e, the Exponential Function

Fudge You

Can anyone explain what e/exponential function/2.71828 informally is???

Ive been on wikipedia, and Ive read my text books. But I can not explain what it is.
I feel confident with logarithms and indices. But I cannot get my head around e.
Any help is much appreciated.

Peemaccee

I just think of it as that number which when raised to the power, x, and differentiated remains the same. ie the rate of change of e^x is e^x. For a function that's cool!

bnt

Another remarkable property of e is its link to complex analysis and fundamental trigonometry, through Euler's Formula. It ties different branches of mathematics together in mind-boggling fashion.

Fudge You

Peemaccee wrote: »

I just think of it as that number which when raised to the power, x, and differentiated remains the same. ie the rate of change of e^x is e^x. For a function that's cool!

I know what you mean Peemacee. And I have learned this in my studies.

What I meant in the op, was, what it is?
Like for example, pi, in school I was thought its 3.14 or 22/7, and it is used for formulas involving circles, and then learned about the radians in trigonometry etc.. But I still never knew what pi was.
But I was told that pi is the ratio of the circumference to the diameter, which is explained easily in wikipedia, and has a nice gif on the top right of the page.

Yakuza

It's also possible to prove that [latex]e^{i \pi} = -1 [/latex].
How's that for a mind-bender?

Ok, so Latex doesn't seem to be working...the above should read e^(i * pi) = -1

ZorbaTehZ

For something concrete you could say that it is the sum of 1+1/1!+1/2!+1/3!+1/4!+... or it is the limit as n tends to infinity of the expression (1+1/n)^n.

sponsoredwalk

If pi being the ratio of the circumference to the diameter is intuitive to you, then e being "the epitome of universal growth" might also be intuitive:

https://www.youtube.com/watch?v=yTfHn9Aj7UM

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

Colonel Sanders

Isn't it also the case that if I randomly select k numbers between 0 and 1 until the sum of these k numbers is >1 then E[k]=e?

Peemaccee

To get a geometically view of where e comes from, the best you can do is to look at the area under the rectangular hyperbola (y=1/x) from an x value of 1.

When the area is 1, we denote that vaue of x as e.



From this simple fact, e spreads itself through out the world of maths.