Friday, March 31, 2006

More on Euler's Identity and Roots of Unity

When Leonhard Euler came up with his Formula and his Identity, he stood on the shoulders of many giants. In the next few blogs, I plan to focus a bit on some of the giants that Euler stood upon including: Euclid, Archimedes, Hipparchus, Ptolemy, Napier and Bernoulli.

Thinking about Euler's identity in the context of Fermat's Last Theorem raises some questions which I think need to be answered:
  • Pi, Sine, and Cosine are based on Euclid's plane geometry. What validity can it have for number theory which is independent of Euclidean or non-Euclidean geometry?
  • How is it possible for a number to be put to the power of an imaginary number? How can this construction possibly have any meaning?
It turns out that trigonometric functions can be defined independently of Euclid (see here). It also turns out that it is possible to use the Maclaurin Series to define a exponents so that they can include any complex power including i (see here)

One of the goals of this blog is to provide a complete set of proofs for each of the propositions that I present or to state those propositions as postulates. Implicit in the use of sine, cosine, and pi is a set of assumptions that are often not thought about:
  • How can we be sure that all right triangles regardless of their size have the same ratio between their sides so that for a given angle, there is one and only one sine or cosine value? (See here for the answer)
  • How can we be sure that pi is really constant? It may be obvious but where's the proof? How can we be sure that for all circles, the ratio of circumference-to-diameter is the same? (See here for the answer)
While these questions are very elementary, I think that it is still important to ask them.

In understanding Roots of Unity and Cyclotomic Integers, I think it is worthwhile to review the achievements of DeMoivre, Taylor, and Maclaurin.

I think it is important and valuable to understand all of these details before exploring Kummer's proof.

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