A worked example: (3+4i) × (1+2i)
Multiplying (3+4i) by (1+2i) gives −5 + 10i — expanding the product term by term and simplifying using i² = −1.
The formula: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Expanding (a+bi)(c+di) gives ac + adi + bci + bdi² — and since i² = −1, the bdi² term becomes −bd, a real number. Grouping the real terms (ac − bd) and imaginary terms (ad + bc)i gives the final result: (3×1 − 4×2) + (3×2 + 4×1)i = −5 + 10i.
Division: (3+4i) ÷ (1+2i)
Dividing the same two numbers gives 2.2 − 0.4i — found by multiplying both the numerator and denominator by the denominator's complex conjugate (1−2i), which eliminates the imaginary part from the denominator entirely.
Why the conjugate trick works
Multiplying (c+di) by its conjugate (c−di) gives c² + d² — a plain real number, since the imaginary cross-terms cancel out. This is exactly why division needs this extra step: turning a complex denominator into a real one makes the whole expression divide cleanly.
What comes after multiplication or division
A multiplication or division result, expressed as real and imaginary parts, can be converted into polar form (magnitude and angle) — a different, often more useful way to describe the same complex number, particularly for repeated multiplication or rotation problems.