Simple consequences of multiplication

Let us take any two real numbers \(a\) and \(b\) and force them into complex representation:

\[a = a + i(0) \text{ and}\hspace{0.5em} b = b + i(0).\]

Following the definition of complex multiplication, we have:

\[ab = (a + i0)(b + i0) = [ab - (0)(0)] + i[a(0) + (0)b] = ab + i(0) = ab.\]

Thus, this boils down to the usual multiplication of real numbers.

In other words, when complex addition and multiplication are restricted to the real numbers, then they will simply be the usual addition and multiplication of real numbers.