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> In what way is e^i*pi = -1 a metaphor for anything? Its a metaphor for taking the unit length vector [1,0] represented by the complex number 1+0i and rotatin
by pvarangot 14y ago
> In what way is e^i*pi = -1 a metaphor for anything?
Its a metaphor for taking the unit length vector [1,0] represented by the complex number 1+0i and rotating it 180 degrees to -1+0i...
> Mathematics is funny because it is presented in
> "reverse", i.e. not the way it was derived.
Its usually presented in both ways in most curricula, sometimes depending on where you read about it or who teaches/tells you about it. Most mathematical books include historical contexts and non-formal accounts of the way results were derived, specially for classic and old results such as Euler's Formula. In most modern topics sometimes the historical context for a theorem is not easy to understand (i.e. discrete signal processing or optimal control) and is only briefly mentioned.
- chubot 14y agoI appreciate what you're saying, but see my response below about semantics. If you are calling it a metaphor, then aren't you calling ALL equations metaphors? That is doing violence to the meaning of the word "metaphor". There is for sure a "relation" (or association) between the symbols e^i*pi = -1 and the picture of a unit vector on a complex plane. But that relation is not a metaphor.
- Evbn 14y agoYou can prove that equality using pure analysis, or pure geometry, using appropriate definitions. The metaphor is the intuition that the two proofs are equivalent in an abstract sense.