e

[th!nkstyle]

alright, ive done many math courses in university yada yada bla bla.

However math never goes deep and philosophical. I need a good mix of not dumbed down to any degree, but does not hide behind its own language of math to explain everything.

I am looking for a real good definition of why the number e comes up everywhere. either via book or webpage.

I am not looking for textbook definitions of how one can get the number e, or how once you figured out what it is, how it can be used to explain all sorts of shapes we see.

I am looking for a mathy/philisophical discussion as to why it should pop-up everywhere - what is the root cause of it coming up everywhere?

Any good recommendations for places to check out for this quest?

[Alchemist]

E iz uh letter, not uh number. Duh, anyone one, even muh ma f'in cousin's Kindergarten kid knows dat. Get yourself o' da gene pool, do society some good, pimp-tight foo'. slap mah fro! E iz uh very common letter, not math.. Please jet back ta first grade with muh beeotch.

(Disclaimer: I'm not this stupid, also I'm incredibly white.. go figure.) (I'll look this up) (sorry) (sorry if your eyes were hurt over seeing this)

[th!nkstyle]

E iz uh letter, not uh number. Duh, anyone one, even muh ma f'in cousin's Kindergarten kid knows dat. Get yourself o' da gene pool, do society some good, pimp-tight foo'. slap mah fro! E iz uh very common letter, not math.. Please jet back ta first grade with muh beeotch.

(Disclaimer: I'm not this stupid, also I'm incredibly white.. go figure.) (I'll look this up) (sorry) (sorry if your eyes were hurt over seeing this)

Good point.

[Rhu]

I am looking for a mathy/philisophical discussion as to why it should pop-up everywhere - what is the root cause of it coming up everywhere?

Have you ever asked yourself how many ways you could divide a number? What would that answer mean?

Contained in e is the answer to that queston. Think of it that way. Think of it as how it contains the summation of every subdivision of a number--that information is extracted and used in various ways depending on what equation it is being used in.

So you could say if you could achieve a greater understanding about an object by breaking it into any number of equal pieces, then you are probably going to have an equation dealing with that object which involves e.

[th!nkstyle]

Have you ever asked yourself how many ways you could divide a number? What would that answer mean?

Contained in e is the answer to that queston. Think of it that way. Think of it as how it contains the summation of every subdivision of a number--that information is extracted and used in various ways depending on what equation it is being used in.

So you could say if you could achieve a greater understanding about an object by breaking it into any number of equal pieces, then you are probably going to have an equation dealing with that object which involves e.

its definitly something to ponder and explore, but it is still.... referencing back to itself you know what i mean?

its seems to me that many people understand it, when in fact they only know how to use it. i would be willing to even bet that the majority of any universtiy science profs wouldnt really be able to talk beyond definitions too much.

im seeking something deeper - something that explores that deepness in many pages. something that asks "Have you ever asked yourself how many ways you could divide a number? What would that answer mean?", but not just in a rhetoric fasion. ive never found anything like this, and am always seeking...

[CosmicDust]

I would guess convenience. Sure, it's an unwieldly little number in itself, but it makes calculus cleaner. They could have used, say, 1-10^x or whatever in the various Maxwell equations and the like, or in the definitions of imaginary sines and cosines, but when you do the calculus you would get all these unwieldly numbers when you take derivatives of exponentiated expressions. So, to avoid accumulating and keeping track of all these unwieldly numbers when you do your calculus, you use e.

[th!nkstyle]

I would guess convenience. Sure, it's an unwieldly little number in itself, but it makes calculus cleaner. They could have used, say, 1-10^x or whatever in the various Maxwell equations and the like, or in the definitions of imaginary sines and cosines, but when you do the calculus you would get all these unwieldly numbers when you take derivatives of exponentiated expressions. So, to avoid accumulating and keeping track of all these unwieldly numbers when you do your calculus, you use e.

Your explaination makes a lot of sense. However, I'm not really seeking what it IS. I've learned its very convienent to have a system where you can have a variable that is the derivative of itself. I've taken many calc. classes, that it has been engrained.

I am not looking for "simple" answers, I am looking for an essay or something that really inspects this issue.

Its funny, I've brought up this topic on other (more science) based forums, and I get the same type of replies. I mean, it's hard to believe that I the only one who thinks that this is really trippy stuff. If anyone really understood this inside out, they could probably understand any concept the universe has to offer. I'm basically admitting that I am tripped out by this number (amoung others); I have taken many classes, and have a degree. But any "authority" that speaks of such matters, speaks about it as if its the easiest thing around, I think just because of that - they are looked at as the authority, so they should know exactly what they are talking about with no doubts whatsoever.

I'm looking for a source that admits that this this weird stuff and does not pretend to know everything about it - and goes on to explore quesitons that are usually rhetorical.

Perhaps this is a futile effort, maybe I'll eventually just have to slowly create what I seek...

[Architectonic]

Its funny, I've brought up this topic on other (more science) based forums, and I get the same type of replies. I mean, it's hard to believe that I the only one who thinks that this is really trippy stuff.

Many people wonder about it, but evidently no one has actually figured it out.....

I assume you know all this stuff:
http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

[Conan]

read the book e:the story of a number

http://www.amazon.com/gp/product/0691058547/qid=1135657728/sr=2-2/ref=pd_bbs_b_2_2/104-0633788-0932725?s=books&v=glance&n=283155

[Xenophon]

It's because physical processes are governed by differential equations, and e happens to be the constant in the solution of a differential equation.

