e is irrational

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Major Section:
Reference
Type of Math Object:
Theorem

Mathematics Subject Classification

11J82 Measures of irrationality and of transcendence
11J72 Irrationality; linear independence over a field

make thhis an example

Maybe it would be a good idea to add this
entry as an example under the "irrational"
node.

Re: make thhis an example

Yes, that seems like a good idea. Have done this now.

how do we know that e is not a natural number?

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?

Re: how do we know that e is not a natural number?

One way to show that e<3 is the following. By definition,

log 3 = \int_1^3 1/t dt.

This integral is larger than its trapezoidal approximation

1/2(1/1 + 1/2) + 1/2(1/2+1/3) = 7/6 > 1,

so log 3 > 1. Hence 3 > e.

Re: how do we know that e is not a natural number?

That is a neat proof, but there is one thing I do not
understand. How do you know that the trapezoidal
approximation is larger than the integral? Although
we have
f(1+1/2) = 2/3 < 3/4 = 1/2(f(1)+f(2))
f(2+1/2) = 2/5 < 5/12 = 1/2(f(2)+f(3))
I don't think it is completely trivial.

One can also show that log 3>1 by approximating:

int_1^3 f(t) > 1/n \sum_{k=1}^2n f(1+k/n)

where f(t) = 1/t. Here the inequality holds as
f is strictly decreasing. (In the sum, the function
is sampled at the right side of each interval.)
Setting n=4 yields numerically 1.01 >1. Although
this gives a sum of 8 terms to compute it is not as
neat as your solution, but still
something one can do by hand.

I added a request for a proof that 2<e<3. It is given
here:

Re: how do we know that e is not a natural number?

> That is a neat proof, but there is one thing I do not
> understand. How do you know that the trapezoidal
> approximation is larger than the integral?

Sorry, my mental picture was slightly off when I wrote that. I had been thinking of the region under y = 1/x as being convex, which of course isn't true.

Re: how do we know that e is not a natural number?

You mean e > 1/0! + 1/1! = 2. As for why e < 3, we can look at e's Taylor expansion again, with the first two terms chopped off:

1/2! + 1/3! + ... + 1/(n+1)! + ...

Compare it term by term with

1/2 + 1/4 + ... + 1/2^n + ...

which is 1.

Use induction to show that 1/(n+1)! < 1/2^n for n>1. So e - 1/0! - 1/1! < 1, or e < 3.

Re: how do we know that e is not a natural number?

It suffices to show that e^(-1)> 1/3

To do so, we use the series representation

e^(-1) = (1-1) + (1/2-1/6) +(1/24-1/120) + ... + (1/(2k)!-1/(2k+1)!)+...

The terms of this series are all positive.

The first two terms add up to 1/3.

Re: how do we know that e is not a natural number?

The simplest solution is to change "1/b < 1" to "1/b <= 1". We don't need < here because there is < in the previous display. This will fix the proof without requiring a separate proof that e is not a natural number.

reference to Euler's proof that e is irrational

Various pages on the web state that Euler proved this.
Either in 1757

or in 1737

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

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.

Re: how do we know that e is not a natural number?

I do not quite follow this. What do you mean with 'previous
display'?

The proof is based on the counterassumption that e is
rational. If e is a natural number, the proof gives that

sum_(k=b+1)^\infty b!/k! is a natural number,
and
0< sum_(k=b+1)^\infty b!/k! <=1

Now there is no contradiction; there is a natural number k
satisfying 0<k<=1.

Re: how do we know that e is not a natural number?

The 'previous display' refers to the inequality which shows

\sum_{k=b+1}^\infty b!/k! STRICTLY LESS than the geometric series
which is = 1/b.

So it is not needed to know 1/b < 1
to get strict inequality (tail) < 1.

// Steve

Re: how do we know that e is not a natural number?

By 'display' I mean a collection of math symbols that is printed on a line by itself, centered horizontally. Steve correctly identified the display that I was referring to.

Re: how do we know that e is not a natural number?

Silly me. Now I get it. Many thanks.