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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!

**Proposition 1**Prove is irrational.

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 »