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The matematical constant e (2.718...)

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Spartacus    122
Can someone give me a explanation of the matematical constant e=2.7182...??? What''s so magical about it and why is it exactly 2.7182...??? If you know of some resources where I can get more information then that would be great too! Thanks!

Real programmers don''t document, if it was hard to write it should be hard to understand

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Guest Anonymous Poster
http://mathworld.wolfram.com/e.html

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uber_n00b    122
::shifty eyes:: you''re a non-believer aren''t you? e shows up nearly everywhere. e and ln (log base e) determine the halflife of an atom, how long it will take an object to cool to a certain temperature, etc (anything where the amount at a time is based on the rate of change of the amount). That is not nearly the limit of where it shows up but I am in a hurry so just go to the link given already. Wikipedia.com also has some good info on it but the site was down when I tried to get onto it this time.

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Sneftel    1788
pi > e

I''m sorry, it had to be done.

"Sneftel is correct, if rather vulgar." --Flarelocke

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PI >> e

I coudn''t resist either, even though it has nothing to do with OP.

You have to remember that you''re unique, just like everybody else.

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Spintwo    358
definiton of e?

e=(1+(1/x))^x where x starts at one and goes forever. use like a large nubmer for x. say...20. You''ll get an answer
the larger x is, more accurate e is.

Charles Hwang -aka oatmeal.net
[Maxedge My Site(UC)|E-mail|NeXe|NeHe|SDL]

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Zipster    2359
I guess he wants to know exactly why e appears everywhere, rather than what it is. Precisely how he said it, why it''s so "magical"

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uber_n00b    122
You know skybin.. e has many other definitions. e can also be defined as the other boundary besides 1 under the graph of ln(x) such that the area under that curve is equal to 1. The second would be the sum of x/x! from 1 to infinity

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Also, the derivative of e^x is..... e^x !!

that sounds pretty cool to me!

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TangentZ    450
My personal favorite "magical" mathematical formula of all time:

e ^ (pi * i) = -1

That''s the "i" as in "imaginary number".

Kami no Itte ga ore ni zettai naru!

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Doc    586
It''s better written as e+1=0, that way you include 0.

Neato formula that includes:
e Euler''s number
i The imaginary number
π Ratio of circumference to diameter of circle
1 The multiplicative identity

The following statement is true. The previous statement is false.
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uber_n00b    122
How is e^(i*pi) magical, although I do agree it's kind of an interesting contradiction to x^y always being > 0. It's very easily derived from the power series expansion of e^(i*x) (which euler showed was equivalent to cos(x) + i*sin(x) or cis(x))

[edited by - uber_n00b on June 7, 2004 1:34:18 AM]

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Guest Anonymous Poster
if you''ve got a population of N radioactive atoms and a fraction l*N of them decays each time unit (the number of particles emitted by decay is the thing measured with long-lived isotopes, not the decrease of the number of atoms), you get a differential equation for the number of atoms at a given time t:

dN/dt = -l*N

which is solved by N = N0 * e^(-l*t).
you see, the thing that makes it important is that the derivative of e^x is again e^x. you can show this by forming the derivative of the taylor series:

e^x = x^0/0! + x^1/1! + x^2/2! + x^3/3! + ....
d/dx e^x =&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;x^0/0! + x^1/1! + x^2/2! + ...

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TangentZ    450
quote:
Original post by uber_n00b
How is e^(i*pi) magical, although I do agree it''s kind of an interesting contradiction to x^y always being > 0. It''s very easily derived from the power series expansion of e^(i*x) (which euler showed was equivalent to cos(x) + i*sin(x) or cis(x))

Errrmmmm, you say it''s "easy", only because Euler did the
hard work for us. That in no way diminishes the "magic" of
the formula.

Imagine showing this to someone living in the Pythagoras-era.

(Huh? What are imaginary numbers?)

