Mathematics · Complex and Fourier
Principal Complex Root Calculator
Calculate the principal nth root of a complex number in polar form.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Magnitude=1; principal phase=3.141592653589793.
- Root magnitude=1^(1/2)=1; phase=3.141592653589793÷2=1.5707963267948966.
- Principal root=6.123233995736766e-17+(1)i.
Problem → model → reason → result
What problem does this model solve?
Calculate the principal nth root of a complex number in polar form.
Why does the model apply?
Taking an nth root divides the phase by n and takes the ordinary nth root of the magnitude.
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
z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)]
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
The principal square root of −1 is i.
Common mistake
A nonzero complex number has n distinct nth roots; this tool displays the principal root only.
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 Principal complex root do?
Calculate the principal nth root of a complex number in polar form.
How does the Principal complex root work?
The calculator applies z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)]. Taking an nth root divides the phase by n and takes the ordinary nth root of the magnitude.
What can I learn from the Principal complex root?
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.