Root Routes

show blue's angle
show red's angle

●

Historically, the search for the square root of minus one,

-1

, gave rise to the complex numbers. Typically we refer to

-1

as

i

. Perhaps the next obvious question is: what is

i

? Do we need to invent another number, or can

i

be found in the complex plane?

The figure shows the complex number plane. The circle is the unit circle with -1 and

i

labeled. As you drag the blue point, the red point shows you its squared value, that is,

red=

2

blue

. So, of course, putting the blue point on

i

puts the red point on -1. Where do you put the blue point to put the red point on

i

? Can you find a second solution? Don't forget every nonzero number has two square roots!