Complex Number Division Calculator
Dividing two complex numbers (a + bi) / (c + di) is handled by multiplying the top and bottom by the conjugate of the denominator. This turns the bottom into the real number c squared plus d squared, giving a quotient with real part (ac + bd) / (c^2 + d^2) and imaginary part (bc - ad) / (c^2 + d^2). Enter the real and imaginary parts of both numbers and this calculator returns the quotient in rectangular form, plus its modulus and argument. The denominator must be nonzero.
Complex division formula
(a + b*i) / (c + d*i) = ((ac + bd) + (bc - ad) * i) / (c^2 + d^2)
Real = (ac + bd) / (c^2 + d^2)
Imaginary = (bc - ad) / (c^2 + d^2)
Modulus = sqrt(Real^2 + Imaginary^2)
Argument = atan2(Imaginary, Real)
Multiplying top and bottom by the conjugate c - di makes the denominator the real value c squared plus d squared, so the quotient is found cleanly.
About complex division
- Division divides the moduli and subtracts the arguments of the two numbers.
- The conjugate trick is the same idea as rationalising a denominator with a surd.
- Dividing by i is the same as multiplying by minus i, a 90 degree clockwise rotation.
- The reciprocal of a complex number is its conjugate over its squared modulus.
- Complex division appears in impedance and transfer function analysis in engineering.
Complex division: frequently asked questions
How do you divide complex numbers?
Multiply numerator and denominator by the conjugate of the denominator. For (a + bi) / (c + di), the result is ((ac + bd) + (bc - ad)i) divided by (c squared + d squared), which clears the imaginary part from the denominator.
Why multiply by the conjugate?
Multiplying the denominator c + di by its conjugate c - di gives the real number c squared + d squared. This turns a division by a complex number into a division by a real number, which is straightforward.
What are the real and imaginary parts of the quotient?
Real part is (ac + bd) / (c^2 + d^2) and imaginary part is (bc - ad) / (c^2 + d^2). This calculator reports both, along with the modulus and argument of the quotient.
When is complex division undefined?
Division is undefined when the denominator is zero, that is when both c and d are zero. The calculator shows n/a in that case, just as dividing a real number by zero is undefined.
What happens to modulus and argument when dividing?
Division divides the moduli and subtracts the arguments: |z1 / z2| = |z1| / |z2| and arg(z1 / z2) = arg(z1) - arg(z2). This mirrors how multiplication adds arguments.
Official sources
- NIST Digital Library of Mathematical Functions: Complex Variables.
- NIST Digital Library of Mathematical Functions: DLMF home.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.