Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 603-613
603
TWO CONSTRUCTIONS OF THE REAL
NUMBERS VIA ALTERNATING SERIES
ARNOLD KNOPFMACHER
,IOHN
and
KNOPFMACHER
Department of Applied Mathematics
Department of Mathematics
University of the Witwatersrand
Johannesburg. Wits 2050, S. Africa
University of the Witwatersrand
Johannesburg, Wits 2050,S. Africa
(Received March i, 1988)
ABSTRACT:
Two further new methods are put forward for constructing the complete
ordered field of real numbers out of the ordered field of rational numbers.
The
mthods are motivated by some little known results on the representation of real
Amongst advantages of the
numbers via alternating series of rational numbers.
methods are the facts that they do not require an arbitrary choice of "base" or
equivalence classes or any similar constructs.
The methods bear similarities to
a
method of construction due to Rieger, which utilises continued fractions.
Key Words and Phrases:
Ordered
oLLono_ numbe,.
AMS (1980) Classification Number: 10A30.
1. INTRODUCTION.
The series of Enge] (1913) and Sylvester (1880) (see Perron [1])for represent:g
real numbers have been studied in some detail.
Much less known is the fact that
there are alternating series representations of real numbers in terms of rationals
corresponding to the above.
The only references to these alternating series that
E Remez [2] and H Salzer [3].
Every real number A has a unique
we are aware of in the literature are in papers of
The series under discussion are as follows:
representation in the form
A
ao + a Ii
1
a a
I 2
1
+
a
(-1)n+1
+
a a
la2a3
I 2
a
+
(ao,a 1,a 2
n
are integers such that
say, where the
ai+ 1 > ai + 1 >_-- 2 for il.
A is rtionZ if and only if A has a finite representation (ao,a I
(Compare this with the expansion of Engel (Perron [i]).)
).
Furthermore.
Corresponding to the series of Sylvester (Perron [1]) we have every real number
A
ao
1
a
1 + 1
a
3
2
a
+
(-1)n+1
a
n
+
a
o,a 1,a 2
)),
A. KNOPFMACHER AND J. KNOPFMACHER
604
say, where the
i are integers defined uniquely by
>_-+
ai(ai 1) for i>--_1. Furthermore, A is
ai+
has a finite representation ((ao,a
an)).
A
such that
rovt.Lonc
al>_--i
I
In many ways, these representations may be compared with that by
continued fractions.
and
if and only if
A
simple
The main purpose of this note is to justify this remark by
deriving some elementary properties of these alternating series representations and
(with these results as an initial motivation) then developing two new methods for
constructing the real number system from the ordered field of rational numbers. These
methods are similar to one recently introduced by G J Rieger [4] for constructing
the real numbers via continued fractions.
The order relations in particular are defined
in an analogous fashion.
The methods share with Rieger’s method the advantage over
other standard
techniques that they do not require an arbitrary choice of a "base", or the use of
(infinite) equivalence classes or similar such constructs.
These properties are
shared as well in the construction of real numbers using ordinary Sylvester and Engel
series, considered in [5]. Two important differences between those and the present
methods are in the definition of the order relations for the series, as well as the
of terminating representations of rational numbers in place of infinite
recurring representations used in [5].
use here
2.
ALTERNATING SERIES REPRESENTATIONS FOR REAL NUMBERS.
For the convenience of the reader, because full previous details may be inaccessible
to many (including the present authors), we prove here the fundamental results concerning
the representation of real numbers via infinite alternating series. It is convenient
to introduce here a more general alternating series, analogous to the positive series
of Oppenheim [6], out of which we can deduce the results for alternating-Engel and
alternating-Sylvester series as special cases. We define the alternating-Oppenheim
algorithm as follows:
Given any real number A, let a
Then we recursively define
a
A
[A] A 1
o
a
n
where
An+
Herein
bi
b
i
--
[1
1
n
a I a2
>_-- 1
An
for
n > 1, A n > 0
(Cn/bn)
i)
a
for
a
c
c
i
0
n >
i
a
a
2
a i)
are positive numbers (usually integers).
1,
The two cases of particular interest to us are those for which b
1,n >___ i (the alternatingn > 1 (alternating-Engel series) and
an,
Sylvester series).
bn cn
CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES
reaZ number A has
Evy
THEOREM 2.1.