Or, imagine showing the proof of Fermat''s Last Theorem to Euler,
who worked hard on it until his death.

(Huh? What are elliptic curves?)

Kami no Itte ga ore ni zettai naru!

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Quaternion interpolation.

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joanusdmentia    1060
quote:

Also, the derivative of e^x is..... e^x !!

And I think this is why it pops up everywhere. Said in english the rate of change depends on the rate of change, which turns out to be a natural part of modelling many things. For example the equation for the way power lines hang between two poles has e in it.

quote:
Original post by uber_n00b
...to x^y always being > 0

err....since when?
(-1)^3 = -1

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uber_n00b    122
To clear a few things up, I meant positive x to some power being less than 0. Also, TangentZ, I am not trying to brag but a few days after I learned power series in math class someone showed me e^i*pi was -1 on a calculator and, being in the power-series mood, was able to come upon the same conclusion as Euler (although I didn't know it at the time). I am not trying to say I'm as good in math as Euler by any stretch of the imagination because that simply wouldn't be true, but am rather trying to explain that that particular proof was NOT his most major contribuation (and by major I mean thought-intensive because e^(i*pi) is very important in engineering). His Zeta function (which was extended to complex numbers by Riemann) was a much greater accomplishment. Second of all, the proof for e^(i*pi) power is MUCH MUCH MUCH easier than the proof for Fermat's last theorem, which was over 150 pages long and presented by Andrew Wiles in the mid 1990s. You can fit the proof for e^(i*x) on one sheet of notebook paper.

EDIT: just so this topic doesn't get deleted because of the digressions, it would be hard to say why the number e is so important, because well 2.71828... just seems like a random number, but like everyone else has mentioned in this thread, it is easy to say why a number like e would be so important since it fairly accurately describes things in nature (including unrestrained population growth, harmonic motion, et al). For some reason, I still like pi more. Shrug.

[edited by - uber_n00b on June 7, 2004 5:48:20 PM]

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Just3D    122
dy/dx of e^x = e^x

Needless to say, the magic of e is also closely related to ln(x), which happens to be the indefinite integral of 1/x. And, of course, you can move ln(x) to the other side of an equation with e^x.

dy/dx = y

1/y dy = dx

S( 1/y dy ) = S( dx )

ln |y| = x

e^ (ln |y|) = e^x

y = e^x

My 2 cents.

(By the way that PI*i equation is cool)!

My apologies if I messed up any of the math...

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quote:
Original post by TangentZ
e ^ (pi * i) = -1

I really don''t see whats magical about that. It seems pretty intuitive to me...

You have to remember that you''re unique, just like everybody else.

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d000hg    1199
Is there any understanding at all of where Pi,e and similar transcendental numbers actually come from and why they take such values? On a purely science-based foundation I mean.

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BitMaster    8651
quote:
Original post by d000hg
Is there any understanding at all of where Pi,e and similar transcendental numbers actually come from and why they take such values? On a purely science-based foundation I mean.

Any decent first semester book on calculus will explain that.

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"Any decent first semester book on calculus will explain that."

Umm... does it explain why pi happens to be 3.14159... instead of just being 3.14?

I suppose previous poster just wondered if there is some universal law, that spits out these strange transcendental numbers to our amusement. At least that would be something I'd like to find out.

[edited by - linkki_ on June 8, 2004 8:02:44 AM]

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higherspeed    230
I would have thought that if pi satisfied a polynomial with integer coefficients that you could ''prove'' a contradictory result about a circle. I''ve done very little on those sort of topics yet, so I''m not really sure.

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Goldfish    128
quote:
Umm... does it explain why pi happens to be 3.14159... instead of just being 3.14?

I suppose previous poster just wondered if there is some universal law, that spits out these strange transcendental numbers to our amusement. At least that would be something I''d like to find out.

You have to remember there''s nothing special about the number itself - there''s no "higher meaning" behind the digits of e. It''s what that number represents that is important.