversion: Rectangular and Polar FormsCalculators Math
- Polar-Rectangular ConverterLearn Math Class
Calculations are based on the listed reference sources. Links open in a new tab.
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