Mathematics · Complex and Fourier
Complex Rectangular to Polar Converter
Convert z=a+bi into magnitude and phase angle.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Magnitude = √(3²+4²)=5.
- Phase = atan2(4,3)=0.9272952180016122 radians.
- Convert phase to degrees: 53.13010235415598°.
Problem → model → reason → result
What problem does this model solve?
Convert z=a+bi into magnitude and phase angle.
Why does the model apply?
Magnitude is the distance from the origin and phase is the direction of the complex number in the plane.
What assumptions does it make?
Complex components use one consistent convention, angles use the units shown, and Fourier samples are equally spaced observations of one signal.
Formula
r=√(a²+b²); θ=atan2(b,a)
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
3+4i has magnitude 5 and phase about 0.9273 radians or 53.13°.
Common mistake
Use atan2(b,a), not plain arctan(b/a), to preserve the correct quadrant.
When does this model not apply?
Small complex-number and four-sample models teach the structure but do not replace full-spectrum analysis, windowing, uncertainty analysis or numerically stable large transforms.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Complex polar form do?
Convert z=a+bi into magnitude and phase angle.
How does the Complex polar form work?
The calculator applies r=√(a²+b²); θ=atan2(b,a). Magnitude is the distance from the origin and phase is the direction of the complex number in the plane.
What can I learn from the Complex polar form?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.