What Exactly Is the Mathematical Constant 'e'?

[Master J]

Could someone give the definition of e in laman's terms? I have always had trouble visiuallizing exactly what e is...

Thank you.

 

[HallsofIvy]

What do you mean by "layman's terms"? e is a number. It is approximately 2.718281...

If this particular layman has taken calculus, he/she should know that the derivative of any function of the form a^x is simply a constant (dependent upon a but not x) time a^x. "e" is that value of a so that constant is 1: that is, the derivative of e^x is just e^x itself.

 

[phlegmy]

yup e is just a plain old number.
i can't tell you where they got it but here's an idea.

if you deposit €1 in the bank at 100% interest for 1 yr you get €2

if they decide to compound the interest every 6 months you'll get:
€1.50 after 6 months, re invested at 50% interest to give you €2.25 at the end of the year

now the bank manager says, "how often would you like me to compound the interest?" and you think, this could be my shot to become seriously rich because obviousy the more often they compound it the more money you seem to get at the end of the year, so you say, "compound it infinatly often" and the nice bank manager says... "well..., for you,, ok"

after 1 year you look at you balance and you get
€2.71828... in other words €e [ in other words ~$10 :D]

 

[Pere Callahan]

2.7 € is not even $5 but never mind ...

 

[phlegmy]

lol it was a joke, i was just tongue in cheek gloating about the current favourable exchange rate thus the :D anyhoo who's to say i didnt mean the singapore dollar :D

 

[lurflurf]

I like this one.
Supose we have something (money, bugs, energy) accumulating exponentially
Thus rate of accumulation/amount=k a constant
in the amount of time it would take the stuff to double if growing at constant rate there is e if the growth is exponential.
suppose we can separate generations
1+1 +1/2+1/6+1/24+1/120+1/720+...
we se each generation is related by
a_n+1=a_n/n
and that the infinite sum converges to a transendential number approximately
2.718281828...

 

[djeitnstine]

e is a graph which has a change of rate of slope which is always the same. Incidentally, the area under its graph is always the same ie the area between x=0 and x=1 is simply e^{1}. e's graph is somewhere between 2^{x} and 3^{x}. This is just one more simple way to describe it

 

[HallsofIvy]

Let's not confuse the man. "e" is not a graph of any kind, it is a number . I assume you are referring to the "graph of the function f(x)= e^x". If so, say that! And in that case, the "change of rate of slope" is not "always the same", it is e^x which is different for different x. It's not clear to me whether by "change of rate of slope" you are referring to the first derivative ("slope") or second derivative ("rate (of change) of slope") or third derivative ("change of rate of slope)- fortunately for the graph of f(x)= e^x they all the same.

And it is definitely not true that "the area under its graph is always the same". The area under the curve, from 0 to 1 is e, but the area under the curve from 1 to 2 is e²- e. What did you mean by that?