Free · Complex number calculators

Complex Magnitude & Phase to a + bi

Convert a complex number’s nonnegative magnitude and phase to its real and imaginary components.

Calculator inputs

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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.

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

Magnitude 2 with phase 90° represents 0 + 2i. Magnitude 5 with phase 180° represents −5 + 0i.

a = r cos φ; b = r sin φ; z = a + bi

Enter these known values and leave the other values blank.

Magnitude r is nonnegative. Phase is measured counterclockwise from the positive real axis, using the selected degrees or radians.
The imaginary field returns the real coefficient b multiplying i. It does not display a second complex number.
This is a forward polar-to-rectangular conversion with read-only components. It does not recover a phase from arbitrary component inputs.
Zero magnitude gives zero real and imaginary components for every phase. The phase of the zero complex number is not uniquely determined.

The numeric output is the coefficient b. Write the final number as a + bi, including the sign of b.

Yes. A nonnegative magnitude can have either signed component because cosine and sine vary with phase.

Bookify requires a nonnegative complex magnitude while retaining signed phase and components.