Euler number "e"

[selpak]

who invented the number e

You can also obtain an approximate value of e by summing the following terms, where n! represents the product of all integers from 1 to n:

1 + 1/1! + 1/2! + 1/3! + 1/4! + . . . + 1/n!

That is a part of a text that i`m reading.
Actually the underlined word "approximate" had my attention, and now i wonder, is it actually "approximate" or is it exact?
i think that this is not approximate, because when he mentioned "n" he meant the last number which means that summation is done infinitely; Am I wrong?



Another thing, can anyone tell me what does the number "e" mean ? "actually i should do it myself, but i researched an i`m asking for more help"
here is what i already know:-
Euler is like Pi, its irrational, its natural, its very useful.
I know the equation of "e" which is mentioned above, and i understand it.
what i want > Can anyone give me an example that directly relate with "e"?



Oh now another question `ve just popped in my mind, can`t we get the precise value of "e" using Limits?

Added after 30 minutes:

plz help as much as u can :S

 

[flatulent]

different ways to find euler number

The equation is an infinite series which will take an infinite time to evaluate. Approximate means that you take enough terms to get the value to as many significant figures as you need.

This number ends up in many places like time constants and alternate forms for trig functions.

 

[Siniša]

e raised to x

Can anyone give me an example that directly relate with "e"?

Example1: step response of simple RC low pass filter

Example2: thermal response of measured body to thermal disturbance

Both will follow 1-e^(-t/{tau})



\]

 

[selpak]

eulers number e raised

i mean an example illustrated, like how did e appear in it and why and blah blah blah... can anyone help,please ?

Added after 1 hours 4 minutes:

whats th relation between logarithms and e?

 

[cherrytart]

the number e hebrew

e is the limit of (1 + 1/n)^n as n approaches infinity

but any value of n other than infinity just gives an approximation

 

[halls]

euler number power of infinity

whats th relation between logarithms and e?

i'm not a matematician, but i guess the e number appeared when matematicians found that the expression \[\lim_{n \rightarrow \infty}\left( 1+\frac{1}{n} \right)^n\] was the solution to many (or appeared in many) equations they were solving, so they started to analyze that expression, thus reaching the conclusion of the irrational number with value 2,71828...

The logarithm is a mathematical operation which is the inverse operation of the power, this is:

If \[\log_{x}{y} = z\], this means that \[x^z = y\]

x is the numeric base of the logarithm. There are two most used cases, when x = 10 and when x = e.

When x = 10, the logarithm is denoted without specifying the base, this is:
\[\log y = z\]

And when x=e, this is called a natural logarithm, denoted as:

\[\ln{y} = z\]

So the relation between number e and logarithms is that:

If \[ln{y} = z\], then \[y = e^z\]

This expression (e raised to some power) is very common in nature, specially when raised to some complex power. For example \[e^{j\cdot{x}}\], is a expression equivalent to: \[e^{j\cdot{x}}=cos(x) + j\cdot{sin(x)}\]

The integration of the differential of a function divided by the function itself, has a natural logarithm as a result, this is:

\[\int {\frac{df(x)}{f(x)}}=\ln{f(x)}\]

And so on... you can check wikipedia, or whatever, i'm sure there are many pages around the logarithm...

 

[selpak]

what is the approximate value for the number e

thank you all, this topic helped me much more than wikipedia and such websites, cause u talk really simply and clearly

Added after 1 hours 17 minutes:

just one more question
how did they know the equation of "e" which is
e = 1 + 1/1! + 1/2! + 1/3! + ..... + 1/n!
?

 

[halls]

proof limit euler number

just one more question
how did they know the equation of "e" which is
e = 1 + 1/1! + 1/2! + 1/3! + ..... + 1/n!
?

Actually, I guess it was the other way. They found a repeating expression in nature which was the limit shown in above posts, called that "e", and calculated its limit value, which seemed to be 2,71828...

But again, it's just a guess, because it's the way i find it more coherent.

 

[las3rjock]

exp e euler

I think a more fundamental answer to this question is that the function
\[ \exp(x) = \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \ldots\]
is needed to solve many mathematical and physical problems. One of the defining characteristics of this function is that
\[ \frac{d(\exp(x))}{dx} = \exp(x) \]
In particular, the value of \[ \exp(1) = e \].

 

[laktronics]

approximation euler number

Hi,
While on this nice thread of discussions, shall I put a little confusion on the topic? Why is it the expressions (1+1/n)^n and ( 1+1/1!+1/2!+.....1/n!) do not match for small values of s? Try substituting n=2, 3 etc. and see for yourselves, the difference decreases and the expressions meet only at n=infinity and that too for a value of 'e'.
Regards,
Laktronics

 

[umery2k75]

integral of the number e

This books is very good "The story of a Number, by Eli Maor's e".This book is written to explain as why 'e' is important for us and how it has taken it's place with us in engineering and science.

 

[Element_115]

does the number e raised to the x equal 1?

What I remember about "e" is:

1)It's and approximation because it can not be written (an infininte number)
just like flatulent and others said.

2) I thing Newton or Euler was empirically looking for a number when raised to the
power of x it's derivative or integral would equal the same number.

https://everything2.com/index.pl?node_id=1417728

Cheers

 

[raygor17]

euler number logarithm

also is a number that satisfies the particular property that e^(x) is exactly its own derivative ie d/dx(a^x) =ln(a)a^x but ln(e)=1 so it is exactly its own dervative

 

[Jone]

e number euler

Take a look at the following simple functions and its derivatives:

f(x) f'(x)
. .
. .
x^2 2x
x 1
1 0
x^-1 -x^-2
x^-2 -x^-3
. .
. .

So, there seem to be no function which has a derivative that goes like 1/x. Certainly there must be processes that change by a rate of 1/x. So we need to invent some function with a derivative 1/x. Simply integrate 1/x from 1 to ∞;

∫(1/t)dt = ln(x).

Here we have invented a new function ln(x). Now ln(x) is one-to-one over ]0, ∞] and so has an inverse there. Invent another function called exp(x) which is the inverse to ln(x). It then satisfies
ln(exp(x)) = x.

When is the integral, or ln(x), equal to 1? This is satisfied for exp(1) = 2.71... Now we can write

exp(x) = exp(1*x) = exp(1)^x.

If we define e = exp(1) we have

exp(x) = e^x.

So the number e is a base for the exponential function.

 

[selpak]

euler number 2.71

you provided information even more than i ever dreamed
thank you all

 

[Duby]

euler number nature

try it with Mathematica 5.1 Software