Hermite (18221901) (
e
) and Lindemann (18521939) (
)
While it is true that Liouville gave, with proof (how else!), the first examples of transcendental numbers, it would  I believe  be generally accepted that the first example of a seriously beautiful transcendental number is due to Charles Hermite, with his proof of the transcendence of
e
, in 1873. (One can only imagine how delighted he must have felt to
know
, with complete certainty (in so far as such a thing exists), that Euler's number,
e
, is
not
a solution of
any
polynomial equation with integer coefficients, with nonzero leading coefficient.)
His proof is actually
quite simple
from a technical point of view (it only required genius to frame it!), and can be easily followed. Here I present only the
ideas
behind his proof (the interested reader may follow the complete details by reading elsewhere, or in my Manchester 19721973 notes when I eventually put them up at my web site).
Basic idea of Hermite's proof of the transcendence of
e
. Suppose that
e
is algebraic. Then
... (i)
for some integers
, with
.
Hermite's idea was to replace every power of
e
in (i) with
simultaneous
rational approximations (meaning they have the
same
denominator) of a
very special kind
; he showed how to construct singlevariable rational number approximations
(with
increasingly large
common denominator
) to the numbers
, such that (i) became
namely
+
... (ii)
Hermite was able to arrange the rational approximations in such a way that
not only
are all the epsilons
small
, and become increasingly smaller as F(0) is made increasingly large (in itself that is an entirely trivial matter in a general setting: choose
any
m
real numbers
, and
any
denominator
q
(and think of
q
as being made larger and larger), then each of those
's is either a rational number with denominator
q
, or lies between two consecutive rational numbers with denominator
q
. Thus there are rational numbers
that are simultaneous rational approximations to
, with, in each case, an error term at most
), but
so small
that
every one
of
is
small
, and become increasingly smaller as F(0) is made increasingly large.
You should now
see
how (ii)
reads
: it looks like (an integer, which varies) + (something small, that's getting smaller) = 0. That, however, would be impossible
if only
one could arrange matters so that the 'integer' is nonzero. In short, that's what Hermite did, but it in a quite complicated way... It was greatly simplified (by Klein?) with an
ad hoc
piece of trickery: arrange for (
)
to be nonzero by choosing a prime number
p
that does not divide
, and arrange the approximations such that
p
does
not
divide the denominator F(0), but
does
divide every one of the numerators
.
A novice reader should seek out the actual detailed proof. Some versions are not for the faint hearted; the one I first read in Hardy & Wright is quite frightful, and certainly does not aim to enlighten. My own recollection of understanding the proof for the first time was reading my schoolbought copy of Felix Klein's (Dover edition, which I've lost)
Arithmetic
; that doesn't seem to be available anymore. I had hoped to type up my Manchester handwritten notes in time for this talk, but that will have to wait. Interested readers ought to find they follow the proof I gave there.
_________
is so well known to everyone that it would not be sensible for me to write about it here. The following are classic, and well known: area of a circle (
), circumference of a circle (
), volume of a sphere
(
), surface area of a sphere (
), etc

:=
+ ... , equals
, as was first proved by Euler.
>
restart;
sum(1/n^2, n = 1..infinity);
>
If you aren't familiar with what a CAS  like Maple  can do, I hope you are impressed with that last evaluation!

the probability (properly defined) that two random integers have greatest common divisor equal to 1 is
(related to the previous infinite sum)

the zeroes of the function of a
complex
variable sin(
z
), with infinite series expansion
+ ...
are ...
, ...
The great Siegel began his classic 1949 Princeton University Press on Transcendental Numbers by writing
The most widely known result on transcendental numbers is the transcendency of
proved by Lindemann in 1882.
If one is acquainted with the classic Greek rulerandcompass construction problems (sadly not a topic that our modern school pupils are exposed to...) of the duplication of the cube, the trisection of a general angle, and the squaring of the circle, then one will know that a solution of the latter reduces to knowing whether or not
is not just algebraic, but is algebraic of a very particular kind...
I think I should be frank and admit that there is no easy proof that
is transcendental. I did include a fairly clear proof of it in my Manchester 197273 course, and hope in time to put it up in the transcendental numbers corner of my web site. I will merely record that the honour of first proving its transcendence goes to Lindemann (1882), and recommend that interested readers consider obtaining a copy of the delightful
Pi
: A Source Book
(see References).