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FROM THE LIBRARY OF
PROFESSOR W.W.BEMAN
AB.1870; AM, 1873
TEACHER OF MATHEMATICS
1871-1922
Sillllll
ATHEMATICS
21,5
W375
24
TRANSCENDENTAL NUMBERS.
An authorized translation by Professor W. W. BEMAN of Chapter XXV.
of Vol. II., of the Lehrbuch der Algebra*
Mar
BY PROFESSOR HEINRICH WEBER.
[Professor Weber's presentation of the recent methods of demonstrating
the transcendency of e and at, especially in sections 205 and 206, is so
elementary that its reproduction in English will be welcomed by many.
For the sake of completeness the whole chapter has been given. W. W. B.]
S 203.
Enumerable Masses.
In the introduction to our work the general number con-
cept was defined and with this general number concept we
have operated, e.g., in the proof of the existence of a root.
In the further course of our investigations, we have dealt
only with algebraic numbers without stopping to inquire
whether the content of the number domain was thus ex-
hausted, or there were non-algebraic numbers besides. The
existence of non-algebraic numbers, also called transcen-
dental, was first demonstrated by Liouville. Other proofs
have been devised by G. Cantor.†
We begin here with the conception, already estab-
lished in the introduction, of a mass or manifoldness,
which means a system of elements of any kind so far de-
fined that in case of any arbitrary object it is possible to de-
cide whether it belongs to the system or not.
We distinguish between finite and infinite masses and
introduce as our first and most important example of an
infinite mass the aggregate of the natural numbers 1, 2, 3, ...
The following definition then holds :
1. Definition. A mass is said to be enumerable [abzählbar
when its elements can be brought into a (1, 1) correspondence with
the whole series of natural numbers or a portion of the same. I
Accordingly every finite mass is enumerable and in the
enumeration the series of natural numbers is used only up
to a certain greatest number. In what follows we specially
consider infinite masses.
Braunschweig, Vieweg und Sohn, 1895 und 1896.
† G. CANTOR, Crelle's Journal, vol. 77 (1873). “Ueber eine Eigen-
schaft des Inbegriffes aller reellen algebraischen Zahlen."
CANTOR, 1. C. The notion of enumerable masses agrees with the notion
of simply infinite systems defined by DEDEKIND without the assumption of
the number system. DEDEKIND, "Was sind und was sollen die Zahlen ?'
26. Braunschweig, 1887. (Second unaltered edition, 1893.)
.
G
-
175
TRANSCENDENTAL NUMBERS.
[Feb.,
1
We may also define an infinite enumerable mass as a mass
of such a nature that to every element a definite number
from the series of natural numbers can be applied as a name,
provided every number of the series of natural numbers is
employed ; that is, a mass in which there is a first, second,
third, . . . hundredth ... element.
Finally, we may also say that an infinite enumerable mass
is one whose elements can be arranged in such an order
that there is a first element, that each element is followed
by a definite other one of the mass, and that every element save
the first is preceded by a definite other element. Such a series
we may call a countable [zählbar] arrangement. It is
clear that an enumerable mass is enumerable not only in one
way, but in infinitely many different ways.
Besides the series of natural numbers which is manifestly
enumerable, we may mention as a second example the system
of rational positive proper fractions which may be enumer-
ated, for instance, in the following manner:
1 1 2 1 3 1 2 3 4 1 5
2 3 3 4 4 5 55 5 5 6 6
5
i.e., so that every greater denominator follows the less
denominator, and that the fractions with equal denominators
are arranged according to the magnitude of the numera-
tors. If, however, the fractions were to be arranged in or-
der of numerical value, we should not have a countable ar-
rangement.
2. The aggregate of all algebraic numbers is an enumerable mass.
To prove this important theorem we recall that every al-
gebraic number @ is the root of one and only one irreduci-
ble equation
(1) f(O)=a, An + a, An-1+...+00 = 0
where do, a, ...0, are integral rational numbers without a
common divisor, and a, is positive and different from zero.
The degree n of the equation (1) is a positive integer and
hence at least equal to 1. Some of the numbers Qy, Q2, ...a
nm 1
may be 0.
tan
.
tan
We will now select the signs so that +0,,#02,
dar •
are not negative, and call the sum
(2)
N=(n − 1)+ a, #ą,+g.
the height of the algebraic number 0. The height is then al-
ways a positive integer.
Now it is easy to see that for a given value of the height
R3
Pirag.W.W. Bus
-üft
1897.]
TRANSCENDENTAL NUMBERS.
176
!
N there is always only a finite number of algebraic num-
bers. For in the first place by (2) n can never be greater than
N, and for given values of N and n the numbers &q, Az, ...am
can be determined in only a finite number of ways. Among
the functions f(0) so determined we retain only the irre-
ducible ones. If now the algebraic numbers be arranged so
that the numbers of less height precede those of greater
height, that among numbers of equal height that one pre-
cedes whose real part is the smaller, that among numbers
of equal height and equal real part the one of smaller
imaginary part precedes, we have a countable arrangement
of the algebraic numbers and it is shown that the aggregate
of all algebraic numbers is an enumerable mass.
