To elaborate slightly; (1 + i) * (1 - i) = 1 - i^2. The left side is somewhere on the complex plane on a circle of radius 2 centered at the origin. The right

To elaborate slightly; (1 + i) * (1 - i) = 1 - i^2. The left side is somewhere on the complex plane on a circle of radius 2 centered at the origin. The right side is somewhere on a circle of radius 1 centered at 1. The circles meet at one and only one place, namely 2. So 2 = 1 - i^2, or i^2 = -1.