This is just a bit of a curiosity of mine, what's up with e? I just got done with Calc II, and I still haven't heard a satisfying explanation of its origins or what it really means. When I first learned about e, I think it was just in the context of interest problems, without much of an explanation as to why it was used in interest problems. Later on, in high school, I learned that it was the limit of (1+(1/x))^x as x approaches positive infinity, which was all fine and good, but didn't really tell me what e was all about.
With the other two more abstract numbers I can think of, like i and pi, I can understand where they've been needed. Pi seemed to be the simplest of them all, with the circumference formula making pi immediately relevant. It also seemed to be fairly logical that a mathematician at some point in time would want to know what the relationship between the radius and the circumference of a circle was.
i was a little stranger (that would sound a bit strange out of context), but still seemed to make sense. It was presented as a solution to the problem of trying to have a square root of a negative number. The finding a solution of sorts to the problem of even roots to negative numbers seemed like a logical step to take.
Unlike pi and i, e has never really seemed to be as logical. Now that I've had Calc I and II, e does seem more useful that it previously did, with e^x being its own derivative, but I still don't get how e came to be, or what motivated is creation.
---
I thought I saw upon the stair a little man who wasn't there.
He wasn't there again today. Oh how I wish he'd go away.
To a degree, what you're looking for, a dead-simple motivation for the use of e, doesn't exist. The closest you can get is the fact that constant multiples of e^x are the only real functions that are their own derivatives. (Before you object "but what about the constant 0 function?", notice that the constant 0 function is indeed a constant multiple of e^x.) By contrast, consider the sequence of derivatives of a^x for arbitrary positive a. That is, let a > 0, and for each n, define c_n such that
c_n * a^x = (d^n * a^x) / (d * x^n)
for all real x. If a = e, c_n = 1 for all n. If a < e, (c_n) converges to 0. If a > e, (c_n) diverges to positive infinity. In this sense, e is the most natural choice for the base of an exponential function.
(GameFAQs sure could use LaTeX support, huh?)
---
The Albino Formerly Known as Mimir
Another property worth mentioning: for any real a and b,
a + bi = r * e^(it)
where (r, t) are the polar coordinates of the complex number a + bi.
---
The Albino Formerly Known as Mimir
(LaTeX support would be nice.)
So yeah, e isn't inherently poignant on its own merits like pi and i are, it becomes so important because it keeps coming up. For example, check all these: http://en.wikipedia.org/wiki/Representations_of_e
Plus things like the derivative thing, or things like e^(pi*i) +1 = 0, giving us those five numbers that can create all others.
---
"To truly live, one must first be born." ~ Evan [aX]
Paper Mario Social: The Safe Haven of GameFAQs. (Board 2000083)
It's also got a sweet power series expansion. But you should already know that.
---
http://img.photobucket.com/albums/v505/UtarEmpire/youaretearingmeapartlisa.jpg
YOU ARE TEARING ME APART LISA
How is e^(pi*i) = -1 ?
---
"A period"
~~This is what I like to add to the end of almost every sentence.~~
Euler's formula
e^(ix) = cos x + i sin x
which can be proved by power series expansions.
---
http://img.photobucket.com/albums/v505/UtarEmpire/youaretearingmeapartlisa.jpg
YOU ARE TEARING ME APART LISA
Basically if I remember correctly the use of e^x ios required in solving dozens of different complex integration equations which is required for many different scientific equations and it can be useful for population equations
---
Being kind may not pay; but it's the right thing to do.