E is the Greek numeral for 5.
In mathematics, e is approximately: (each clump is 20 digits) 2. 7182818284590452353 6028747135266249775 7247093699959574966 9676277240766303535 4759457138217852516 6427427466391932003 0599218174135966290 4357290033429526059 5630738132328627943 4907632338298807531 9525101901157383418 7930702154089149934 8841675092447614606 6808226480016847741 1853742345442437107 5390777449920695517 0276183860626133138 4583000752044933826 5602976067371132007 0932870912744374704 7230696977209310141 6928368190255151086 5746377211125238978 4425056953696770785 4499699679468644549 0598793163688923009 8793127736178215424 9992295763514822082 6989519366803318252 8869398496465105820 9392398294887933203 6250944311730123819 7068416140397019837 6793206832823764648 0429531180232878250 9819455815301756717 3613320698112509961 8188159304169035159 8888519345807273866 7385894228792284998 9208680582574927961 0484198444363463244 9684875602336248270 4197862320900216099 0235304369941849146 3140934317381436405 4625315209618369088 8707016768396424378 1405927145635490613 0310720851038375051 0115747704171898610 6873969655212671546 8895703503540212340 7849819334321068170 1210056278802351930 3322474501585390473 0419957777093503660 (source: Mathematica: N[E,1100])
/ 1 \ lim | 1 + - | ^ n n->oo \ n /
oo --- \ 1 | --- / n! --- n=0
1 + 1 _______ 0 + 1 _______ 1 + 1 _______ 1 + 1 _______ 2 + 1 _______ 1 + 1 _______ 1 + 1 _______ 4 + 1 _______ 1 + 1 _______ 1 + 1 _______ 6 + ...
___________________________________ | | | oo | | --- | | \ 1 | | | --- = lim ( 1 + 1/n )n | | / n! n->oo | | --- | | n=0 | |___________________________________|
n --- \ n! (a+b)n = | ----------- * an-k * bk / k!*(n-k)! --- k=0
tn (k = n) ___________________________________________________ | | | n! | | (1/n+1)n = lim ----------- * (1/n)n-n * (1)n | | n->oo n!*(n-n)! | |___________________________________________________|
_____________________________________________________________ | | | n! | | (1/n+1)n = lim ------------------- * (1/n)n-(n-1) * (1)n-1 | | n->oo (n-1)!*(n-(n-1))! | |_____________________________________________________________| n(n-1)! (1/n+1)n = lim ------------- * (1/n)1 * (1)n-1 n->oo (n-1)!*1! (1/n+1)n = lim n * (1/n) * 1 = 1 n->oo
______________________________________________________________ | | | n! | | (1/n+1)n = lim ------------------- * (1/n)n-(n-2) * (1)n-2 | | n->oo (n-2)!*(n-(n-2))! | |______________________________________________________________| n(n-1)(n-2)! (1/n+1)n = lim -------------- * (1/n)2 * (1)n-2 n->oo (n-2)!*2! n(n-1) 1 (1/n+1)n = lim -------- * --- * 1 n->oo 2! n2
1 n(n-1) (1/n+1)n = lim --- * lim -------- * 1 n->oo 2! n->oo n2
oo --- 1 1 1 1 \ 1 lim ( 1 + 1/n )n = --- + --- + --- + --- + ... = | --- n->oo 0! 1! 2! 3! / n! --- n=0
O OO ====== |||1|| ------ |23||| ------ |||||| ------ |||||| ------ EBEGBE # Notes: 1 : E 3 : G# 5 : B
Everything Guitar Project : The Everything Guide to Guitar Chords: E
High and funky, strictly an E/B chord:
-12- -9- -9- -9- -X- -X-
Definition: e is the unique number such that dex/dx = ex.
Proof of uniqueness and existence: Suppose that e1, e2 both have the desired property. Consider the function f(x) = e1xe2-x. f'(x) = 0, so e1/e2 = f(1) = f(0) = 1, and thus e1 = e2. So e is unique. The existence is ascertained by the propositions below. QED
How do we find the value of e? Well, we know that ex describes exponential growth where the growth rate per unit time is equal to the current value. We can approximate this exponential growth by dividing a unit time interval into n equal subintervals. In each subinterval the increase is approximately by a factor 1 + 1/n. Thus we would expect the expression (1 + 1/n)n to give a good approximation of e for large n. Hence we are led to
Proposition: e = limn→∞(1 + 1/n)n
Proof: First we need to show that the limit exists.
(1 - 1/(n+1)2)n+1 > 1 - 1/(n+1) ⇒ (1 + 1/(n+1))n+1 > (1 + 1/n)n
so the sequence is increasing. Using the binomial expansion we find that
(1 + 1/n)n < SUM(k = 0, n)(1/k!) < SUM(k = 0, ∞)(1/k!)
for all n, so the sequence is bounded above. Thus the limit exists, and we may call it d. Consider f : R -> R+, f(x) = dx. This is a continuous strictly increasing function, and therefore has a continuous inverse which we call ln. Since ln is the inverse we have
1 = ln d = ln (limn→∞(1 + 1/n)n) = limn→∞ n*ln(1 + 1/n))
Hence as h → 0, (ln(1+h))/h → 1 and (dh-1)/h → 1. Differentiating f from first principles gives
f'(x) = (dx+h - dx)/h = dx(dh-1)/h = dx
as h → 0. So d satisifes the condition that defines e, and hence e = d = limn→∞(1 + 1/n)n. QED.
