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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!
Here is a proof using a traditional method (See Euclid’s Elements Book X which incorporates Theatetus work on incommensurable numbers. It includes a proof that is irrational (Proposition 22), and ends with a proof that there are infinitely many distinct irrational numbers (Proposition 115): Read the rest of this entry »