Enter Complex Number
Formula Used
For a complex number z = a + bi, the polar form is based on modulus and argument.
- Modulus: r = √(a² + b²)
- Argument: θ = atan2(b, a)
- Trigonometric form: z = r(cos θ + i sin θ)
- Exponential form: z = re^(iθ)
The atan2 function is used because it detects the correct quadrant.
How to Use This Calculator
- Enter the real part of the complex number.
- Enter only the imaginary coefficient.
- Select degrees or radians.
- Choose principal angle or positive angle output.
- Select decimal precision.
- Press the submit button.
- Review the result below the header.
- Download the result as CSV or PDF when needed.
Example Data Table
| Complex Number | Modulus | Argument | Quadrant | Polar Form |
|---|---|---|---|---|
| 3 + 4i | 5 | 53.1301° | Quadrant I | 5(cos 53.1301° + i sin 53.1301°) |
| -3 + 4i | 5 | 126.8699° | Quadrant II | 5(cos 126.8699° + i sin 126.8699°) |
| -5 - 12i | 13 | -112.6199° | Quadrant III | 13(cos -112.6199° + i sin -112.6199°) |
| 6 - 8i | 10 | -53.1301° | Quadrant IV | 10(cos -53.1301° + i sin -53.1301°) |
Complex to Polar Form Guide
A complex number can be written as a real part plus an imaginary part. Rectangular form is useful for addition and subtraction. Polar form is better when you need size, direction, multiplication, division, roots, or powers. This calculator changes a value like a + bi into r(cos θ + i sin θ) and re^(iθ).
Why Polar Form Matters
The modulus r tells how far the point is from the origin. The argument θ tells the angle from the positive real axis. Together, these two values describe the same point on the complex plane. This view is common in signals, electronics, control systems, vectors, and advanced algebra.
Reading the Result
The result shows the modulus, the argument, the quadrant, and both common polar notations. Degrees are easier for many learners. Radians are often required in calculus and higher mathematics. The calculator can also show a positive angle from 0 to 360 degrees, or a principal angle from -180 to 180 degrees.
Accuracy and Precision
Small rounding changes may appear when decimal inputs are used. You can choose the number of decimal places. More decimals are helpful for engineering checks. Fewer decimals make classroom answers easier to read. The exact formulas remain the same.
Practical Uses
Polar form makes multiplication simple because moduli multiply and angles add. Division works by dividing moduli and subtracting angles. Powers use repeated angle addition. Roots divide the angle into equal parts. This is why polar form appears in circuit analysis, wave motion, phasors, rotations, and geometry.
Best Input Tips
Enter the real part in the first box. Enter the imaginary coefficient in the second box. Do not type the letter i. For 3 - 4i, enter 3 and -4. For a pure imaginary number, enter 0 for the real part. Check the selected angle mode before copying the final answer.
Common Mistakes
The most common mistake is using ordinary arctangent without quadrant correction. A number in quadrant two or three can receive the wrong angle. The atan2 method fixes that problem by reading both parts together. Another mistake is forgetting that zero has no defined angle. When both parts are zero, only the modulus is meaningful. Treat that case with care always.
FAQs
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 usually written as r(cos θ + i sin θ) or re^(iθ).
What is the modulus?
The modulus is the distance of the complex number from the origin. For z = a + bi, it is calculated as √(a² + b²).
What is the argument?
The argument is the angle made by the complex number with the positive real axis. This calculator uses atan2(b, a) to find the correct quadrant.
Should I use degrees or radians?
Use degrees for basic interpretation and classroom work. Use radians for calculus, advanced algebra, physics, and most technical formulas involving trigonometric functions.
Why is atan2 better than tan inverse?
Ordinary tan inverse can miss the correct quadrant. The atan2 method reads both the real and imaginary parts, so it returns a better angle direction.
What happens when both parts are zero?
The modulus is zero, but the argument is not unique. Every angle points to the same origin, so the angle should be treated as undefined.
Can polar form be converted back?
Yes. Use a = r cos θ and b = r sin θ. Then write the rectangular form as a + bi.
Why is polar form useful?
Polar form makes multiplication, division, powers, roots, rotations, and phasor calculations easier. It is common in engineering, signals, circuits, and higher mathematics.