We have, for example:
o
1, ao
1, di
+1,
+ 2,
£1,
N=1, n = 1, do = 1, ay = 0,
N=2, n=1, a = 1, ay
N= 3, n=
do = 2, ay
n=2, a
2, ao = 1, az
1, az = 0, a, = 1,
and the beginning of an orderly series of algebraic num-
bers becomes
1
1
0, — , – ✓
2'
, -2
2,...
in
Every portion of the series of natural numbers is an enu-
merable mass: for we have only to arrange the numbers of
such portion in order of magnitude to obtain a countable ar-
rangement.
Thus it appears that every portion of an enumerable
mass is itself an enumerable mass. Hence it follows among
other things that the mass of real algebraic numbers is
enumerable.
§ 204.
Unenumerable Masses.
We now proceed to the proof of the theorem that there
are number masses which are not enumerable and in parti-
cular we shall show that:
The aggregate of all real numbers even when we are restricted to
a finite interval is not enumerable.
For this purpose consider any enumerable mass of unequal
real numbers which in a countable arrangement (2) may
be designated:
177
[Feb.,
TRANSCENDENTAL NUMBERS.
(9)
W1, W2, 03, 04, • •
(where it is to be observed that the elements may not be
arranged in order of magnitude).
For brevity we shall speak of that one of two elements of
the series 2 which has the smaller index as the earlier, the
one with greater index as the later.
We now select any two real numbers a and B, a <B, and
show that in the interval '=(a, b) there is at least one
number not found in the series 2. As soon as we have
shown the existence of one such number for every interval,
there must be infinitely many, since we can apply the same
argument to every portion of the interval (a, B).
First, it is clear that our proposition is correct if only a
finite number of numbers of the series 2 is contained in any
finite portion of the interval 8, and we may, therefore, pass
immediately to the case where an infinite mass of numbers
of the series 2 lies in every part, no matter how small, of
the interval o.
Designate by ,,, the two earliest numbers of the series 2
which lie in the interval o, assume a, < B, and put d=ß, - Qy,
so that 0,=(Q7, B,) is a portion of the interval .
Now designate similarly by a, b, the two earliest num-
bers of the series 2 which lie within the interval 8, (exclu-
sive of the limits), assume a, <B, and put 0;=ß, - az
We can proceed in this way and obtain an unlimited
series of intervals
0, 0, 0, 0, ...,
each of which includes all that follow, and two series of
numbers :
a, a,, A2,...,
B, B1, B2, ...,
all of which, with the possible exception of the first an B,
belong to the series 2. The numbers a, a,, Q.2, . , form an
increasing series, the numbers B, B1, B2, ..., a decreasing
series, and at the same time every a is less than every B.
1. Hence it follows that the numbers a, have an upper limit a, the
numbers B, a lower limit b and that a Eb (see Vol. I., $ 38, 1.)
Possibly we may have a=
From the method of formation of the intervals o, 0, 023 ...
it follows that:
2. If any number w of the series 2 falls in the interval ', , eii-
cluding its limits Qy , Bu, then w occurs later in the series 2 than
the pair of numbers av , By , belonging to the same series.
26.
ו ע
1897.]
178
TRANSCENDENTAL NUMBERS.
For ay , By were the two earliest numbers of the series 2
occurring in the interval 0,-1. Hence it follows that:
3. The greater the index v, the later are the two numbers ay , By
in the series 2, and as we have assumed the series of intervals o,
not to terminate; we can advance Qy , B, as far as we please in the
series 2 by increasing v sufficiently.
Now it follows very readily that no number g which co-
incides with one of the numbers a, b, or, if a and b are un-
equal, lies between them, can belong to the series 2.
For the number g lies within each one of the intervals .
Now if we assume that g occurs in l, and by (3) increase
v so that a, , B, come later in 2 than g, then by (2) g cannot
lie in the interval 2, and thus the absurdity of our hypoth-
esis is demonstrated.
Hence it follows that the aggregate of all the numbers of an in-
terval a, B does not form an enumerable mass.
This theorem may be demonstrated in another way
which
is simpler in some respects and may be briefly indicated.
We do not restrict the generality if we contine ourselves to
the interval from 0 to 1. We shall imagine all numbers of
this interval represented by decimal fractions with an in-
finite number of terms. Finite decimal fractions are in-
cluded if we make all the digits after a certain one equal
to zero.
To render this representation by decimal fractions
unambiguous, it must be agreed that for a finite decimal
fraction this representation must always be chosen, so that,
for example, 0.4999 ... must not be written for 0.5000...
We will now assume that these decimal fractions form an
enumerable mass. They may then be arranged in a count-
able series, represented as follows:
0.a
(2)
0,
*)
02
0.01
(1) (1) (1)
aya, "Az
(2) (2)
(2)
1
(3)
3
2
dos
do2
door
0.a.
2, (3)
(3)
M
where the alv) represent digits of the decimal system.
But it is very easy to form a decimal fraction (or indeed,
as many as we please) which is not contained in the series
12. We have only to form
n=0.B, B,Bz..
where the B, are digits of the decimal system, satisfying the
one condition that for every v, B, is different from alp). This
179
[Feb.,
TRANSCENDENTAL NUMBERS.
number n, which also belongs to the interval (0, 1) cannot
be a number of the series 82.
