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Complex Number Polar Form Calculator

Enter the real and imaginary parts to get the polar form.

How it is calculated

For z = a + bi, the real part a and the imaginary part b (the number multiplying i, not including the i) are read as plain numbers. The modulus is r = √(a² + b²). The argument is θ = atan2(b, a), converted from radians to degrees. Using atan2 rather than a simple arctan(b/a) puts the angle in the correct quadrant. The argument lies between −180° and 180°, so a number in the lower half-plane gets a negative angle, and a negative real number gets exactly 180°. Values are rounded to two decimals, and the polar line is shown as r ∠ θ°. For 0 + 0i the angle is undefined, but the tool displays 0.00°. Angles are in degrees only, not radians.

Worked example

Convert z = −3 + 4i. Here a = −3 and b = 4. Modulus: √(9 + 16) = √25 = 5, shown as 5.00. Argument: atan2(4, −3). A plain arctan(4 / −3) would give −53.13°, but the point is in the second quadrant (negative real, positive imaginary), so the correct angle is 180° − 53.13° = 126.87°. The tool shows the polar form 5.00 ∠ 126.87°, which means 5 (cos 126.87° + i sin 126.87°).

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Common mistakes

  • Using arctan(b / a) on a calculator. It cannot tell −3 + 4i from 3 − 4i and gives −53.13° for both; only atan2 returns the right quadrant.
  • Entering the imaginary part as 4i or with the i letter. Only the number goes in the field, 4.
  • Reading a negative argument such as −126.87° as an error. It is the same direction as 233.13°, since the tool reports angles from −180° to 180°.
  • Feeding the angle into a radians-based formula or function. The tool gives degrees, so convert (multiply by π / 180) before using it in a spreadsheet or program.
  • Dropping a minus sign on the real part. Compare −3 + 4i (126.87°) with 3 + 4i (53.13°): the modulus is the same, the angle is not.

About this calculator

Every complex number a + bi can be rewritten in polar form as r(cos θ + i sin θ), where the modulus r = √(a² + b²) is its distance from the origin on the complex plane, and the argument θ = atan2(b, a) is the angle it makes with the positive real axis. This form turns multiplication and division of complex numbers into simple addition/subtraction of angles and multiplication/division of magnitudes — far easier than working directly with the rectangular a + bi components.

Electrical engineers rely on polar form constantly for phasor analysis of AC circuits, where voltage and current are represented as complex numbers with magnitude and phase angle; it's equally central to signal processing (Fourier transforms), control systems, and any physics involving oscillations and rotations.

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Comments (1)

  • Editorial team

    The angle comes out between -180 and 180 degrees, but many textbooks and exam questions want 0 to 360. For a negative result just add 360, so -53.13 degrees becomes 306.87 degrees. If you need radians, multiply the degrees by pi and divide by 180; 126.87 degrees is about 2.2143 rad. Both describe the same point, so match whatever convention your course or software expects. Which convention does yours use?

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