Dividing and the multiplicative inverse

Every non-zero z = a + ib has a multiplicative inverse z⁻¹ = 1/z = a/(a² + b²) − i b/(a² + b²), for which z × z⁻¹ = 1. The number 0 has no inverse.

Worked example: for z = 3 + 4i, a² + b² = 25, so z⁻¹ = 3/25 − (4/25)i. Check: (3 + 4i)(3 − 4i)/25 = 25/25 = 1.

Division by a non-zero z₂ is multiplication by its inverse: z₁/z₂ = z₁ × (1/z₂).

In practice, multiply the top and bottom by the bottom's partner with the opposite imaginary sign; the bottom becomes the real number c² + d², and the fraction splits into real and imaginary parts.

Worked example: (5 + i)/(2 − 3i) = (5 + i)(2 + 3i)/(4 + 9) = (10 + 15i + 2i + 3i²)/13 = (7 + 17i)/13 = 7/13 + (17/13)i. Check: (2 − 3i)(7 + 17i) = 14 + 34i − 21i + 51 = 65 + 13i, and dividing by 13 gives 5 + i.

Since i × (−i) = −i² = 1, the inverse of i is −i: 1/i = −i. For example 3/i = −3i.