The formation of n may be made still more general by
arbitrarily selecting the B's as far as we please and then ap-
plying the law, By Žaly).
Since it has thus been shown that the real algebraic num-
bers form an enumerable mass and that in every interval
there are numbers which do not belong to a given enumer-
able mass it follows immediately that:
In every real interval there are transcendental numbers.
$ 205.
Transcendency of the Number e.
It is a far more difficult task to decide whether any given
number is algebraic or transcendental. Here interest has
chiefly been concentrated upon the two numbers which oc-
cur so frequently in analysis, e, the base of the natural sys-
tem of logarithms and the Ludolphian number 4, the ratio
of the circumference of the circle to the diameter.
For the number e the question was answered by Hermite
in a celebrated memoir,* which has been made the basis of
the later investigations upon the number 7. The solution
for the number t, which was specially interesting on ac-
count of its relation to the famous problem of the quadra-
ture of the circle, for a long time, however, presented insur-
mountable difficulties. Finally, Lindemann succeeded in
demonstrating that u also belongs to the transcendental
numbers. But Lindemann's proof still presented great dif-
ficulties. These, however, have gradually been so com-
pletely removed by the investigations of Weierstrass, Hil-
bert, Hurwitz and Gordant that the demonstration can now
be made with quite elementary means and in the most sim-
ple manner.
If we represent by II (n) the product
II (n)= 1. 2. 3...n
then, for any a,
* HERMITE, “Sur la function exponentielle." Comptes rendus, vol.
LXXVII, 1873.
† LINDEMANN, "Ueber die Zahl 7." Mathem. Annalen, vol. 20, 1882.
WEIERSTRASS, " Zu Lindemann's Abhandlung über die Ludolph'sche
Zahl.” Sitzungsbericht der Berliner Akademie, December 3, 1885. The
articles by HILBERT, HURWITZ and GORDAN are all three found in vol.
43, Mathem. Annalen (1893), the first two also in the Göttinger Nachrich-
ten of 1893.
1897.]
180
TRANSCENDENTAL NUMBERS.
an
(
1)
1l(n)
as n increases indefinitely, approaches the limit zero and
that more rapidly than the terms of a decreasing geometri-
cal progression. For if k is an integer > 1 x | then is
XC
li
always a proper fraction if h 2k. Consequently
xn
in-k
Il(n)
Il(K) k +1 k +2
Hence it follows that the infinite series
2c2 203
(2)
e* = 1 + x +
+
+...
II(2) II(3)
converges for all values of x, and by it we define the ex-
ponential function es, whose simplest properties we here
assume as known from analysis. We have, in particular,
the relation
paty e eg
for any two values x,y; whence it follows that, for n a posi-
tive integer, en is the nth power of the number
1
1
(3) e=2+ + + ...= 2.718281828459...
I
(2)
11 (3)
If now r represents any positive integer, formula (2) may
be written
II
(4) II (r) e* =II (r) +
x 2 + ... + x + x" U,
II (9)
x +
II (1)
Il (2)
.
2
zos
where
x²
(5) U=
n+1+ (+ 1)(-+2)+(n+1)(-+2)(n+3)+
t...
and this formula holds for p= 0) if we assume II (0)=1.
Designating | x | by 5, it follows from the theorem that
the absolute value of a sum is never greater than the sum
of the absolute values of its parts, that
Ę
$3
$?
| US1+
p+1+(n+1)(r+2)+(+1)(r+2)(-+ 3)
t...
and therefore
.
181
[Feb.,
TRANSCENDENTAL NUMBERS.
83
ہم رله
62
+
1.2
+
U, 1<| 1+i
If we put
tie
1-2-3
=
- es.
U, -
qr
es
then
ar is a function of x of which we know that 1 q1 < 1.
We now substitute in formula (4)r= 0,1,... n, and write
out the terms in reverse order as follows:
}
nl (n) €* = 9m 24 % + 2* + nxn-1 + n(n-1) **-? +
+ II (n)
(6) Il(n-1) e* = In–, 29-10% + x-1 + (n-1) 29-2
+(n-1).(n 2) 24–3+.
6* = 9, es + 1.
Now assume an arbitrary integral function of x of the nth
degree:
(7) f(x) = Cm " + Cn-yan-1 + 09-2 3-2 + ... + Co,
and its derivatives
f'(x) = nc, **-1 + (n-1)(n-1 33–3
2n1 -+ ...
(8) F"(x) = n(n-1) 22-2 + (n-1)(n--2) Cn~182~3 + ...
ll Com
(9)
xn_1
f(n) x = (n)
and for brevity put
F(x) = f(x) + f'(x) + f'(x) + ... + f(n)(x).
Then F(x) is an integral function of x of degree n.
Finally assume
(10) Q(x) = Cn9n 2" + Cn-1 (n-1 +...+ Co 90
(11) P= C, I (n)+ C, II (n-1)+...+ c.
Q(x) depends upon x, though not expressible rationally
in terms of 2; P, however, is independent of x.
