Root Routes

Historically, the search for the square root of minus one,, gave rise to the complex numbers. Typically we refer to as . Perhaps the next obvious question is: what is ? Do we need to invent another number, or can be found in the complex plane?

The figure shows the complex number plane. The circle is the unit circle with -1 and labeled. As you drag the blue point, the red point shows you its squared value, that is, red=blue². So, of course, putting the blue point on puts the red point on -1. Where do you put the blue point to put the red point on ? Can you find a second solution? Don't forget every nonzero number has two square roots!