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°.
Understand Complex polar form
One idea, three depths
Choose how deeply to explain Complex polar form
Complex polar form: Convert z=a+bi into magnitude and phase angle.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Complex polar form to answer this question: convert z=a+bi into magnitude and phase angle? Enter Real part a and Imaginary part b; the calculator shows Magnitude r. For example: 3+4i has magnitude 5 and phase about 0.9273 radians or 53.13°. The answer tells you Magnitude r.
Age 15Explain it to a 15-year-oldConnect it to the formula
Magnitude is the distance from the origin and phase is the direction of the complex number in the plane. The rule is r=√(a²+b²); θ=atan2(b,a). Its input values are Real part a, Imaginary part b, and the main result is Magnitude r. For example: 3+4i has magnitude 5 and phase about 0.9273 radians or 53.13°.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated complex polar form relation over the valid real-number domain stated below. The implemented relation is r=√(a²+b²); θ=atan2(b,a), evaluated from Real part a, Imaginary part b to produce Magnitude r. Magnitude is the distance from the origin and phase is the direction of the complex number in the plane. Use atan2(b,a), not plain arctan(b/a), to preserve the correct quadrant.
Inputs and valid domain
- Real part a must be a finite real number.
- Imaginary part b must be a finite real number.
Important boundary: Use atan2(b,a), not plain arctan(b/a), to preserve the correct quadrant.
The formula
r=√(a²+b²); θ=atan2(b,a)
How the calculator works through it
It substitutes Real part a, Imaginary part b into the formula and exposes every numerical step above. The main output is Magnitude r, accompanied by Phase in radians, Phase in degrees.
Read the result correctly
The Magnitude r is the direct answer to “convert z=a+bi into magnitude and phase angle.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
3+4i has magnitude 5 and phase about 0.9273 radians or 53.13°.
Where this model stops being reliable
Use atan2(b,a), not plain arctan(b/a), to preserve the correct quadrant.
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 Complex polar form works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Complex polar form uses r=√(a²+b²); θ=atan2(b,a). 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 Complex polar form correctly.
Review this foundation about 7 min
Optional enrichment
- Functions and periodic behaviour
A function viewpoint connects Complex polar form 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.
- Evaluate the principal relationship: r=√(a²+b²); θ=atan2(b,a).
- Return Magnitude r and check the domain conditions described above.
Python
from math import *
def complex_polar_form(a, b) -> float:
return sqrt(((a * a) + (b * b)))
assert abs(complex_polar_form(3, 4) - 5) < 1e-6 * max(1.0, abs(5))
C
#include <assert.h>
#include <math.h>
double complex_polar_form(double a, double b) {
return sqrt(((a * a) + (b * b)));
}
int main(void) {
const double expected = 5;
const double actual = complex_polar_form(3, 4);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double complex_polar_form(double a, double b) {
return std::sqrt(((a * a) + (b * b)));
}
int main() {
constexpr double expected = 5;
const double actual = complex_polar_form(3, 4);
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_polar_form(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global complex_polar_form
section .text
complex_polar_form:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-16]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-40]
addsd xmm0, [rbp-48]
movsd [rbp-32], xmm0
sqrtsd xmm0, [rbp-32]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = complex_polar_form(a, b)
result = sqrt(((a * a) + (b * b)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := Sqrt[((a * a) + (b * b))];
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). Complex Rectangular to Polar Converter. MW SysArc Tools. https://math.mwsysarc.com/complex-fourier/rectangular-polar-converter
MLA 9
MW SysArc. “Complex Rectangular to Polar Converter.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/complex-fourier/rectangular-polar-converter. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Complex Rectangular to Polar Converter.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/complex-fourier/rectangular-polar-converter.
Harvard
MW SysArc (2026) ‘Complex Rectangular to Polar Converter’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/complex-fourier/rectangular-polar-converter (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_complex_polar_form_2026,
author = {{MW SysArc}},
title = {Complex Rectangular to Polar Converter},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/complex-fourier/rectangular-polar-converter},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Complex Rectangular to Polar Converter
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/complex-fourier/rectangular-polar-converter
N1 - Published July 21, 2026
ER -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 . Calculations tested .