If we now multiply equations (6) in order by C-1, ...,
and add, we get
(12)
€* P= F(x) + € 0 (x).
n
$
1897.]
182
TRANSCENDENTAL NUMBERS.
This formula will serve as the foundation for the follow-
ing deductions.
If we assume e to be an algebraic number it must satisfy
an equation of degree m (say)
(13)
C + C, et C, e + ... + Cem = 0
whose coefficients Co, C, C, ... Con are integral rational num-
bers,* and C. and Cm are not zero. It is required to show
the absurdity of this hypothesis.
To this end we substitute in (12) for x successively the
integers 0, 1, 2, ... m, so that becomes identical with 2,
multiply by C, C, ... C and add.
Then by (13), since P is independent of x,
(14) 0=C, F(0)+C, F(1)+...+ Cm F(m)
+ CoQ (0)+ CeQ(1)+...+ Cmem Q (m)
and now by a suitable choice of the still arbitrary function
F(x) we are to show that (14) is impossible.
We select a prime number p, which is greater than met
and put
2c2–1 (1 — x)" (2 — x)”... (m
x)?
(15) f(x)
II (p-1)
so that the degree n of f(x) is equal to (m + 1) p-1 and
we now prove two things :
(1) CF(0) + CF (1)+...+ C F (m) is an integer dif-
ferent from zero and, therefore, disregarding sign, at least
equal to 1.
(2) CQ(0) + Ce Q(1) + ... + Ce Q (m) is less than 1,
both upon the assumption that p has been suitably dis-
posed of. If these two points are established, we recognize
* The following definitions are given by Weber in % 133:
An algebraic number o is called an integral algebraic number when it
satisfies a rational equation
(1)
OM + A, OM-It... + Am–10 + Am=0
whose coefficients A1, A2, ..., Am are integers.
Integral algebraic numbers include as a special case ordinary integers,
which, for distinction, we call integral rational numbers. W. W. B.
† That there is always a prime number p greater than any given num
ber ll was already demonstrated in EUCLID (Elements, Book IX., Prop.
XX., vol. 2 of HEIBERG's edition). The proof is simply this, that the in-
teger II (u)+1 which is obviously greater than u is divisible by no prime
number, which is not greater than j, because this number gives the remain-
der 1 when divided by the numbers 2, 3, ...ll. Hence it is impossible
that there be no prime number above lo
i
183
[Feb.,
TRANSCENDENTAL NUMBERS.
the impossibility of equation (14) and therefore of equation
(13), from which (14) was derived.
If we arrange the numerator of f(x) in powers of x we
get an expression of the form :
Ap-
t...
2-129–1 + A, 27" + Ap+1 22+1
(16) f (x2)
Il (241)
where Ap_1, Ap, A2+1 ... are integers and A,-=[ll (m)]”, and
hence is certainly not divisible by p. . Comparing (16)
with Taylor's Formula
x?
f(x) = f(0) + x f'(0) + 1 (2)
II (2) f”(0) + ...,
we have
f(0) = f'(0) = f'(0) = ... = f(-2) (0)=0,
fro-1) (0) Ap-i, f(P)(O)=p Apyf (+1) (0)=P(+1)/2+1, ...
Hence
F(0) = Ap-1 + p 4, +p(p+1)^2+1 +...
is an integer not divisible by p. If, however, we arrange f(x)
in powers of (x - 1), we get
B, (x - 1)” +Be+1 (x - 1)p+1 + ..
f(a)
11 (p-1)
where B,, Bp+1, ... are again integers. Thence it follows as
above by comparison with
-
f(a)=f(1) + (x - 1)f' (1) +
II (2)
)
that
F(1)= p B, +p (p+1) B,+1+...
Consequently, F(1) is an integer divisible by p, and similarly
we can show that F(2), F(3),...F (m) are integers divisi-
ble by p. Since p can be chosen so large that C, is not
divisible by p, it follows that
C.F(0) + C, F(1)+...+Cm F (m)
is an integer not divisible by p and hence does not vanish;
thus (1) is demonstrated.
If we can show further that by increasing p, Q(a) can for
any positive x be made as small as we please, it will follow
that for a sufficiently large value of p the expression (2) can
certainly be made less than 1, and our proof is completed.
1897.]
184
TRANSCENDENTAL NUMBERS.
و7..
Returning to the expression (10) for Q(x), let Yn1,
denote the absolute values of the coefficients en Ci—1, .Co
of f(x). Since que In1, . . . are less in absolute value than
1, it appears that for every positive x, IQ (x) | is less than
(17)
(2) = 1,21" + Yn–21-1 + ... +ro.
The coefficients Com Cn1, Co of the function f(x) in (15)
differ from the rm in-1, ...y, only in sign. If we replace x
by — x, thus forming the function
29–1 (1 + x)” (2 + 2)"...(m + x)”
f(-x) =
II (p - 1)
this function has the same coefficients as f(x) but with all
the signs positive. It is therefore nothing else than the
function & (oc).
If we put
X= x (1 + 2) (2 + x) ... (m + x),
then
X Xp-1
(18)
Ų (x)
Il (p-1)
and, as shown at the beginning of this section, with p in-
creasing indefinitely this approaches the limit zero.
