Free · Complex numbers calculators

Complex Real/Imaginary Parts to Polar Form

Calculate magnitude and principal phase from real and imaginary components, with degree/radian angle display.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

For 3 + 4i, magnitude is 5 and phase is approximately 53.1301°.

r = √(a²+b²); φ = atan2(b,a); z = r × exp(iφ)

Worked example

Enter these known values and leave the other values blank.

Real component a
3
Imaginary component b
4
Magnitude r
5
Principal phase φ
53.13 deg

Assumptions and limitations

  • The phase uses both components and the correct quadrant via atan2.
  • For the zero complex number, magnitude is zero and phase is not unique. The retained numerical convention displays zero as its representative phase.
  • The exponential-form note uses radians, even when the separate phase display is converted to degrees.
  • The captured outputs are one-way; enter real and imaginary components to obtain the polar representation.
  • Arithmetic uses 64 significant decimal digits, with separate rounding for displayed values.

Common questions

Why does atan2 differ from arctan(b/a)?

atan2 distinguishes quadrants and handles an exactly zero real component without dividing by it.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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