Mathematics · Complex and Fourier
Complex Number Addition Calculator
Add two complex numbers by combining their real and imaginary components.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Real part: 3+-1=2.
- Imaginary coefficient: 2+5=7.
- Result = 2+(7)i; magnitude = 7.280109889280519.
Understand Complex addition
One idea, three depths
Choose how deeply to explain Complex addition
Complex addition: Add two complex numbers by combining their real and imaginary components.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Complex addition to answer this question: add two complex numbers by combining their real and imaginary components? Enter First real part a, First imaginary part b, Second real part c, and 1 other input; the calculator shows Result real part. For example: (3+2i)+(−1+5i)=2+7i. The answer tells you Result real part.
Age 15Explain it to a 15-year-oldConnect it to the formula
Real and imaginary directions form independent axes, so complex numbers add component by component like two-dimensional vectors. The rule is (a+bi)+(c+di)=(a+c)+(b+d)i. Its input values are First real part a, First imaginary part b, Second real part c, Second imaginary part d, and the main result is Result real part. For example: (3+2i)+(−1+5i)=2+7i.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated complex addition relation over the valid real-number domain stated below. The implemented relation is (a+bi)+(c+di)=(a+c)+(b+d)i, evaluated from First real part a, First imaginary part b, Second real part c, Second imaginary part d to produce Result real part. Real and imaginary directions form independent axes, so complex numbers add component by component like two-dimensional vectors. Only like components combine: real with real and imaginary with imaginary.
Inputs and valid domain
- First real part a must be a finite real number.
- First imaginary part b must be a finite real number.
- Second real part c must be a finite real number.
- Second imaginary part d must be a finite real number.
Important boundary: Only like components combine: real with real and imaginary with imaginary.
The formula
(a+bi)+(c+di)=(a+c)+(b+d)i
How the calculator works through it
It substitutes First real part a, First imaginary part b, Second real part c, Second imaginary part d into the formula and exposes every numerical step above. The main output is Result real part, accompanied by Result imaginary coefficient, Result magnitude.
Read the result correctly
The Result real part is the direct answer to “add two complex numbers by combining their real and imaginary components.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
(3+2i)+(−1+5i)=2+7i.
Where this model stops being reliable
Only like components combine: real with real and imaginary with imaginary.
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 addition works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Complex addition uses (a+bi)+(c+di)=(a+c)+(b+d)i. 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 addition correctly.
Review this foundation about 7 min
Optional enrichment
- Functions and periodic behaviour
A function viewpoint connects Complex addition 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 First real part a, First imaginary part b, Second real part c, Second imaginary part d.
- Evaluate the principal relationship: (a+bi)+(c+di)=(a+c)+(b+d)i.
- Return Result real part and check the domain conditions described above.
Python
from math import *
def complex_addition(a1, b1, a2, b2) -> float:
return (a1 + a2)
assert abs(complex_addition(3, 2, -1, 5) - 2) < 1e-6 * max(1.0, abs(2))
C
#include <assert.h>
#include <math.h>
double complex_addition(double a1, double b1, double a2, double b2) {
return (a1 + a2);
}
int main(void) {
const double expected = 2;
const double actual = complex_addition(3, 2, -1, 5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double complex_addition(double a1, double b1, double a2, double b2) {
return (a1 + a2);
}
int main() {
constexpr double expected = 2;
const double actual = complex_addition(3, 2, -1, 5);
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_addition(double a1, double b1, double a2, double b2)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global complex_addition
section .text
complex_addition:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd xmm0, [rbp-8]
addsd xmm0, [rbp-24]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
leave
ret
MATLAB
function result = complex_addition(a1, b1, a2, b2)
result = (a1 + a2);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a1_, b1_, a2_, b2_] := (a1 + a2);
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 Number Addition Calculator. MW SysArc Tools. https://math.mwsysarc.com/complex-fourier/complex-number-addition
MLA 9
MW SysArc. “Complex Number Addition Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/complex-fourier/complex-number-addition. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Complex Number Addition Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/complex-fourier/complex-number-addition.
Harvard
MW SysArc (2026) ‘Complex Number Addition Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/complex-fourier/complex-number-addition (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_complex_addition_2026,
author = {{MW SysArc}},
title = {Complex Number Addition Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/complex-fourier/complex-number-addition},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Complex Number Addition Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/complex-fourier/complex-number-addition
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Complex addition do?
Add two complex numbers by combining their real and imaginary components.
How does the Complex addition work?
The calculator applies (a+bi)+(c+di)=(a+c)+(b+d)i. Real and imaginary directions form independent axes, so complex numbers add component by component like two-dimensional vectors.
What can I learn from the Complex addition?
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 .