I wrote this note in 2008 to introduce complex numbers. It is posted for the Math Teachers at Play blog carnival.
What is the ?
In the set of Real Numbers, , the solution is undefined. We must consider the set of Complex Numbers, . Complex numbers are numbers of the form of , where and are real numbers: and is the imaginary unit. Note is defined by or equivalently .
Thus, we have:
We now have . So we must ask "What is the ?". To evaluate we will use Euler’s formula, So by substituting , giving
Note we have isolated and now arrived at the following equality: . Taking the square root of both sides gives
Thus, we have now have and following through application of Euler’s formula to , gives
So going back to the original problem and substituting , gives
Hence, the solution, , is a complex number!