Calculator Input
Enter rectangular parts, or switch to polar input. The calculator will convert, analyze, graph, and export the result.
Example Data Table
| Rectangular number |
Modulus |
Argument |
Polar form |
| 3 + 4i |
5 |
53.1301° |
5[cos(53.1301°) + i sin(53.1301°)] |
| -3 + 3i |
4.2426 |
135° |
4.2426[cos(135°) + i sin(135°)] |
| -5 - 12i |
13 |
-112.6199° |
13[cos(-112.6199°) + i sin(-112.6199°)] |
| 7 - 2i |
7.2801 |
-15.9454° |
7.2801[cos(-15.9454°) + i sin(-15.9454°)] |
Formula Used
The calculator uses atan2. This places the argument in the correct quadrant. It also supports signed and positive angle ranges.
How to Use This Calculator
- Select rectangular or polar input.
- Enter the real and imaginary parts, or enter radius and angle.
- Choose degrees or radians.
- Select the argument range and precision.
- Add a power, root degree, or second complex number if needed.
- Press Calculate to view the result above the form.
- Use the CSV or PDF buttons to save the result.
Guide to Polar Form of Complex Numbers
Understanding the Idea
A complex number can be drawn on a plane. The horizontal value is the real part. The vertical value is the imaginary part. Rectangular form uses these two parts directly. Polar form describes the same point with distance and direction. The distance is the modulus. The direction is the argument. This makes many complex operations easier to read.
Why Polar Form Helps
Multiplication is simple in polar form. The moduli multiply. The arguments add. Division is also direct. The moduli divide. The arguments subtract. Powers become faster through De Moivre’s rule. Roots become organized because their angles spread evenly around the origin. This is why polar form is common in algebra, signals, circuits, and vector work.
Reading the Argument
The argument must match the correct quadrant. A simple tangent ratio can miss this detail. This calculator uses the two-input arctangent method. It checks the signs of both parts. Then it returns a reliable direction. You may view the answer in degrees or radians. You may also choose a signed or positive angle range.
Using Advanced Results
The calculator shows rectangular, trigonometric, exponential, and phasor forms. It also gives the conjugate and reciprocal. You can test powers and roots. The root table lists each branch. The graph shows the complex vector, its conjugate, the chosen power, and the roots. This visual check helps confirm angle direction and symmetry.
Practical Notes
Always check the zero case. The modulus is zero when both parts are zero. In that case, the argument is not unique. For nonzero numbers, polar form gives a compact summary. Use higher precision for engineering or numeric work. Use fewer decimals for teaching, homework, and quick comparison.
Frequently Asked Questions
What is polar form of a complex number?
Polar form writes a complex number using its distance from the origin and its angle from the positive real axis. It is commonly written as r[cos θ + i sin θ] or re^(iθ).
How is the modulus calculated?
The modulus is the distance from the origin to the complex point. For z = a + bi, it is found with √(a² + b²).
How is the argument found?
The argument is the angle of the vector. This calculator uses atan2(b, a), so the result is placed in the correct quadrant.
Can I use radians instead of degrees?
Yes. Select radians in the angle unit field. The calculator will show the main argument, powers, and roots using radians.
What happens when the complex number is zero?
The modulus is zero. The argument is not unique because the point has no direction from the origin. The calculator explains this case.
Why are roots shown in a table?
Complex roots usually have multiple values. The table lists each root index, angle, polar form, and rectangular form for clear comparison.
What is exponential form?
Exponential form writes the number as re^(iθ). It is equivalent to the trigonometric polar form by Euler’s identity.
Can I export my answer?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable result summary.