By Tony Ho (P1942) on Monday, December 18, 2000 - 03:09 pm :

I know that e = 2.71828...
However, what defines e, and why is it so important in mathematics?

Tony

By Kerwin Hui (Kwkh2) on Monday, December 18, 2000 - 07:41 pm :

One of the proper definitions of

e

is:

$e = lim_{n \to \infty} (1 + 1 / n)^{n}$

One of property of $e$ which is the reason behind virtually every case of its importance is that $d / dx (e^{x}) = e^{x}$ .

There are other definitions of $e$ , for example:

$\exp (x) = 1 + x + (1 / 2) x^{2} + (1 / 3!) x^{3} + \dots$

$e = \exp (1)$

Kerwin

By Brad Rodgers (P1930) on Monday, December 18, 2000 - 08:03 pm :

e is defined as the limit of (1+1/n)ⁿ (or some books define it by a series, but the above definition is more useful - at least for my current intents; and the series definition is hard to remember [at least I can't remember it right now], hence why I'm not going to use it). This is useful because if y₁ =e^x_1 , then we can see that

dy₁ /dx₁ =e^x_1 =y₁

(see if you can figure out why by letting let e=(1+dx)^1/dx )

If we let y₁ =x₂ and x₁ =y₂

then

dx₂ /dy₂ =x₂

So we can say that for x=e^y , from which we can say y=lnx, that dy/dx=1/x. This is important because it allows us to have an integral for 1/x.
If you don't understand any of that, just post back, I know that I found e and ln (especially ln)very hard to understand when I first came across them. (there are a few discussions on this board already on e, I'll post a few of them later).

e also has a ton of significance towards imaginary numbers and trigonometry (see in the complex numbers section of Asked NRICH for more on this). There are also a number of functions concerning e: among them the catenary, which happens to be the way that a chain hangs and the way the St. Louis arch is built.

Brad

By Brad Rodgers (P1930) on Monday, December 18, 2000 - 08:04 pm :

Oops, didn't see your post Kerwin.