Free · Complex numbers
Complex Magnitude & Phase to Real and Imaginary Parts
Turn a complex number’s nonnegative magnitude and phase into its real and imaginary components, with degree and radian inputs.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
Loading calculator…
How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- 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
A complex number with magnitude 5 and phase 0° has real part 5 and imaginary part 0. At 180° it lies on the negative real axis, with real part −5.
a = r cos(θ); b = r sin(θ); z = a + bi
Worked example
Enter these known values and leave the other values blank.
- Nonnegative magnitude r
- 5
- Phase angle θ
- 0 deg
- Real component a
- 5
- Imaginary component b
- 0
Assumptions and limitations
- Magnitude is the nonnegative distance from the origin. The real and imaginary components may be negative.
- Phase is measured counterclockwise from the positive real axis. Check whether the angle selector says degrees or radians.
- Adding a full turn to phase gives the same components. This tool computes rectangular components from magnitude and phase; both component fields are read only.
- At magnitude zero, every entered phase gives the zero complex number. Zero itself does not have a unique phase.
- Trigonometric results use finite numerical precision; tiny residuals near axis angles may be displayed as zero.
Common questions
What does the imaginary output represent?
It is the real coefficient b in a + bi, where i² = −1. An output of −3 means the imaginary term is −3i.
Can the magnitude be negative?
No. Magnitude is a distance. Use the phase angle to describe direction, including points with negative real or imaginary components.
References
Bookify requires a nonnegative complex magnitude.