[Conan]

It's because physical processes are governed by differential equations, and e happens to be the constant in the solution of a differential equation.
dy/dx(e^x)=e^x

[Xenophon]

dy/dx(e^x)=e^x

just a semantical point, that should just be d/dx. There is no y in that equation.

[Melody]

re: op

i think e pi etc are examples of entities which are the manifestations of the structures and their residues of the conceptual systems of mathematics

for example, both e and pi appear in constructs relating to curvature

an analogy i can make is that: if an engine is curvature, the noise it makes is pi. we cannot clearly see how the engine works, but we can see/hear/touch/lick it on the outside, those sensations corresponding to numbers or other mathematical entities

its like a spike in the spectrum of the system as in frequency analysis, but where the 'temporal' domain = the structure of the system, and the 'frequency' representation = the numeric superficialities we see

or something


alternatively: we want to precisely explain what curvature is, so we apply a mathematical system (in this case such being merely a linear numeric system) over it. because these systems are not congruent, there will be residual artifacts resulting from their meshing. e is the numeric manifestation of such an artifact


there be my heuristics


....reminds me of 'the matrix,' where the architect tells neo he is a 'remainder' aka residue
Your life is the sum of a remainder of an unbalanced equation inherent to the programming of the matrix. You are the eventuality of an anomaly, which despite my sincerest efforts I have been unable to eliminate from what is otherwise a harmony of mathematical precision. While it remains a burden assiduously avoided, it is not unexpected, and thus not beyond a measure of control. Which has led you, inexorably, here.how oddly relevant

[Hypnos]

* A cute survey (http://sprott.physics.wisc.edu/Pickover/trans.html) of transcendental numbers.

* I don't have any compelling ideas for the deeper structure of e, or pi, or any of them. You can say that d/dx e^x = e^x is a coincidence from its Taylor Expansion, so it is convenient as a base, and whence you also get Stirling's approximation for the factorial.

* The identity $e^(i \pi) = 1$ was offered as proof of the existence of God in the court of Catherine the Great. Precisely because it defies the kind of easy explanation I gave above.

[Serotonin]

* A cute survey (http://sprott.physics.wisc.edu/Pickover/trans.html) of transcendental numbers.




ii = 0.207879576... (Here i is the imaginary number sqrt(-1). Isn't this a real beauty? How many people have actually considered rasing i to the i power? If a is algebraic and b is algebraic but irrational then ab is transcendental. Since i is algebraic but irrational, the theorem applies. Note also: ii is equal to e(- pi / 2 ) and several other values. Consider ii = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for ii.
Here is how you can compute the value of ii = 0.207879576...
1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2.
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
3. Therefore e^(iPi/2) = i.
4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2).
5. Therefore i^i = e^(-Pi/2) = .207879576...

That is wicked cool.

Once I impressed my maths teacher by differentiating y=x^x.

Actually (this could keep me occupied for hours) what does the graph of y=x^x look like*?


*Edit: For x<1

[lexiphanic]

I remember 'e' being something describable as the slope (or somesuch) of any held piece of rope or string as it curves from one end to the other.

[Edmond Zedo]

I memorized the first 50 decimal places of e in an hour when I was in high school. For extra credit. Didn't do the assignment that day.

I don't remember what it represents. :)

[Xenophon]

I remember 'e' being something describable as the slope (or somesuch) of any held piece of rope or string as it curves from one end to the other.

That's not surprising, as a rope or string can only exhibit force along the direction of tension.

ie dy/dx = K*F_y/F_x

Since F_y and F_x are functions of, among a great many things the position what part of the string we are looking at. this means that

dy/dx = f(x,y)

Therefore we have a differential equation and there will be an e in the solution.

The fact of the matter is, the rate of change of the variables in any system are almost always based on the state of the system in some way. And as soon as that happens, you have a differential equation and e is bound to show up.

[Serotonin]

That is wicked cool.

Once I impressed my maths teacher by differentiating y=x^x.

Actually (this could keep me occupied for hours) what does the graph of y=x^x look like*?
[/indent]
*Edit: For x<1

Don't mind me, I've answered my own question.

http://upload.wikimedia.org/wikipedia/en/thumb/6/68/Tetration_large.png/800px-Tetration_large.png

Obviously, tetration for x<0 only exists in the complex plane.

http://upload.wikimedia.org/wikipedia/en/f/f5/Tetration_period.gif

[Hypnos]

Once I impressed my maths teacher by differentiating y=x^x.

Actually (this could keep me occupied for hours) what does the graph of y=x^x look like*?
[/INDENT]

*Edit: For x<1
x >> 1 is interesting. x^x = exp(x ln x) ~ exp(x!)

For x < 1, you have x^x=1 at x=1 (obvious) and at x=0 (L'Hopital's rule). Then you can easily show that there is precisely one maximum at 1/e.

5 minutes of work, tops.

[Serotonin]

x >> 1 is interesting. x^x = exp(x ln x) ~ exp(x!)

For x < 1, you have x^x=1 at x=1 (obvious) and at x=0 (L'Hopital's rule). Then you can easily show that there is precisely one maximum at 1/e.

5 minutes of work, tops.

Minimum, you mean. 1/e, of course, now it's all coming back.:banana:

[Hypnos]

Minimum, you mean. 1/e, of course, now it's all coming back.:banana:
yes -- I was thinking concave up.

[Melody]

How many people have actually considered rasing i to the i power?oooh me me!!


ja, i just realized the algebraic definition of transcendental numbers [those which are not roots of any polynomial over the rationals,] 'along with' their obtuse nature, fits my theoremizationings