The other approach to take when exploring exponentials is to use power series.
Proposition: e = SUM(k = 0, ∞)(1/k!)
Proof: Define exp: R -> R by exp x = SUM(k = 0, oo)(xk/k!). Termwise differentiation gives that exp' x = exp x. Consider f(x) = (exp x)(exp a+b-x). f'(x) = 0, so (exp a)(exp b) = f(a) = f(0) = exp a+b. We can consider this as a functional equation, and using that exp is continuous we find that exp x = (exp 1)x. Thus exp 1 satisfies the condition that defines e and e = exp 1 = SUM(k = 0, ∞)(1/k!). QED.
Mark's teenage years were troubled, leading to drugs and trouble with the law. When Mark was nineteen, his father died of a heart attack. A year later, Mark packed up everything and moved to L.A. where he would begin his music career.
Taking the name 'E,' he went on to release two albums, A Man Called (E) in 1992, and Broken Toy Shop in 1993. These albums sold poorly, and after a three year span of writing songs, E teamed up with drummer Jonathan Norton and bassist Tommy Walters to form Eels.
Eels released their first album, Beautiful Freak in 1996, which found success and airplay for the single Novocaine for the Soul.
At the same time, tragedy struck E, his sister committed suicide, and his mother became terminally ill with lung cancer. In 1998, Eels released the powerful album Electro-shock Blues, both sad and beautiful, an expression of his feelings about his departed family.
Eels' most recent releases have been Daisies of the Galaxy (2000), and Souljacker (2001).
source: http://eels.artistdirect.com/biography/index.shtml
By definition, for all real numbers x, f(x)=ex is the function whose rate of change (derivative) is equal to f(x), i.e., f'(x)=f(x) for each x in the real numbers. Thus a Taylor polynomial would be easy to construct (and use to approximate f(x)), if we were given any specific information about the value of f(x) for some real number x. So let f(0)=a, an arbitrary (non-zero) real number. Then the nth Taylor polynomial is:
(n) (n) f'(0) 1 f''(0) 2 f (0) n T (x,0) = f(0) + -------·(x-0) + --------·(x-0) + ... + ---------·(x-0) 1! 2! n! a 1 a 2 a n = a + ---·(x-0) + ---·(x-0) + ... + ---·(x-0) 1! 2! n! n --- / i \ \ | x | = a · / | -- | --- \ i! / i=0
Now, it would be nice if f(0)=1, because then a=1 and there are no extra constants floating around. Notice that a=0 is possible, but highly uninteresting, since then the function and all its derivatives are zero. It is also worth noting that the above formula is only an approximation; the real value of f(x) can only be determined by adding up the countable number of terms in the sequence.
ex can also be defined as the inverse of ln(x), the natural logarithm of x. All the properties of powers of numbers apply to ex, which can be easily shown using the definition of ln(x).
E is a <deep breath> cross-platform scripting language with capabilities-based security and builtin support for distributed programming. It's called E because, while there are languages called A, B, and C, there is none called D†; thus, the original designers decided that D must be bad luck, and moved on to E. The current implementation is written in Java, and comes with ELib, which lets Java programs interact and cooperate with ones written in E. It is closely related to the CapIDL and EROS projects, among several others, and is worked on by some very smart people.
I'm not going to even try to explain most of the language in this writeup; instead I'm going to explain a few core concepts which will hopefully get some few of you interested enough to learn the language.
One of the neat things about E is it's mechanisms for deadlock-free distributed programming, which are inspired by real life. Let's say that you have to get a document from Bob over in accounting. You walk over and find that Bob hasn't finished it yet, so what to do? In many distributed systems, you would stand there and wait for him to finish it (block). Instead, of course, you tell him you'll be back later, and go do something else. This is how E works.
The key operation is the 'eventually operator', which is <-. To tell a car object to move‡, you might say:
car <- moveToHere(5,7) // The car will eventually move, but not right now.
But we don't even have to have an actual thing to make calls on with the eventually operator. We can do that just based on the promise that, at some point in the future, we will have the appropriate object:
// carVow is a promise that the car maker will send us a car def carVow := makeCar <- ("Skyline") // once we get the car, tell it to move carVow <- moveTo(2,3) // but we keep going, even if the car isn't built yet
You can also call in on a promise, waiting until something really is done:
when (carVow) -> done(car) { println("Right on, we've got wheels!") car <- goTo("The 7-11") } catch(prob) { // Something bad happened, and now we don't get our car println("Dude, where's my car? Answer: " + prob) }
It's impossible for me to write too much more without just replicating everything in E in a Walnut, so I will direct you there for all further information, including the 'normal' programming stuff for single process systems. E really is cool; check it out.
† Alas, StrawberryFrog informs me that there is a programming language named D (http://www.digitalmars.com/d/). In defence of both myself and the E programming team, E existed before D. :P
‡ Most examples either stolen from or heavily inspired by ones in E in a Walnut.
References:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
See also: Æ, æ, Œ, œ, È, è, É, é, Ê, ê, Ë, ë, ∃, ∈, ∋, €, Ε, ε, Ē, ē,
