Remarks regarding complex multiplication

1. When \(r_2 = 1\) and \(\theta_2 = \alpha\), we see that

\[z_1 z_2 = r_1 [\cos(\theta_1 + \alpha) + i \sin(\theta_1 + \alpha)].\]

This means that the resultant complex number \(z_1 z_2\) has the same modulus (length) as \(z_1\) but its argument has been added by \(\alpha\). This means that \(z_1 z_2\) represents the vector obtained by rotating the vector \(\overline{OZ_1}\) through an angle of \(\alpha\) radians about the origin \(O\).

2. In particular, when \(\theta_2 = \alpha = \frac{\pi}{2}\), the corresponding complex number \(z_2 = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = i\). This means that multiplying the complex number \(z_1\) by \(i\) results in an counterclockwise rotation of \(90{}^\circ\) about the origin. One can deduce similarly that multiplying by \(-i\) results in a clockwise rotation of \(90{}^\circ\) about the origin.

3. One can now accurately interpret the geometric content of multiplying by \(-1\), i.e., it corresponds to a rotation of \(180{}^\circ\) about the origin (the sense of rotation does not matter now). For the first time, one attains the full perspective of why \((-1) \times (-1) = 1\) -- more precisely, two rotations of \(180{}^\circ\) performed one after another return the (complex or real) number to where it was.