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[parent] e is irrational (Theorem)

From the Taylor series for $ e^x$ we know the following equation:

$\displaystyle e=\sum_{k=0}^{\infty}\frac{1}{k!}.$ (1)

Now let us assume that $ e$ is rational. This would mean there are two natural numbers $ a$ and $ b$, such that:
$\displaystyle e=\frac{a}{b}.$
This yields:
$\displaystyle b!e\in\mathbb{N}.$
Now we can write $ e$ using (1):
$\displaystyle b!e=b!\sum_{k=0}^{\infty}\frac{1}{k!}.$
This can also be written:
$\displaystyle b!e=\sum_{k=0}^{b}\frac{b!}{k!}+\sum_{k=b+1}^{\infty}\frac{b!}{k!}.$
The first sum is obviously a natural number, and thus
$\displaystyle \sum_{k=b+1}^{\infty}\frac{b!}{k!}$
must also be natural. Now we see:
$\displaystyle \sum_{k=b+1}^{\infty}\frac{b!}{k!}=\frac{1}{b+1}+\frac{1}{(b+1)(b+2)}+...< \sum_{k=1}^{\infty}\left(\frac{1}{b+1}\right)^k=\frac{1}{b}.$
Since $ \frac{1}{b}\leq 1$ we conclude:
$\displaystyle 0<\sum_{k=b+1}^{\infty}\frac{b!}{k!}<1.$
We have also seen that this is an integer, but there is no integer between 0 and 1. So there cannot exist two natural numbers $ a$ and $ b$ such that $ e=\frac{a}{b}$, so $ e$ is irrational.



"e is irrational" is owned by mathwizard.
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See Also: $e^r$ is irrational for $r\in\mathbb{Q}\setminus\{0\}$, natural log base


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Attachments:
proof that e is not a natural number (Proof) by CWoo
e is irrational (Proof) by rspuzio
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Cross-references: irrational, integer, sum, natural numbers, rational, equation, Taylor series
There is 1 reference to this entry.

This is version 10 of e is irrational, born on 2002-03-20, modified 2007-10-01.
Object id is 2795, canonical name is EIsIrrationalProof.
Accessed 6867 times total.

Classification:
AMS MSC: 11J72 (Number theory :: Diophantine approximation, transcendental number theory :: Irrationality; linear independence over a field)
 11J82 (Number theory :: Diophantine approximation, transcendental number theory :: Measures of irrationality and of transcendence)
Pending Errata and Addenda
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Discussion
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reference to Euler's proof that e is irrational by matte on 2006-02-13 15:53:16
Various pages on the web state that Euler proved this.
Either in 1757

 http://www.bath.ac.uk/~ma2mrm/generalinterest.html

or in 1737

 http://numbers.computation.free.fr/Constants/Miscellaneous/irrationality.html

Currently the whole production of Euler is available on
the web. Here:

 http://www.math.dartmouth.edu/~euler/

Does anyone have a reference to the original proof? It would
be nice to have a link to this in this entry. However,
as the works are written in latin, it is not completely trivial
to find a specific result.
[ reply | up ]
how do we know that e is not a natural number? by matte on 2006-02-04 12:32:28
How do we know that 1/b<1 in this proof? In other words,
how do we know that b cannot be 1, and e a natural number?
Are there some simple way of deducing this? For example,
it is easy to see that

 e< 1/0! + 1/1!=2,

but is it easy to see that e<3?

[ reply | up ]
make thhis an example by igor on 2002-05-20 03:09:34
Maybe it would be a good idea to add this
entry as an example under the "irrational"
node.
[ reply | up ]
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