TG we-Calculate logoTG we-Calculate
All Calculators
Intermediate

Divide Complex Numbers Calculator

Divide any two complex numbers of the form (a+bi)÷(c+di) by multiplying both numerator and denominator by the conjugate of the denominator. Get the real part, imaginary part, and magnitude of the result instantly.
In (a + bi)
In (a + bi)
In (c + di)
In (c + di)
Result magnitude |z|
2.2361

Magnitude of the quotient complex number

Real part (Re)
2.2
Imaginary part (Im)
0.4i
Result
2.2 + 0.4i
|numerator| / |denominator|
5 / 2.2361
z₁z₂z₁/z₂Complex plane: numerator (z₁), denominator (z₂) and quotient (z₁/z₂) as vectors
Step by step
  1. 1

    Denominator squared |c+di|² = c²+d²

    2² + 1² = 5
    Multiplying by the conjugate (c−di) clears the imaginary part from the denominator.
  2. 2

    Real part Re = (ac+bd) ÷ (c²+d²)

    (4×2 + 3×1) ÷ 5 = 2.2
  3. 3

    Imaginary part Im = (bc−ad) ÷ (c²+d²)

    (3×2 − 4×1) ÷ 5 = 0.4
  4. 4

    Magnitude |z₁/z₂| = √(Re²+Im²)

    √(2.2² + 0.4²) = 2.2361
Results are estimates for general information only and are not professional advice — always verify important results independently before relying on them. Read the full disclaimer.
Quick answer

How does this calculator work?

To divide (a+bi) by (c+di), multiply top and bottom by the conjugate (c−di) to get Re=(ac+bd)/(c²+d²) and Im=(bc−ad)/(c²+d²). The magnitude of the result is |z₁|/|z₂|. Enter the real and imaginary parts of each number to see the quotient instantly.

Formula
(a+bi)÷(c+di) = [(ac+bd)/(c²+d²)] + [(bc−ad)/(c²+d²)]i
How this is calculated

To divide complex numbers, multiply both the numerator and the denominator by the complex conjugate of the denominator. The conjugate of (c+di) is (c−di), so the product (c+di)(c−di) = c²+d², which is a real number. This eliminates the imaginary part from the denominator.

After multiplying out, the numerator (a+bi)(c−di) = (ac+bd) + (bc−ad)i. Dividing each component by the real denominator c²+d² gives the real part Re = (ac+bd)/(c²+d²) and imaginary part Im = (bc−ad)/(c²+d²).

The magnitude of the quotient equals |z₁|/|z₂| — the ratio of the two moduli. Division is undefined when the denominator is exactly zero (c=0 and d=0). All arithmetic here is exact floating-point; for very large or very small numbers, rounding errors may affect the last few digits.

Frequently asked questions

Multiplying numerator and denominator by the conjugate (c−di) converts the denominator to the real number c²+d², making it straightforward to separate the result into real and imaginary parts by simple division.

The magnitude of the quotient is |z₁|/|z₂| — the modulus of the numerator divided by the modulus of the denominator. This is one of the key properties of complex multiplication and division.

Division by zero is undefined in complex numbers just as with reals. If c=0 and d=0, the calculator shows a warning instead of a result.

Also known as

divide complex numbers
complex number division
complex division calculator
imaginary number division
complex quotient formula
how to divide complex numbers
divide a+bi by c+di

APA

TG we-Calculate Editorial Team. (2026). Divide Complex Numbers Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/divide-complex-numbers-calculator

Chicago

TG we-Calculate Editorial Team. "Divide Complex Numbers Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/divide-complex-numbers-calculator.

IEEE

TG we-Calculate Editorial Team, "Divide Complex Numbers Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/divide-complex-numbers-calculator

BibTeX

@misc{wecalculate_divide_complex_numbers_calculator, title = {Divide Complex Numbers Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/divide-complex-numbers-calculator}}, year = {2026}, note = {TG we-Calculate} }

Did this calculator help you?

Related calculators

Fraction Calculator

Add, subtract, multiply or divide two fractions and get the simplified result plus its decimal value instantly.

Beginner

Polar to Rectangular Converter

Convert polar coordinates (r, theta) to rectangular (x, y) coordinates and plot the resulting point.

Intermediate

Rectangular to Polar Converter

Convert rectangular (x, y) coordinates to polar form (r, theta) using magnitude and the atan2 angle.

Advanced

Ratio Calculator

Solve proportions of the form a:b = c:d for the missing value, and simplify any ratio to its lowest terms.

Beginner

2×2 Matrix Calculator

Find the determinant, trace, rank, eigenvalues and inverse of any 2×2 matrix in one step, with an animated transformation.

Intermediate

2D Distance Calculator

Calculate the straight-line Euclidean distance between two points on a plane, plus midpoint, slope and angle.

Beginner