Root Routes

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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?

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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, . 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!

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Contributed by: John Kiehl (March 2011)
Open content licensed under CC BY-NC-SA

Details

Snapshot 1: The blue point at is the square root of the red point at -1.

Snapshot 2: Here is the second square root of -1.

It is remarkable that only is needed to allow you to take any root of any complex number to get a complex number. Even more: over the complexes, every polynomial equation has a solution.

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