
We know e is defined as e= lim(x>0)(1+x)^(1/x).
Why is e often used as base while taking log, though it's exact value is not known like pi?
grandtheftauto wrote:  We know e is defined as e= lim(x>0)(1+x)^(1/x).
Why is e often used as base while taking log, though it's exact value is not known like pi? 
because e=mc² ?
<gd&r>
Kopernikus wrote:  grandtheftauto wrote:  We know e is defined as e= lim(x>0)(1+x)^(1/x).
Why is e often used as base while taking log, though it's exact value is not known like pi? 
because e=mc² ?
<gd&r> 
e in your example is Energy but e with logaritmes is a constant (2.718).
Adri
I'm guessing he probably knew that
e is one of the few transcendental irrationals that we know about (although obviously, as Cantor shows, there are a hell of a lot of them 'out there').
The limit definition you gave kinda says why e is important: e is related to multiplying small numbers lots of times. Continuous things, like motion, space, time, waves, circles, all can be expressed using small numbers (for example, a circle is a polygon with small sides), and multiplication and exponentiation are kinda primitive things, so e happens a lot.
I'm not sure but I've heard 'e' being referred to as 'Euler's Constant'. So, was it given the letter 'e' because of exponentiation or because of Euler?
And why would it be called Euler's constant? Correct me if I'm wrong but I doubt that Euler discovered it.
No it isn't Euler's constant  that is a different thing.
e is sometimes called 'Euler's Number' just to really confuse things...
Bikerman wrote: 
e is sometimes called 'Euler's Number' just to really confuse things... 
Aha! Confirmation.
I knew mathematicians sometimes did things just for the perversity of making things more confusing!
e is email email is important too for today world.haha~~
but actually electronic is very important for our modern world.
e is extremely important and it creeps up everywhere. It is basically the building block of differential equations. I find many things interesting about it. Let me share a few and then ask a question that I've always wondered and then see if anyone knows the answer (are there any mathematics historians here?).
We all know that compound interest is as follows: P(1+r/n)^(nt). Now, if we take the limit as n > infinity then you'll, of course, get Pe^(rt). Now when I was a student in high school I thought that this was interesting because what it essentially told me was that the rate that the function was increasing was directly proportional to the value of the function. I intuitively understood this when I was introduced to this concept at the age of 14 or 15. I intuitively understood that if n got higher and higher, it would approach a value to where the rate of change was directly proportional to its value...
About a year or two later, I was taking calculus. We then, of course, learned that the derivative of e^x was itself! It was fascinated... Why was I so fascinated? I was fascinated because this would fit in to what I had understood a couple years ago... If the derivative was itself, then of course the rate of change would be directly proportional to it's value... it would equal its value!
So then I began to wonder something... This is the only question that my teacher wasn't able to answer for me... When Euler developed e, was he looking at exponential functions and trying to find out a limit? Or was he looking at graphs and trying to find out a graph where its derivative is itself? Because these are related.. Or was he doing something completely different and this is all coincidence? I'm inclined to think that he was looking at graphs and trying to find a graph where it's slope equaled it's value and then exponential functions came from that because anytime the rate of change of a function is directly related to it's value, e will tend to creep in.
Because the derivative of e^x is e^x, e tends to creep in a whole bunch in differential equations (and many other branches of mathematics) where you deal with a lot of derivatives....
If you really want to know about the mysterious constant 'e', have a read of Eli Maor's great little book, e: The Story of a Number.
This little transcendental number arose long ago in compound interest equations.
One such equation is S=(1+1/n)^n. Where S= a compounded balance using 100% interest beginning with $1, and n = the number of years of compounding.
Now examine how the balance, with increasing years approaches ...... you guessed it "e"! In fact, in the limit, S=(1+1/n)^n as n approaches infinity, S=e.
This ubiquitous little number shows up then in finance, mathematics (esp differential and integral calculus), engineering and all over science.
e is pi's little cousin!
e = 2.718281828459045
It's easy to memorize that far
Its not just e that's important, in fact Pi and i (imagery value) are also very important.
And for some reason they have this special relationship:
e^(i*pi) = 1
even though their origin comes from different branches of mathematics they have this neat relationship.
It is also fun
One simple way to calculate e is:
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
Bikerman wrote:  It is also fun :)
One simple way to calculate e is:
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ... 
Ahhh... Aren't Taylor Series interesting? I didn't like them so much when I first encountered them in high school... But it's interesting how things become more interesting when your mathematical skills increase...
Wow
Simply I wanna say it is super duper discussion about e.
Let me tell you e is really e in terms of anything. [Kidding]
Anyway, thanks for such a good discussion.
PureReborn wrote:  Its not just e that's important, in fact Pi and i (imagery value) are also very important.
And for some reason they have this special relationship:
e^(i*pi) = 1
even though their origin comes from different branches of mathematics they have this neat relationship. 
Well that's just because of the identity:
e^(ix) = cos(x) + isin(x).
So if you plug in pi for x, you get:
cos(pi) + isin(pi) = 1. So it's pretty trivial to show that. If you don't believe the identity that I just gave you, then it's also very trivial to prove the identity. Just consider the Taylor Polynomials of e^x, sin(x) and cos(x).
The Taylor series for e^x, as already posted by Bikerman, is e^x = 1 + x + x^2/2! + x^3/3! +....+x^n/n! as n approaches infinity.
The Taylor series for sin(x) is xx^3/3!+x^5/5!... and for cos(x) it's 1x^2/2!+x^4/4!...
So, given the info I just provided, plug in ix for x in the Taylor series for e^x, and you'll see that you get the Taylor series for cos(x) plus i times the Taylor series for sin(x)...
Most branches of mathematics connect together in some way or another... It really shows the legitimacy of mathematics if you can use mathematics developed hundreds of years apart to prove other things... Of course once Complex Analysis was being developed, I'm sure the mathematicians went to look for complex analysis applications in other branches of math... And of course, Euler's Identity was developed by Euler after Taylor developed Taylor Series and so it only makes since that he would have tried looking at the the Taylor Series for his his constant, e, and then realized what happens when you plug in other values of x...
Cliffer wrote:  e is email email is important too for today world.haha~~
but actually electronic is very important for our modern world. 
I strongly suspect whether you really read the post
i don't really know, but it's important for me because it's the default logarithmic base for actionscript. Idk for other languages tho.
