Exploring the Mystery of Euler's Number "e

[DerFelix]

Hi!
This is my first post, and maybe there is already a thread about this, but I couldn't find one.
I found this forum through StumbleUpon.

My question is, what exactly is e? How did Mr. Euler get to this number?
I am from germany and recently graduated in a "gymnasium". Somewhat like the american high school. So I used e before, and I get how it is used and what for. However I don't understand the thing itself. How he found it and what it really is... (other than "natural growth").

I really hope I get a conclusion out of this. I always found maths interesting and never had any problems in school. This is the only thing I just could never understand.

Thank you for your attention!

 

[arildno]

It's just another number.

 

[HallsofIvy]

It can be shown, fairly easily, that the derivative of a^x is C_a a^x where C_a is a constant: it depends on a but not b.

It is not too difficult to show that C₂ is less than 1, and C₃ is larger than 1. And since it can also be shown that C_a depends "continuously" on a, there must be some number between 2 and 3 so that constant is equal to 1: e is defined as that number. In other words, e is chosen so that the derivative of e^x is just e^x itself.

I did not show here those things I declared to be "easy" or "not to difficult" because I do not know how much Calculus you would understand.

 

[DerFelix]

It can be shown, fairly easily, that the derivative of a^x is C_a a^x where C_a is a constant: it depends on a but not b.

It is not too difficult to show that C₂ is less than 1, and C₃ is larger than 1. And since it can also be shown that C_a depends "continuously" on a, there must be some number between 2 and 3 so that constant is equal to 1: e is defined as that number. In other words, e is chosen so that the derivative of e^x is just e^x itself.

I did not show here those things I declared to be "easy" or "not to difficult" because I do not know how much Calculus you would understand.

That's allright. This actually makes sense to me. So e was deliberately chosen just for this purpose?
In school they told us some story about interest and how if you calculate interest wrong you get to this number.
Your explanation is much better. Thanks a lot!

 

[symbolipoint]

Yes you can start with exponential growth in money interest as a practical example. Look at the growth in interest in one year for an investment if compounding is only done one time each year. Then try the same example but change to compounding done two times each year. Redo the example now using compounding maybe every three months (or four times per year). Try this again using compouning each month (or twelve times each year). ... Try this with compouning each week (or 52 times per year). Try the same example again but with compouning every day.
What if the compounding were continual?

Actually, my description is vague because no Algebra is being shown, so you probably want a clearer derivation. I found one a few weeks ago in an intermediat algebra book. As the compounding goes to infinity, some particular value (base) approaches a lmit called e which is approximately 2.18...?

Try to go through the process which I described above and you can see where the pattern will go.

 

[robert Ihnot]

A very important fact is that e is the base of the "Natural logarithm," which occurs for the
$$\int_{1}^e \frac{1}{x} = In (e) =1$$

Or by using the derivative, we obtain $$\frac{In(x+h)-In(x)}{h} =In((1+h/x)^\frac{1}{h}$$

Then substituting n for 1/h as $$h\rightarrow 0$$ We obtain the definition of e^(1/x), from the form (1+1/nx)^n as n goes to infinity.

Thus $$In(e^\frac{1}{x}) = 1/x.$$ Actually, engineers seem to need e all the time, since the natural log frequently occurs.

 

[prasannaworld]

I believe the easiest definition is that if you have a constant "e" and:

f(x) = e^x (for all xER)
then:
f'(x) = e^x

I vaguely try to explain this here... Please just PM me with criticizms :shy:
http://prasannaworld.byethost4.com/mathematics_Calculus4.html

 

[robert Ihnot]

Looking at it from my standpoint and considering the position of prasannaworld, we have

Y=e^x; InY=x; dy/y =dx, thus dy/dx = Y=e^x.

Also, derivative of e^cx = ce^cx. This has considerable ramifications, especially with complex exponents, since it can be shown e^(ix) = cos(x)+isin(x).

We find $$isin(x) = \frac{e^{ix} -e^{-ix}}{2}$$ The derivative, which is easy to obtain, becomes the cos(x) =$$\frac{e^{ix} +e^{-ix}}{2}}$$

So that many diverse things can be tied together using e. Particularly the very famous equation: $$e^{i\pi}=-1.$$ (Though, you are not expected to understand all this for now.)

 

[gmax137]

If you are really interested, look for this book:
"e the story of a number" By Eli Maor. It explains a lot of the "e" stuff and logs and why we care about the natural logs (base e).

 

[Jame]

Just as one can imagine the number e isn't something that first came to the mind of Leonhard Euler, but gradually became more apparent with time since it inevitably shows up every here and there when dealing with continuous change and exponential functions. For an easily comprehensible and good read: http://www-history.mcs.st-andrews.ac.uk/HistTopics/e.html

One of the IMHO greatest presentations of the number e and the exponential function is to be found in the calculus books of Richard Courant. With minimal effort he makes it very clear why there is a need for e, and how it ties things together.