Therefore e is a transcendental number.
X
-1,
$206.
Transcendency of the number 1.
By the same methods we may now show the transcendency
of the number . As definition of this number we use the
property that it is the least positive number which satisfies
the equation
(1)
eit
when ex is defined by the formula $ 205 (2).
If we assume now that t and consequently i n also is an
algebraic number, then is in one of the roots of an irreduci-
ble rational equation z(x)= 0 whose coefficients are inte-
gral rational numbers.
If we denote the various roots of this algebraic equation
by Pi, Pn, ... By and the coefficient of xv in z by a, then
(2) x(x)= a (x – 3)(x – B)... (a – B, ),
and the products a Bı, a B, ... a B, are integral algebraic num-
!
185
[Feb.,
TRANSCENDENTAL NUMBERS.
bers and their integral symmetric functions therefore in-
tegral rational numbers ($ 133).
As the number i must occur among the B's, we must
have by (1):
(1 + eB.) (1 + eB2)...(1 + eBv)= 0,
or expanding
1 + Ieki + Eeßi+Br + E Pi+Bu+B* + ...=0.
Among the exponents in this sum the number 0 may oc-
cur several times; suppose (C-1) times so that C is a
positive integer at least =1. The remaining exponents
B, B: + BB:+ B +Px..., some of which may be equal to one
another, we will designate by An, Q.2, ... Qy, so that the equa-
tion becomes .
(3)
C + ea, + +...+eQue = 0.
The quantities a, a,, ... are algebraic numbers which
multiplied by the rational integer a become integral alge-
braic numbers. The symmetric functions of the various
sums By, B:+BP:+Bu+Px... are at the same time symmetric
functions of B,... Bu, and therefore rational numbers. These
sums are therefore the roots of a rational equation and since
the root 0 can be removed as often as it occurs, the quantities
An, Q.2, ... Au are also roots of a rational equation. The funda-
mental symmetric functions of the quantities a a,,a, ...a
are integral rational numbers.
The absolute values of the numbers
α α
au
a, agg...au
we shall designate by
a,, ,, ...0
If in equation (12) of the previous section we put
x = 0, ang ang ..., and apply equation (3), we obtain
(4) 0=CF(0) + F(a) + F(0) + ... + F(Q)
1+CQ (0) + eas Q (q) +ear Q (Q) + ... + eu Q (Qu),
and we have to show the impossibility of this equation by
a suitable choice of f(x).
Let p again be a prime number which may increase with-
out limit, and put
1897.]
186
TRANSCENDENTAL NUMBERS.
P
Orde )
a?
M
aug
aup+r–22–1 (x – 0,)” (a — ,)”... (x
(5) f(x)
II (p - 1)
so that f(x) is a function with rational coefficients.
As in the previous section we arrange in powers of x:
f(x)=49–1 202 + 4, xº + Ap+1 2+1
" + ...
Il p 1)
where Ap-1, A,, Ap+19 ... are integral rational numbers and
A2-1 = (-1)" aup+P-1 Qal, ...
If we assume p greater than either of the rational integers
a, ama, 4,...
4,-1 will not be divisible by p and hence
F(0) = 49–1 + p A, +p (p+1) Ap+1 + ...
will be an integer not divisible by p. Likewise if p is taken
sufficiently large C is not divisible by p.
On the other hand, arranging the numerator of f (x) in
powers of a (x—a,) we obtain
B, a? (—a,)” + Bp+1 ap+1 (x-a,) P+1 +..
f(x)
Il (p-1)
are no longer rational but are integral al-
gebraic numbers, since the numerator of f(x) is an integral
function of ax, aung •
.
From this it follows as in the last section that
F%) = p B, a" + p (p+1) B,+1 Q2+2 +...
If similarly we form F(Q), F (az),...F(Qu) and observe
that the sum
B, (a) + B, (a) +...+ B, (QM)
is an integral rational number, it follows that
F(a) + F(Q) + ... + F)
is an integral rational number divisible by p. Hence finally
we conclude that
CF(0) + F(a) + F(Q)+...+F (QM)
is an integral rational number not divisible by p, and hence
not equal to zero; it must therefore, disregarding sign, be
at least equal to 1.
where B, B2+1,..:
a au
187
[Feb.,
TRANSCENDENTAL NUMBERS,
we
1-)
One by
If now finally it can be shown that with the present
choice of f(x) the function Q(x) can be made as small as
we please, for any value of x, by a suitable increase of
P,
may show, as before, the impossibility of equation (4).
This may be seen as follows: Instead of the function
(6)
f(x) = 0,2* + 09–127–1 + ... +
consider the function
a up+2–1 22–1 (x + ay) ” (x + ay)!... (x + ay )”
4(x)=
Il (p-1)
=Yn x * + Yamla t
+...+ You
which has all its coefficients positive, (though not necessa-
rily rational). The coefficients On: Cn-19 · are formed by
multiplication and addition from the numbers a, -y, — ,
Qu, and the corresponding coefficients Yu Yn-1, ...
are obtained from these on replacing —Q,, -,,
their absolute values ay, day Que ; whence by the theorem
previously mentioned it follows that the coefficients Yra Yn-1,
are certainly not less than the absolute values of the cor-
responding coefficients One Cn—1,
Now for every finite x whose absolute value is $, the ab-
solute value of Q(x) by $ 205 (10) is not greater than
Yn + Yn-1 3-1 +...+y=4 (6),
and that 4() can be made as small as we please by in-
creasing p sufficiently appears from the form
X XP-1
4 (x)=
ax ' II (p-1)
where
X= a +1 x (x + a) (x + a)... (x + au).
