The Hunh? Function

I   
was playing around with Euler's famous equation while reading the book by Eli Maor

e : The Story of a Number

I had been looking for a book about "e" (2.718281828...) for some years that would compliment Peter Beckman's Pi book. And there it was in Barnes & Noble.

Comments about the below conundrum are appreciated. Did I do something really stupid?



What goes?


Of course, this "thing" is just a less transparent form of
x = -1 . . . x-1 = -1-1 . . . 1/x = -1 = x
I include it here for demonstration. And what I wish to demonstrate is the difference between symbolic mathematics and quantitive mathematics. By symbolic I mean slinging variables around according to the rules of equations using letters and functions as opposed to evaluating the equation (arriving at an actual number ... a quantity).

In Euler's famous equation, we have a quantity e^pi = 23.1406... raised to the i power. Yielding what appears to be a quantity (-1). I dispute this and think of -1 as an entity having dual citizenship. It is a number (we can acquire this quantity by adding or multiplying other numbers in infinite variations, i.e. -.526 + -.474 = -1) ... and ... it is a qualitive symbol in a small family (1, -1, i, 0, infinity). These are the numbers one must watch out for when calculating quantities because of the logical trouble they may cause if handled incorrectly.

Now, examining Euler's equation, (e(pi)i = -1), we can see that e^pi = 23.1406, (which is most definitely a quantitive number), has been transformed into a symbol. It has been stripped of its numeric clothing and equated to the symbol -1.

One might complain that here -1 appears as a quantity but I disagree.

If we replace 23.1406... with, say, 23.1407... will -1 change to ... hmmmm .... .999998...?

I suspect rather that e^pi, might be replaced with any quantity and the equality would be preserved, i.e. (some other x) i = -1 . If this is so, Euler's equation is rendered trivial or even false.

Some equations involving i give an infinite number of values with one of them being the "principal" value.
Where then might a mistake have occurred? Euler obtained it from ... exi=cosx - i sinx ... by replacing x with pi ... then ... sinpi = 0 and thus ... isinpi =0 ... (being that anything multiplied by "0" equals "0" ... right?).

But, by what reason does one accept that ... i times 0 = 0 ? This is a symbolic argument not a numeric one. There is no precedent for it other than previous rules. These rules are simply extended to i without any further justification.

We cannot be sure that they hold here since we are dealing with a completely abnormal animal ... the square root of -1 ?!

I'm not impressed or mystified by this equation because it looks like an unidentified error. I have the feeling that the entire subjuct of complex numbers may be similar to the Planck scales of physics ... they may have no intrinsic logical meaning at all (no matter how useful they may be ... or seem to be).

All those who look for new truths (including myself) implicitly expect everything that logic leads them to ... is yet another truth.

I'm looking over my shoulder for another stray dog on this one ... for another ass chewing.



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