Turn rectangular values into clean polar notation fast. See radians, degrees, and exponential form clearly. Save steps, export data, and verify every conversion easily.
Rectangular form: z = a + bi
Modulus: r = |z| = √(a² + b²)
Argument: θ = atan2(b, a)
Polar form: z = r(cos θ + i sin θ)
Exponential form: z = reiθ
Conjugate: z̅ = a - bi
Reciprocal: 1/z = (a - bi)/(a² + b²), for z ≠ 0
| Rectangular form | Modulus | Angle (degrees) | Polar form |
|---|---|---|---|
| 3 + 4i | 5 | 53.1301° | 5(cos 53.1301° + i sin 53.1301°) |
| 1 + i | 1.4142 | 45° | 1.4142(cos 45° + i sin 45°) |
| -2 + 2i | 2.8284 | 135° | 2.8284(cos 135° + i sin 135°) |
| -5 - 5i | 7.0711 | -135° | 7.0711(cos -135° + i sin -135°) |
| 0 + 6i | 6 | 90° | 6(cos 90° + i sin 90°) |
A complex number has a real part and an imaginary part. Polar form rewrites that number using distance and angle. This calculator converts rectangular input into polar notation fast. It also shows trigonometric and exponential forms. That helps students check algebra, geometry, and phasor work.
Polar form is useful in many maths topics. It simplifies multiplication, division, powers, and roots. Magnitudes multiply cleanly. Angles add or subtract directly. This makes several long calculations shorter and easier to verify.
The tool reports the modulus, the principal argument, and the selected angle unit. It also identifies the quadrant or axis. You can review the rectangular form, the normalized polar form, and Euler style notation in one place. Extra outputs include the conjugate and reciprocal when possible.
You can enter values as separate real and imaginary parts. You can also type a compact expression such as 3+4i, -5i, or 7-2i. This flexibility reduces input mistakes. It also speeds up repeated homework checks.
Use this page for classroom examples, exam practice, and signal analysis exercises. It works well for vector style interpretation too. Engineers often read modulus as amplitude and argument as phase. Students often use the same ideas in De Moivre based problems.
Arguments can be shown in radians or degrees. The range can stay in principal form or shift to a zero to two pi style interval. This is important because equivalent angles describe the same point. Clear normalization keeps answers consistent across textbooks.
The result box appears above the form after submission. That placement makes comparison easier. You can export the output as CSV for records. You can also save the result as PDF for notes or client work. The example table below shows common conversions and expected patterns.
When the complex number is zero, the modulus is zero and the argument is undefined. The calculator handles that case safely. It avoids misleading angle output and explains why the polar expression changes. This makes the page practical for both beginners and advanced users.
Because polar coordinates describe direction clearly, they also help with graphing. Teachers often switch between forms in lessons. A calculator that shows both views reduces confusion and improves answer checking.
Polar form writes a complex number using magnitude and angle. It is usually shown as r(cos θ + i sin θ) or reiθ.
Take the square root of the sum of squares of the real and imaginary parts. That is √(a² + b²).
The argument is found with atan2(b, a). This method uses both signs and returns the correct quadrant automatically.
atan2 is more reliable because ordinary arctan cannot fully determine the quadrant from the ratio alone.
Its modulus is zero, but the argument is undefined. The calculator shows that clearly to avoid a false angle result.
Yes. Choose degrees in the form. The calculator can also display radians for textbook or advanced maths work.
Exponential form is reiθ. It is based on Euler’s relation and is common in engineering and higher mathematics.
Angles differing by full turns are equivalent. Adding 360° or 2π does not change the point in the complex plane.
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.