Multiplying complex numbers

The product is defined by (a + ib)(c + id) = (ac − bd) + i(ad + bc). It is what you get by multiplying out the brackets as in ordinary algebra and then replacing i² by −1.

Worked example: (2 + 3i)(4 − i) = 8 − 2i + 12i − 3i² = 8 + 3 + 10i = 11 + 10i. The formula agrees: ac − bd = 8 − (3)(−1) = 11 and ad + bc = −2 + 12 = 10.

Multiplication obeys closure, the commutative law z₁z₂ = z₂z₁, the associative law (z₁z₂)z₃ = z₁(z₂z₃), and the distributive laws z₁(z₂ + z₃) = z₁z₂ + z₁z₃ and (z₁ + z₂)z₃ = z₁z₃ + z₂z₃.

The multiplicative identity is 1 = 1 + i0, since z × 1 = z for every z.

A number times its partner with the opposite imaginary sign is real and never negative: (a + ib)(a − ib) = a² + b². For example (3 + 4i)(3 − 4i) = 9 + 16 = 25, because the two cross terms 12i and −12i cancel.

Multiplying by i swaps the parts and changes one sign: i(a + ib) = −b + ia. So i(3 + 4i) = −4 + 3i. On the Argand plane this turns the point a quarter turn anticlockwise about the origin, keeping its distance from O.

Multiplying by a real number k scales both parts: k(a + ib) = ka + ikb, so 3(2 − 5i) = 6 − 15i.