The idea
A complex number can be written as a real part plus an imaginary part. The imaginary unit i satisfies i squared equals negative one. Ordinary distributive multiplication still applies, but products containing i squared must be simplified before collecting terms.
Before you begin: Expand two brackets and collect like terms, including negative coefficients.
Multiply and express the result in Cartesian form.
| × | 1 | −2i |
|---|---|---|
| 2 | 2 | −4i |
| 3i | 3i | 6 |
Multiply each term in the first bracket by each term in the second.
Replace i squared by negative one. The final term becomes positive six.
Collect the real terms and imaginary terms separately.
A mistake to avoid
The square of i is −1, not 1. Also, the imaginary part of 8 − i is the real coefficient −1; its imaginary term is −i.
Your turn.
Multiply a complex number by its conjugate.
Show the solution
The imaginary terms cancel. Multiplying by the conjugate gives the square of the modulus, a nonnegative real number.