Why e? Exploring the Mystery of Nature's Constant

[quasar987]

oops there's an extra m in my topic title, I was going for why e.

I believe most the the exponential equations in physics come from the fact that a^[f(x)] = a^[f(x)] * lna * df/dx but the book I had in my first calculus class didn't had a proof for that.

Does anybody have one? And most importantly, why e? Does that number represent anything special; is it a certain ratio like pi or anything like that? It really seem to be coming out of nowhere for me. The only definitions I've seen are all unintuitive: "e is defined as the integral from there to there of this" or "e is the number such that [such and such]", etc. But why does it appear in nature so often??

(If you know a similar thread exists, tell me because I didn't find one.)

 

[matt grime]

It appears because it is the solution to the differential equation

dy/dx=y

which appears in lots of forms in 'naturally' occurring examples.

if you took any function, a^x, and differentiated it, what you get back is a constant times a^x, ie d/dx of a^x = l(a)a^x, where l(a) is some constant dependent on a. e happens to be the number where the constant is 1.

given that e^x is now this important function one can use its properties to work out what e is, perhaps by integration or differentiation, looking for its taylor series.

 

[HallsofIvy]

Well, yes, a "similar thread" exists at https://www.physicsforums.com/showthread.php?t=45253

I'm surprized you couldn't find that since you posted it! It also contains the incorrect equation "a^[f(x)] = a^[f(x)] * lna * df/dx". What you meant was that the derivative of a^f(x) is a^f(x)(ln a)(df/dx).

The proof of that certainly is in most calculus books. Just write a^f(x) as
e^f(x)ln(a) and use the chain rule.

If you are talking about a proof that d(e^x)/dx= e^x, that depends upon how you define e^x itself,.

 

[chakotha]

e...

A function such that the value of the function at any point equals the rate at which the value of the function is changing at that point

Is that right?

Or

A curve f(x) such that a tangent drawn at a point (x,y) on the curve will have a slope of y.

 

[Zurtex]

e...

A function such that the value of the function at any point equals the rate at which the value of the function is changing at that point

Is that right?

Or

A curve f(x) such that a tangent drawn at a point (x,y) on the curve will have a slope of y.

Erm actually generally speaking e is a number not a function

 

[Omid]

e: The Story of a Number
The book can be a good help for you.
Here you can read the first chapter :
http://pup.princeton.edu/chapters/s5342.html

 

[T@P]

e is just a number... just like pi or other constants there are some infinite sequences that approximate pi. The reason it is so useful is because of what chakotha said, the derivative of e^x is e^x
hope that helps...

 

[Omid]

!

e is just a number... just like pi or other constants there are some infinite sequences that approximate pi. The reason it is so useful is because of what chakotha said, the derivative of e^x is e^x
hope that helps...

You made me remember my algebra teacher
When I asked: "Where do logarithms come from? Why are they so important? ". He simply repeated the definition of logarithms . Then was the time I really hated math, because I thought math is created just to bother me in the exams
Logarithms are more than a definition and e is much more than just a number. If they weren't so, it wouldn't take us so many years to find them.
Now quasar987 should decide to get to know e better or feel satisfied with the arguments given here.
Thanks

 

[pig]

When I asked: "Where do logarithms come from? Why are they so important? ". He simply repeated the definition of logarithms . Then was the time I really hated math, because I thought math is created just to bother me in the exams

This reminded me of my high school math teacher, she wrote "DERIVATIVES" in big letters (well, not in english) on the board and the derivatives of elementary functions, the (f*g)' = ... and the other couple of rules underneath. Then she showed us how to calculate them. She made no mention of limits even though we learned limits earlier that year, no mention of what derivatives actually are, or how they can be used... I even asked my parents to ask their math teacher friend to explain all of this to me, even though I knew how to calculate what I needed to know, simply because not understanding what it is incredibly bugged me

 

[quasar987]

Thanks a lot for the link Omid.