Hence it is proved that:
The number z is a transcendental number. The quadrature of
the circle can not be effected by any geometric construction in
which only algebraic curves and surfaces are employed.
1
$ 207.
Lindemann's General Theorem Regarding the Exponential
Function,
The transcendency of the numbers e and t, just demon-
strated, is contained as a special case in a very general
theorem regarding the exponential function announced by
1897.]
188
TRANSCENDENTAL NUMBERS.
To prove
ang 0.22
a
Lindemann in the memoir above mentioned, of which an
extended proof was given by Weierstrass in his memoir.
We propose to demonstrate this theorem here by the use
of the same methods. as in the two special cases. The
theorem may be stated as follows:
I. An equation of the form
(1)
Cei + Cee's +...+ Cepn=0
is impossible if the coefficients C1, C2, ... Care algebraic numbers
and the exponents z...mare distinct algebraic numbers, unless
all the coefficients C,C,, ... Cm are equal to zero.
this we establish first a lemma.
Let
... a
be any real or imaginary but distinct quantities, and
A,, A2, ... A
likewise arbitrary quantities which do not all vanish.
Similarly for the two series of quantities
B, B, ...,
B, B,,...B
Designate by
11/29 - ..Yi
the distinct sums among the rs sums a: + Bx and put
A= A, ea: + A, eas +...+A, ear,
(2)
B= B, eß. + B, eß. +...+Bess,
(3)
AB= C, ey + Cey: +...+ C, en
The lemma to be proved may then be stated as follows:
1. The coefficients C, C,,... can not all vanish.
In the proof of this theorem we may obviously assume
that no one of the coefficients A1, A,, ... A, B, B,, ... B, van-
ishes, since those that vanish may be omitted.
For brevity we shall say for the moment that of two dif-
ferent complex numbers a, b, a is less than b (a<b) when
the real part of a is less than the real part of b, or if the
real parts are equal, when the imaginary part of a is less
than the imaginary part of 6.
If, in this sense, a <b, b< c, then is a < cand if a <b,
c<d, then is a + c < b + d.
In every finite series of distinct complex numbers there
is then a definite minimum and hence if ay is the least
189
[Feb.,
TRANSCENDENTAL NUMBERS.
among the a's, B, the least among the B's, then is az + B, the
least among the y's, and this sum is not equal to any other
of the sums a: + BxTherefore C= A, B., and C, is differ-
ent from zero, which was to be proved.
This theorem may now be generalized by mathematical
induction.
2. If
A' =
A" =
A,' caz' + A,' eaz! +
- A,"eaz!! + A, ea.z!! + ...
A,'" ,!!! + A," dag!!! to..
A"
.. the
are any number of sums of the form (2), and Yu, Ya,
distinct sums among the expressions
a! +a" + " + ...,
then in the product
(4)
A' A" A"...
= C, ey + C, ex + ...
not all the coefficients C, C,... are equal to zero.
This lemma enables us in the proof of theorem (1) to
make the simplifying hypothesis that the coefficients C,
C,, ... Cm are integral rational numbers.
Having assumed the existence of an equation of the form
(5)
X,e"+ X,0*2 + ...=0
with algebraic coefficients X, X, ... and distinct exponents
27, , ... we may always suppose the X, X, ... to belong to
the same algebraic corpus [Körper]* 2.
* The following résumé of WEBER'S definitions take from
MATHEWS's review in Nature, vol. 55 (1896), pp. 25-28: The notion of
a corpus, which is of the most fundamental character, is due to Dedekind,
and is as follows : Let us take a finite or infinite system of elements
a, b, Y, etc., concerning which nothing is assumed except that they can
enter into rational combination according to the rules of ordinary algebra;
then the totality of all rational functions of a, B, Y, etc., except those
which involve division by zero, constitutes a corpus denoted by
2 (a, , 7,...).
The simplest corpus is that of all rational numbers. This is contained
in every other corpus; for if w be any element of the corpus, then by defi-
nition the corpus contains w I w, that is, unity; and from this all other
numbers may be derived by rational operations only.
If the elements of a corpus are all numbers it is called a numerical
corpus, but the elements may be independent variables, or even variables
subject to algebraical conditions.
If 2 (a, b, y, ...) is any corpus, and x any quantity not contained in it,
we
1897.]
190
TRANSCENDENTAL NUMBERS.
Designate by U, U,,
variables and construct the norm
of the linear form X, u, + X, u, t...
(6) N (X + X, , +...)= 4 (4, 4,...),
which is an integral homogeneous function of the variables
U with rational coefficients, whose degree is equal to the de-
gree of the corpus 2.
