Q:

About the definition of complex multiplication

Some people say that the complex product is the way it is to respect the distributive law of multiplication. However, the distributive law acts in the whole number, like:
$$(a+b)(c+d) = ac + ad + bc + bd$$
The multiplication for complex numbers would be something in the form:
$$(z_1+z_2)(z_3+z_4) = z_1z_3 + z_1z_4 + z_2z_3 + z_2z_4$$
Where $z_n$ is a complex number in the form $a+bi$
I don't see why the distributive law must act inside the number:
$$(a+bi)(c+di) = ac + adi + bci - bd^2$$
What's the real reason for this definition?

A:

Let $z_1 = a, z_2 = bi, z_3 = c, z_4 = di$.
Then the rule
\begin{equation*}
\tag{$\spadesuit$}(z_1 + z_2)(z_3 + z_4) = z_1 z_3 + z_1 z_4 + z_2 z_3 + z_2 z_4
\end{equation*}
tells us that
\begin{align*}
(a + bi)(c + di) &= ac + adi + bci + bd i^2 \\
 &= ac - bd + (ad + bc)i.
\end{align*}
So if we want ($\spadesuit$) to be true, we are forced to use the standard formula for multiplication of complex numbers.

