Complex Polar Form CalculatorFree online math calculator — instant results, no sign-up
Convert a nonzero complex number to modulus and principal argument.
About the Complex Polar Form Calculator
The Complex Polar Form Calculator converts a nonzero complex number from its real and imaginary parts into its modulus and principal argument in degrees. For the default number 3 + 4i, it returns a modulus of 5 and an argument of 53.1301023542°, describing the number’s distance and direction from the origin in the complex plane.
- Real part
- The real part a is the horizontal coordinate of the complex number a + bi; it is unitless here and defaults to 3.
- Imaginary part
- The imaginary part b is the real-valued coefficient of i and the vertical coordinate of a + bi; it is unitless here and defaults to 4.
Formula and method
z = a + bi; r = √(a² + b²); θ = atan2(b, a) × 180/π, normalized to −180° < θ ≤ 180°. Polar form: z = r[cos(θπ/180) + i sin(θπ/180)]. With a = 3 and b = 4: r = √(3² + 4²) = 5 and θ = atan2(4, 3) × 180/π ≈ 53.1301023542°.
- zThe nonzero complex number being converted.
- aThe real part of z.
- bThe imaginary part of z, meaning the coefficient of i.
- iThe imaginary unit, defined by i² = −1.
- rThe modulus, or distance of z from the origin.
- θThe principal argument in degrees, measured from the positive real axis.
- atan2(b, a)The two-argument arctangent, which determines the angle in radians using the signs of both coordinates.
- πPi, used to convert between radians and degrees.
Read the two inputs as a and b and verify that they are finite real numbers and not both zero. Compute the modulus from the square root of the sum of their squares. Use atan2(b, a) to determine the correct quadrant, convert the angle from radians to degrees, and express it in the principal range. For the default inputs, these calculations give 5 and 53.1301023542°.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Real part3
- Imaginary part4
Results
- Modulus5
- Argument (degrees)53.1301023542
How to use the Complex Polar Form Calculator
- 1Enter the horizontal coordinate in Real part; leave it at 3 to follow the worked example.
- 2Enter the coefficient of i in Imaginary part; leave it at 4 for the worked example.
- 3Confirm that the two inputs are not both zero, then run the calculation.
- 4Read Modulus: the default inputs produce 5.
- 5Read Argument (degrees): the default inputs produce 53.1301023542°, placing 3 + 4i in the first quadrant.
Reading your result
Modulus reports the nonnegative distance from the origin; for any valid nonzero input it is positive, and for 3 + 4i it is 5. Argument (degrees) reports the principal direction relative to the positive real axis: positive angles turn counterclockwise and negative angles turn clockwise. The default argument, 53.1301023542°, combines with the modulus to describe the polar form of 3 + 4i.
Limitations
The zero complex number has modulus zero but no defined argument, so this conversion requires at least one nonzero input. This documentation uses the principal-argument convention (−180°, 180°]; some systems instead use [−180°, 180°), differing only on the negative real axis. Displayed angles are rounded, and very large or very small inputs may be affected by numerical precision or implementation limits. Inputs are interpreted as real-valued coordinates, not as angle measurements.
Common questions
Why does 3 + 4i have modulus 5?+
Its coordinates form the legs of a right triangle, so the distance from the origin is √(3² + 4²) = √25 = 5.
Why use atan2 instead of arctan of the imaginary part divided by the real part?+
atan2 uses the signs of both parts to identify the correct quadrant and handles a zero real part without division by zero. For the default inputs, atan2(4, 3) gives 53.1301023542° after conversion to degrees.
What makes the reported argument a principal argument?+
Angles differing by a whole number of full turns describe the same complex direction. A principal argument selects one representative; the convention used here is greater than −180° and at most 180°.
Can I convert a purely real or purely imaginary number?+
Yes, provided it is nonzero. A positive real number has argument 0°, a negative real number has argument 180° under this convention, and purely imaginary numbers have argument 90° or −90° according to the sign of their imaginary part.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.