If in (6) we put
U, = e*1 , U
Uz
e92
.
every product of the variables u becomes a quantity of the
form ex, where z is a sum of numbers x, and 0(U7, Uz, ...)
becomes an expression of the form Cei + C,e2 +. whose
coefficients are rational numbers. If the X, X,... do not
all vanish, C, C, ... by (2) are not all zero, even when
the equal terms in the function (e*i , e*2 .) are combined
into a single term Ce. But if equation (5) holds, then
(e*u, e92, ...) = 0 and hence
Cei + Cest...=0.
If any of the coefficients C, C, ... are fractions, they can
be converted into integers by multiplying the equation by
the lowest common denominator.
the corpus 2 (x, a, b, y,) is said to be derived from 2 (a, b, Y,...) by the
adjunction of x.
If z is an undetermined variable, the polynomial
f(2)= do zm tay xm-it...tam
is said to be a function in 2, when all the coefficients do, di, am belong
to 12.
When 12 is given, we may, if we like, regard all the quantities belong-
ing to it as rational : for this reason Kronecker calls a corpus a domain of
rationality.
A function in 2 is reducible (in 2) if it can be resolved into the prod-
uct of two functions in 2. A function which is irreducible in 2 may be
reducible in a corpus derived from by adjunction.
Let f(x) be an irreducible function in 2 of the nth degree in z; then the
equation
f(x)=0
is assumed to haven conjugate roots 21, 22,
If 12 is a numerical cor-
pus, the roots have actual numerical values ; but this is really immaterial
so far as the general algebraic theory of the corpus is concerned. By the
separate adjunction of the roots, we obtain the conjugate corpora 2 (21),
12 (22), ... 12 (zn), each of which is called an algebraical corpus of the
nth degree. These conjugate corpora are not necessarily all different; they
may, in fact, be all identical, and the corpus is then called a Galoisian or
normal corpus.
W. W. B.
an.
191
[Feb.,
TRANSCENDENTAL NUMBERS.
3. We need then only to show that equation (1) can not hold
for any system of integral rational numbers C, C, ... Co which
do not all vanish.
We assume, therefore, the existence of an equation
(7)
A, e*i + A, e*2 + ...=0
whose coefficients A1, A2, ... are integral rational numbers,
not all = 0, the exponents x,, %,, ... being distinct algebraic
numbers. Designate by 2 a normal corpus [Normalkörper]
to which all these numbers X1, X, ... belong, and by a
primitive number of this corpus; further by
o'= (0,8'), 0' =(0,0"), ...
the substitutions of the corpus l. If by one of these substi-
tutions, s', the numbers xn, 22, pass into xı', x;', ... then
these also must be distinct.
Let now u be a variable and put
3
(8)
U= A, exity + A, e, t...
Apply to this function all the substitutions o', o"
.. and
designate the functions thus obtained by U', U", ... Al-
though we are not dealing with algebraic numbers simply,
we may call the product of all these functions the norm of
U. This product has the form
(9)
N(U) = C, eur. + C, e"?, +...,
where the C, C2, ... are likewise integral rational numbers.
The quantities z, , ... are numbers of the corpus 2, which,
if we combine terms with like exponents into a single term,
we may assume to be distinct. At the same time we have
on account of (7)
(10)
Cei + Cep: +...= 0.
The expression in the second member of (9), as shown by
its formation as norm, has the property of remaining un-
changed after any one of the substitutions o', o'', ... If we
expand (9) in powers of u, every term of the expansion must
possess this invariant character; hence, if h is an arbitrary
positive exponent, the expression
C%;" + CZ +...
must admit the substitution ; but this sum is a number of
the corpus 2, and hence à rational number. This may be
1897.
192
TRANSCENDENTAL NUMBERS.
summed up as follows: If g (2) designate any integral
function of z with rational coefficients, then is
C9 (2) + 0,9 (2) +...
a rational number.
4. The proof of theorem I is thus reduced to the proof of the fol-
lowing special case : Given an equation
(11)
Cei + C, 12 +...+ Cem= 0
: are distinct algebraic numbers, while
C, C,... and the combinations
(12) Cg (2.) + 0,9 () +...+ Cm 9 (mm)
are rational numbers, if g(2) be any integral function with ra-
tional coefficients, then must C, C,... Call vanish.
To prove this theorem we determine first a positive in-
tegral rational number c such that
in which 21,227
2
m
- C2
29
= C2
m
(13)
X = C219 X 2
become integral algebraic numbers. We assume the co-
efficients C, C, ... to be integral rational numbers and
form an integral function
(14)
4 (2) = ax + a,2-1+...+ a
with integral rational numerical coefficients, which pos-
sesses the following properties :
(1) The numbers X1, X2, ...Xm are found among the roots
of 4 (a)
0.
(2) The sum
C%'(x) + C," (X) + ... + Cm 9' (xm)=k ,
which by hypotheses (12), (13) is a rational integer, is dif-
ferent from zero.
To see that such a function as y(x) always exists take
first a function z(0), which satisfies only condition (1),
and the second condition that no one of the numbers
X1, X23 •..2m is a double root of y(x). Such a function mani-
festly exists. We may, for example, to get a function >
take the norm of the product (x - 2) (x x) ...(x -- 20m)
freed of any divisors that it may have in common with its
derivative.
