Change values into clear polar notation quickly. Review modulus, angle, radians, degrees, and quadrant details. Useful for study, verification, teaching, and daily problem solving.
| Complex Number | Modulus | Angle (Degrees) | Polar Form |
|---|---|---|---|
| 3 + 4i | 5 | 53.1301° | 5(cos 53.1301° + i sin 53.1301°) |
| -2 + 2i | 2.8284 | 135.0000° | 2.8284(cos 135.0000° + i sin 135.0000°) |
| -5 - 5i | 7.0711 | -135.0000° | 7.0711(cos -135.0000° + i sin -135.0000°) |
| 0 + 6i | 6 | 90.0000° | 6(cos 90.0000° + i sin 90.0000°) |
For a complex number z = a + bi, the modulus is r = √(a² + b²).
The argument is θ = atan2(b, a).
The polar form is z = r(cos θ + i sin θ).
You may also write the same value as z = r cis θ.
Enter the real part in the first field.
Enter the imaginary part in the second field.
Choose how many decimal places you want.
Select the angle range and display preference.
Click the convert button.
Read the modulus, angle, quadrant, and polar forms.
Use the export buttons to save the result and example table.
Complex numbers appear in algebra, trigonometry, signals, and engineering. Polar form gives another way to express them. Instead of writing a + bi, you write r(cosθ + i sinθ). This format highlights magnitude and direction. It also simplifies multiplication, division, powers, and roots.
Polar form describes how far a complex number sits from the origin. That distance is the modulus. It also shows the rotation from the positive real axis. That rotation is the argument. Together, these values explain the number clearly in the complex plane.
For a complex number z = a + bi, the modulus is r = √(a² + b²). The angle is θ = atan2(b, a). The atan2 function is useful because it selects the correct quadrant. After that, the calculator reports θ in radians and degrees. It can also normalize the angle to a preferred range.
Polar notation is useful in circuit analysis, wave modeling, vectors, and transformations. It reduces repeated algebra when multiplying complex values. You multiply moduli and add angles. For division, you divide moduli and subtract angles. These steps are faster than expanding long rectangular expressions.
Enter the real part and imaginary part. Choose the decimal precision that fits your work. Select the angle normalization you want. Then submit the form. The calculator returns modulus, argument in radians, argument in degrees, quadrant, and the polar expression. You can also download the results as CSV or PDF for records, homework, or reports.
If the number is 3 + 4i, the modulus is 5. The angle is about 53.13 degrees. The polar form becomes 5(cos 53.13° + i sin 53.13°). This tells you the point is five units from the origin and rotated into Quadrant I.
When both parts equal zero, the modulus is zero and the angle is undefined. This calculator handles that edge case clearly. It also uses quadrant-aware logic, so negative real or imaginary values still produce the correct argument without manual correction for most users.
Polar form writes a complex number using its modulus and angle. Instead of a + bi, it becomes r(cos θ + i sin θ) or r cis θ.
atan2 uses both the real and imaginary parts. That helps the calculator place the angle in the correct quadrant without extra manual adjustments.
The modulus becomes zero. The angle is undefined because the origin has no unique direction. The calculator shows that case clearly.
Yes. In the principal range, the argument can be negative. You can also choose a positive range so the angle stays between 0 and 2π.
Different courses and applications prefer different angle units. Showing both helps with homework, checking solutions, and switching between algebra and trigonometry contexts.
Quadrants help verify the angle direction. A correct modulus with a wrong quadrant can lead to the wrong polar form and later calculation errors.
Yes. The calculator accepts positive values, negative values, and decimals for both parts of the complex number.
Polar form is common in complex analysis, electrical engineering, wave problems, vectors, and any work involving repeated multiplication, division, powers, or roots.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.