Profound ?

About: Euler's Identity

e ^(?i) +1 = 0
it is not really all that amazing to find simplicity as a consequence of simplifying
Naaaa ... it's just in the definition of e

Tags

  1. eulers identity
  2. mathematics
  3. formula
  4. seth cartoons
  5. pi
  6. profound

Comments


Seth says
M 2006-05-09 06:12:25 3516
M 2006-05-09 06:10:29 3516
I used to use that as my screen saver !
BTW, it could be the definition of i , pi, 1 or 0
i don't think so ... all of those have "natural definitions" ... the odd ball is e ... and has a wierd definition.   You cannot define 1, 0, or π to be any different without turning math on it's ear.  But you can say that the value of e is a consequence of legislating this formula.  If we legislated another formula, ... say e ^ (iπ) + 3 = 0 ... then the value of e would just be different, and the natural log tables would be different. 

Mark de LA says
Well, just because we both like to argue, consider this:
Napier lived 1550-1617 inventor of logarithms (probably)
Euler lived 1707-1783 did a lot of other stuff including the identity equation of Euler (?maybe)
The base of natural logarithms is e. See the Wikipedia stuff on natural logarithms. (see the section on "Reason for being Natural" - it wasn't arbitrary!)
                   I would say that Napier defined "e" and natural logarithms before Euler came along!
Another nice feature of e is that the plot of the function e^^x crosses the ordinate at 1 .
So I would say that the definition came with logarithms. Euler's equation either defines each of them or none of them. It shows an exquisite relationship amongst these well know numbers.


Seth says
M 2006-05-09 06:27:26 3516
It's an equation, man, can define any term just by rearranging . 
M, i think i knew that.  you need to look deeper.

Seth says
Let me be more specific.  The definition of 0, 1, and ? are not in question.  They are a fixed part of a huge body of mathmatical know how.  The base of natural logorithms was chosen arbritrarily as the most convenient value that it could possibly be.  Now Euler's Identity can be looked at as the consequence of that choice, or it can be looked at as the choice itself.   ... hmmm what is the ming for QED ?


Seth says
M 2006-05-09 07:29:55 3516
Natural" - it wasn't arbitrary!)
                   I would say that Napier defined "e" and natural logarithms before Euler came along!
You are right, of course.  I am playing fast and loose with the definition of arbitrary.  Point is that something was chosen here.  That choice was kind of a legislation.  It could have been chosen differently. But Napier chose to make his formulae simple but choosing this to be the value of the base.  The reasoning could have proceeded from Euler's Identity ... that could have been the starting point, rather than the consequence.  

This formula is so very simple because Napier chose to simplify ... and not quite so paradoxical as quoted in the Wikipedia:

source: Kasner and Newman p.103
After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."
I too was amazed by this formula.  But if you think about how it came to be, it is not really all that amazing to find simplicity as a consequence of simplifying.


Mark de LA says
As far as answering the question - profound ?



Seth says
M 2006-05-25 12:20:56 3516
As far as answering the question - profound ?
naaah.   The clerk can call out "suare root of minus one" as an ordinal number just as easily as she can call out "37".  Now if the machine was issueing a currency with an cardinal denomination, ... it would not only be profound but also fraudulant.




Mark de LA says
Well, then let me buy some goods from the speaktome catalog for an imaginary number of dollars

Seth says
seth 2007-04-10 22:59:55 3516

When you first learned about pi, didn't you have to kind of squinch up your mind to accept that it was based on the diameter rather than the radius ? ... i did.

Mark de LA says
seth 2007-04-10 23:18:36 3516
seth 2007-04-10 22:59:55 3516

When you first learned about pi, didn't you have to kind of squinch up your mind to accept that it was based on the diameter rather than the radius ? ... i did.
Well this is cute.  I would have to study the derivation to see how one switches to a minus if the "pi" changes to refer to a radius.  OTOH, I think that the traditional pi makes sense when you look at a figure like the one from below:
Because of the curvature I can see that the circumference is at least twice the diameter & probably some more - 3.1415926535897932384626433832795 ought to do it for now.


Seth says
M 2007-04-11 13:05:24 3516
 I think that the traditional pi makes sense when you look at a figure like the one from below:
Do you mean that the ratio ~ 1 to 3 is almost visible in the diagram?   For me that was not the criteria ... me, i knew that the center point and radius was what determines the circle and that the radium appears in many more formula than the diameter.  So i was dissapointed when i learned that pi was calculated based on the diameter.

Seth says



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