Euler's Identity - "Euclid looked on beauty bare", indeed.

This is one of Analyst_for_Life's favorite mathematical truths, asserting an identity that relates the additive identity, zero; the multiplicative identity, one; the ratio of circumference to diameter in any circle, PI; the imaginary unit, i (aka "the square root of minus one"); and the base of the natural logarithms, e (aka "Euler's Number").

An article at this link:

Euler Identity

does a decent job of explaining the essential background to the identity, pointing the reader to an array of elementary and not-so-elementary topics that are tied together in this remarkable fact. Don't miss the links to extended discussions of the properties and significance of the numbers PI and e .

The fascination with the ability of mathematical investigations to pack massive quantities of beautiful relationships into tiny expressions like this was part of what got Analyst_for_Life hooked on the subject early on, seducing him away from the further study of physics which had been his father's hope for him: a fully crendentialed physicist laboring away at some prominent center of research and graduate study.

But he found physics was far less coherent, indeed he became convinced that physicists had lost their way. Whenever he sought for a lucid, coherent, consistent account of the physical reality behind the confusingly contradictory experimental evidence which is the domain of "modern physics", the answers boiled down to "accept the mathematical description, which resolves the apparent contradiction, and don't try too hard to intepret it or you will just get confused". You might say he finally lost faith in the explanatory power of physical theory. But the mathematics itself was fascinating, beautiful, and apparently went as far as possible in "making sense" of the evidence. So that, for him, became "where it's at" (one of the irritating, ungrammatical youthful code expressions of that time).

In due time, he came up with a few neat new insights (actually just one little insight, combined with some big hairy insights borrowed from earlier students of the subject), packaged it up in a short paper entitled, "On the Transcendence of Certain Exponential Products", and persuaded his review committee to confer a Ph.D for the work just in time to go off to a small Christian liberal arts college and begin professing his favorite subject. The year was 1975: a very good year for Analyst, which also saw the arrival of our middle son. The roaring turbulence in the world around us faded into dim background noise in the face of these outpourings of gracious blessings from our Father, who we had only come to know a short time before.

Perhaps some time there will be a visitor who will laugh at how trivial Euler's identity is. That being is referred to an even more marvelous construct known as the "j-function". Analyst_for_Life will be pondering the content of this for "duration_of_franchise", or later if the issues of eternity allow. That more exalted being may digest the j-function and "move on" to bigger and better things. May it be so, provided that one is well connected to the Creator. http://mathworld.wolfram.com/j-Function.html

Note: I ended up at the above link while searching for some background on the "Monster Group", the foundational "simple group" of greatest cardinality discovered in the multi-decade search for a comprehensive classification of "simple" finite groups, which ended before the start of the 21st century. Apparently this search is still in a controversial stage. A good reference for the "latest and greatest" news on this project is:

http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups