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.
Understand Principal complex root
One idea, three depths
Choose how deeply to explain Principal complex root
Principal complex root: Calculate the principal nth root of a complex number in polar form.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Principal complex root to answer this question: calculate the principal nth root of a complex number in polar form? Enter Real part a, Imaginary part b, Root degree n; the calculator shows Principal root real part. For example: The principal square root of −1 is i. The answer tells you Principal root real part.
Age 15Explain it to a 15-year-oldConnect it to the formula
Taking an nth root divides the phase by n and takes the ordinary nth root of the magnitude. The rule is z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)]. Its input values are Real part a, Imaginary part b, Root degree n, and the main result is Principal root real part. For example: The principal square root of −1 is i.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated principal complex root relation over the valid mixed integer and real-number domain stated below. The implemented relation is z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)], evaluated from Real part a, Imaginary part b, Root degree n to produce Principal root real part. Taking an nth root divides the phase by n and takes the ordinary nth root of the magnitude. A nonzero complex number has n distinct nth roots; this tool displays the principal root only.
Inputs and valid domain
- Real part a must be a finite real number.
- Imaginary part b must be a finite real number.
- Root degree n must be an integer, at least 1.
Important boundary: A nonzero complex number has n distinct nth roots; this tool displays the principal root only.
The formula
z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)]
How the calculator works through it
It substitutes Real part a, Imaginary part b, Root degree n into the formula and exposes every numerical step above. The main output is Principal root real part, accompanied by Principal root imaginary coefficient, Root magnitude, Root phase.
Read the result correctly
The Principal root real part is the direct answer to “calculate the principal nth root of a complex number in polar form.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
The principal square root of −1 is i.
Where this model stops being reliable
A nonzero complex number has n distinct nth roots; this tool displays the principal root only.
Learn it by changing one value
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.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Principal complex root works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Principal complex root uses z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)]. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Complex numbers and components
Real and imaginary components provide the notation needed to interpret Principal complex root correctly.
Review this foundation about 7 min
Optional enrichment
- Functions and periodic behaviour
A function viewpoint connects Principal complex root to signals, periodicity and transformations.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Real part a, Imaginary part b, Root degree n.
- Evaluate the principal relationship: z^(1/n)=r^(1/n)[cos(θ/n)+i sin(θ/n)].
- Return Principal root real part and check the domain conditions described above.
Python
from math import *
def complex_nth_root(a, b, n) -> float:
return (pow(sqrt(((a * a) + (b * b))), (1.0 / n)) * cos((atan2(b, a) / n)))
assert abs(complex_nth_root(-1, 0, 2) - 6.123233995736766e-17) < 1e-6 * max(1.0, abs(6.123233995736766e-17))
C
#include <assert.h>
#include <math.h>
double complex_nth_root(double a, double b, double n) {
return (pow(sqrt(((a * a) + (b * b))), (1.0 / n)) * cos((atan2(b, a) / n)));
}
int main(void) {
const double expected = 6.123233995736766e-17;
const double actual = complex_nth_root(-1, 0, 2);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double complex_nth_root(double a, double b, double n) {
return (std::pow(std::sqrt(((a * a) + (b * b))), (1.0 / n)) * std::cos((std::atan2(b, a) / n)));
}
int main() {
constexpr double expected = 6.123233995736766e-17;
const double actual = complex_nth_root(-1, 0, 2);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double complex_nth_root(double a, double b, double n)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern pow
extern atan2
extern cos
global complex_nth_root
section .text
complex_nth_root:
push rbp
mov rbp, rsp
sub rsp, 112
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-16]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-64]
addsd xmm0, [rbp-72]
movsd [rbp-56], xmm0
sqrtsd xmm0, [rbp-56]
movsd [rbp-48], xmm0
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-88], xmm0
movsd xmm0, [rbp-88]
divsd xmm0, [rbp-24]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-48]
movsd xmm1, [rbp-80]
call pow wrt ..plt
movsd [rbp-40], xmm0
movsd xmm0, [rbp-16]
movsd xmm1, [rbp-8]
call atan2 wrt ..plt
movsd [rbp-112], xmm0
movsd xmm0, [rbp-112]
divsd xmm0, [rbp-24]
movsd [rbp-104], xmm0
movsd xmm0, [rbp-104]
call cos wrt ..plt
movsd [rbp-96], xmm0
movsd xmm0, [rbp-40]
mulsd xmm0, [rbp-96]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = complex_nth_root(a, b, n)
result = ((sqrt(((a * a) + (b * b))) ^ (1.0 / n)) * cos((atan2(b, a) / n)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, n_] := ((Sqrt[((a * a) + (b * b))] ^ (1.0 / n)) * Cos[(ArcTan[a, b] / n)]);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Principal Complex Root Calculator. MW SysArc Tools. https://math.mwsysarc.com/complex-fourier/principal-complex-root
MLA 9
MW SysArc. “Principal Complex Root Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/complex-fourier/principal-complex-root. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Principal Complex Root Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/complex-fourier/principal-complex-root.
Harvard
MW SysArc (2026) ‘Principal Complex Root Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/complex-fourier/principal-complex-root (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_complex_nth_root_2026,
author = {{MW SysArc}},
title = {Principal Complex Root Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/complex-fourier/principal-complex-root},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Principal Complex Root Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/complex-fourier/principal-complex-root
N1 - Published July 21, 2026
ER -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 . Calculations tested .