Complex Number Division Calculator

Work out complex number division instantly with clear inputs, formula shown and shareable results.

About Complex Number Division Calculator

Dividing multiplies numerator and denominator by the conjugate of the denominator, which makes the denominator the real number c² + d². The moduli divide and the arguments subtract.

Formula

Complex division

(a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)

Worked example

With the default values below, Complex Number Division Calculator returns Quotient of -1 + 2i. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numerator real part: 5
Numerator imaginary part: 10
Denominator real part: 3
Denominator imaginary part: -4

Results

Quotient: -1 + 2i
Modulus of the quotient: 2.23606798
Argument of the quotient: 116.565051 °
c² + d²: 25.000000

Step by step

1Values used
2Complex division
3Quotient= -1 + 2i
4Modulus of the quotient= 2.23606798
5Argument of the quotient= 116.565051 °
6c² + d²= 25.000000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Divide 5 + 10i by 3 − 4i.

The denominator's modulus squared is 25, giving (15 − 40)/25 + (30 + 20)/25·i = −1 + 2i.

Why multiply by the conjugate?

Because (c + di)(c − di) = c² + d² is real, which clears the imaginary part from the denominator.