How to Add, Subtract, Multiply and Divide Complex Numbers

Adds, subtracts, multiplies or divides the complex numbers a + bi and c + di. Multiplication uses i² = −1 to tidy the result, and division multiplies top and bottom by the conjugate c − di so that the denominator becomes a real number.

This adds, subtracts, multiplies or divides the complex numbers $a + bi$ and $c + di$ , where $i$ squares to $-1$ .

Addition and subtraction simply combine the real parts and the imaginary parts. Multiplication expands the brackets and tidies up with $i^2 = -1$ .

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Division multiplies both top and bottom by the conjugate $c - di$ . The denominator becomes the real number $c^2 + d^2$ , and what is left is division by a real number.

\dfrac{a + bi}{c + di} = \dfrac{(ac + bd) + (bc - ad)i}{c^2 + d^2}

Example

Take $(3 + 4i)(1 - 2i)$ . The real part is $3 \times 1 - 4 \times (-2) = 11$ and the imaginary part is $3 \times (-2) + 4 \times 1 = -2$ , so the answer is $11 - 2i$ . Expanding by hand agrees: $3 - 6i + 4i - 8i^2 = 3 - 2i + 8 = 11 - 2i$ . The absolute value is $\sqrt{121 + 4} = \sqrt{125}$ , about 11.18.

Why the conjugate works

$(c + di)(c - di) = c^2 - (di)^2 = c^2 + d^2$ , and $i$ disappears. It drives $i$ out of the denominator, in the same way that rationalising clears a $\sqrt{2}$ from the bottom of a fraction.

Notes

With $c$ and $d$ both 0 this divides by zero, so nothing is calculated.