We know e (exponential) is a irrational number

[newton1]

we know e (exponential) is a irrational number...
how can we prove it??

 

[emu]

The way we calculate e can be similar to Pi.
One way yields an infinite series of non-repeating rational numbers. The sum is therefore irrational.

Try proving the sqrt(5) is irrational.

 

[HallsofIvy]

Emu, do you know of any way of proving directly that the digits in the decimal expansion of e are NOT repeating? I'm not saying it can't be done, only that I think it's easier to prove e cannot be written as a fraction.

One standard method is to use the Taylor's series (which may be what emu meant): e= 1+ 1/2 + 1/6+ ...+ 1/n!+ ...
If j is any positve integer then e*j!= integer+ 1/(q+1)+ 1/(q+1)(q+2)+ ... which is not an integer so e cannot be written as a fraction with denominator j for any j.

A theorem I saw years ago was this: If c> 0, and there exist a function f(x), continuous on [0,c], positive on (0,c) and such that f(x) and its iterated anti-derivatives can be taken to be integer valued at both 0 and c, the c is irrational!

Taking f(x)= sin(x) in this theorem shows that pi is irrational.

It can also be used to prove: If c is a positive number other than 1 and ln(c) is rational, then c is irrational.

Since e is a positive number, not equal to 1, and ln(e)= 1 is rational, it follows that e is irrational.

 

[Frac]

http://antwrp.gsfc.nasa.gov/htmltest/gifcity/e.2mil

Write that as a fraction!

 

[Paradox]

While e cannot be written as a fraction, e to its first 2 million decimal places can. I'm just not going to.

 

[Hurkyl]

e*j!= integer+ 1/(q+1)+ 1/(q+1)(q+2)+ ... which is not an integer

That's not obvious... I don't see why the infinite sequence there cannot add up to an integral value.

Hurkyl

 

[bogdan]

Let An=1+1/2!+...+1/n!;
It's quite simple to prove that 1/(n+1)!<e-An<1/(n!*n);
Let's suppose e is rational, so it's equal to p/q, where p and q are integers.
1/(n+1)!<e-(1+1/2!+...+1/n!)<1/(n!*n); | *n!;
1/(n+1)<n!*p/q-n!*(1+1/2!+...+1/n!)<1/n;
But between 1/(n+1) and 1/n is no integer...
n!*p/q must be an integer because for n big enough n! is a multiple of q;
So e is not rational...
QED

 

[climbhi]

that's kind of a neat prove, thanks for posting it. I don't think I would've caught that last part about for n big enough...

 

[newton1]

thank you...:)

 

[Hurkyl]

e-An<1/(n!*n)

That bit isn't obvious either... but the ordinary taylor remainder formula gives e/(n+1)! for that term which is sufficient for the proof. Don't tell me how to get that end of the inequality, it'd be a good exercise to figure it out myself!

Hurkyl