Now put
$(x) =
* x(),p'(,)= a* (2,), Ç” (c,)= , x(z,),..
193
[Feb.,
TRANSCENDENTAL NUMBERS.
Then the sum
C14'() + ... + Conso" (x)=C7' (2,).x," + ... + Cmx' (cm) x mm
certainly cannot vanish for every exponent h, since other-
wise, contrary to our hypothesis, C, x'(x), ...Cmx'(m) would
all equal zero.
Now that the existence of a function 4(x), such as was de-
sired, has been established, we apply the formula $ 205 (12),
e* P(x) = F(x) + e Q (C).
In this we put x = 71,722 •
2m and designate the absolute
values of 21,
Z by 51, 52
Then by (11) we obtain
(15) O=C, F(x) + C, F(%) + ... +CF(2m)
+ C, e$i Q(2) + C, e62 Q() +...+ Cm esm F(m),
22
mº
and if f(x) is an arbitrary integral function of the nth degree,
we have
(16) F(x) = f(x) + f(x) +...+ f(n) (2).
Designating by pa natural prime number which may be
made as large as we please, we put x=cz [according to
(13)] and
9 (2)P-16'(x)
(17)
f(x) =
11(p-1)
where y (x) is the function satisfying the conditions
(1), (2).
Arranging in powers of (x --X,) we may write
! (x)?? ¢'(x) = ç'(x)(x—3,)+ A, (x) (x—2,)
+ Ap+1(x,) (x--<,)p+1 +...
where A, (2), Ap+1(2),... are integral algebraic numbers
and rational functions of x. On the other hand by Taylor's
theorem:
f(2)=f(2) + (2—2) f'(2) +
(2—2)?
f" (2.) +...
1:2
and c (2-2) = x-X. The comparison shows that
f(27) - 0,f'(2)= 0, ... f(-2)(22) = 0, f(p-1) (2,)=(P-10'(x,)”,
f) (2.) =po A, (x,), f(+1) (2.) =p (P+1) c2+1 4,+1(x,),...
and hence
1897.]
194
TRANSCENDENTAL NUMBERS.
F(2.) = 02-10'(x,)” + p C” A, («)+P (P + 1)c2+1 Ap+1(x)+...;
in this x,, , may be replaced by X,,,, or by Xg, 73, etc.
Accordingly the integral rational number
C, F(27) -+ C, F() +...+ Cm F(zm)
is congruent, to the modulus p, to
03- [C,& (<,)” + C, 9' (x,)” +...+ Cm Vol (2 m)'].
Since C, C, ...,C, are integral rational numbers, it follows
that Cº=C1, CP= C,, ... (mod. p), and by an application of
the polynomial theorem,
C&'(x) + C, (x,)” +...+ Cmo (n)"=
[C%'(x) + C,&'(x) + ... + Cm 8' (2.)] = k (mod. p).
Accordingly we have the congruence
(18) CF(2) + C, F(z)+...+ Cm F(zm) = -1 kop (mod. p).
Now the numbers c, k are independent of p, and hence we
can take p so large that it will not be contained in cand in k.
5. Hence the sum C F (2.) + CF ()+...+CF() is an
integral rational number different from zero and therefore at least
equal to 1.
We designate by y, (2) the integral function derived from
4(20) on replacing any negative coefficients among a, a,,a...
by their positive values. The function
8 (2) -781' (x)
fi(2)
(=
Il(p-1)
which thus results from (17) will have only positive coeffi-
cients and these coefficients are in absolute value certainly
not less than the corresponding coefficients of f(2), because the co-
efficients of fi(x) are derived by addition from the same
numbers which in the formation of the coefficients of f(2)
are partly added, partly subtracted.
Hence from formula $ 205 (10) letting = 1 * , we
have
19(2)] = $(5)P-! «,!(5)
Il(p-1)
And therefore for any finite value of = Q(x) can be made as
small as we please by a sufficient increase of p.
By 5 and 6 it has been shown that if p be taken large
enough equation (15) is impossible. Hence 4 and conse-
quently the whole theorem I. is proved.
UNIVERSITY OF MICHIGAN
3 9015 04207 7530
195
TRANSCENDENTAL NUMBERS.
[Feb.,
This theorem now admits of manifold applications. It
gives us first the transcendency of e, if we regard C1,C2, ...
21, 22, ... as integral rational numbers. It gives further the
transcendency of T. For from the equation 1 + ein = 0 it
follows by I. that it, and consequently 7, can not be alge-
braic.
It follows further that
For every algebraic number x, except x = 0, X= ex is a trans-
cendental number.
For every algebraic X, except X= 1, every natural logarithm
x = log X is a transcendental number.
For every arc which stands in an algebraically expressible rela-
tion x to the radius, except x =
sin x is a transcendental
number.
This follows from I, since 2 i X = pix - eix,
The same is true for the other trigonometric lines cos x,
1
tan x, and for the chord sin To add one more re-
2 2
sult: The transcendental equation tan x = a x for a alge-
braic has, except 0, only transcendental numbers for roots.
0, X
UNIV, OF MICH.
JAN 21 1924
BOUND

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