605
a unique representation in the
orm
l=o+l-!+
aI
a2 a3
where
az+
>
a% +i) al>
i
reag numbe A has a unique
Fu)uthermore every
1
A=ao+l_
a
ala2
I
eprettion in the
6orm
1
+
ala2a3
where
>
%+
a
a
o
o
+A 1
+
bl
1
a
a2
c1
1
=o + i___
aI
Now
c
l.!_ +
blb2
i
2
ClC 2
a3
a
I
< A n <__ 1
a
n
1
An+l (_.1
a
n
for 0 < A
1
a+l
n
In particular by setting
Furthermore
{an}
>
b
i.
an(an
+ i)
1
a (a +1)
is strictly increasing.
if
if
Ar
> 0 for i
- 0
as
0 < A < 1
n
n we obtain
for all
1
cn
n
)(cn/bn)
Cn/bn)
an(an+l
i
Ck- 1
A n )(Cnlb n)
(-n1
<
An+ 1
An+ 1 <
blb2..:bk-i Ak
ClC2
n
an+l
Thus
+ (-i) k-i
impl es
an
an+l
i.
Repeated application of the alternating-Oppenheim algorithm yields
PROOF.
A
% +lal>
n
,
It follows that
since
<
n
al
1
and the sequence
A has an alternating-Sylvester
expansion
A
a o + 1___ 1__ +.1__
a
a
a
3
2
I
((ao,al,a2
which may perhaps terminate.
Secondly, by setting
cn
an
b
n
1
for all
n we obtain
))
A. KNOPFMACHER AND J. KNOPFMACHER
606
1
an+
a
A
Thus
1
a +
n
<
An+l
and so
a
la 2
>
An+ 1
a
n
as
0
+ 1
if
=,
n
A.c > 0
since
a
n
for i
I
n
>--_ 1
has the alternating-Engel expansion
i
A:ao+l_
a
I ala 2
1
+
(a
ala2a 3
o,a 1.a 2
which also may terminate.
Uniqueness of the representations follows from Proposition 2.3 on order below.
{Various other interesting special cases of the alternating-Oppenheim algorithm will
be treated in a separate article.)
We deduce now an important result on the alternating series expansions for
rational numbers.
PROPOSITION 2.2
t
a
The alternting-Sylvester and aternting-Engel sies tminate
terms i an oy i A is rational.
fbte numb
o
PROOF.
Clearly any number represented by a finite expansion is rational.
Conversely, since Ai,i
1, is rational let A
pi/qi, (pi,qi) 1. Now
i
since for either algorithm
]>
ai
--.
1
1
it follows that
qi
<
Pia
Pi
In the alternating-Sylvester case we now obtain
r+ 1
eL
qi/l
aiqi
Pi+I qi- Piai < Pi Since {pi} is a strictly decreasing sequence
of non-negative integers we must eventually reach a stage at which
O, whence
Thus 0_<_--
Pn+1
(-1) n-1
a
n
+ 1___ 1__ +
+
a
I a2
The result for the alternating-Engel series follows similarly from
A
a
o
i+i
qi+l
qi-Piai
qi
We note that for rational numbers there is a possible ambiguity in the final term,
analogous to that for continued fractions. We eliminate this as follows"
We replace the finite sequence
((ao,a
an_2,an_1+l)) in the case a n an_l(an_l+1).
CONVENTION 1.
((ao,a
(ao,a I
,a n
we identify
by
A(I
(ao,a I
an_2,an_l+l)
in the case
with its finite expansion
(Uo,al
an)) by
Similarly we replace
an_l+1 Furthermore
a n) or ((ao,a
an)),
i
a
n
respectively.
In order to be able to compare finite sequences of different lengths in size
we introduce the symbol m with the following properties" For any r(I,
oo+ K to
r < co
CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES
607
Now we can represent finite sequences by infinite sequences as follows-
CONVENTION 2
For every
hence A
(ao,al
A
(ao,a I
an,m,m
A
((a
o,a I
PROPOSITION
A
((ao,a I
an))
a
(I let
n
a#
Similarly represent
).
(1
2.3 (On Order)
))
LeA: A
((bo,b I )).
B
((ao,a I
A
by
j > n
for
m
an,,
and
)).
B
), or
(bo,b
In both o/ these case, the coditon A<B
(ao,a
l equivalent to"
a2n <
a2n+l
(i i)
that
a.
b2n or
2n+1
.
>
A
2n
or i
We shall use the notation
PROOF.
A
where i
((ao,a 1,a 2
)),
).
(ao,al,a 2
1_
An
and
2n+1
1
an
1__
A’n
a
+
arian+ 1
is the
1
fit index
+
for
an+ an+ 2
n
1
-...
anan+ l an+ 2
such
iO
for
(Note that we do not assume at this stage that
An
A n as
dfined by either algorithm
Now suppose (i) holds.
A
+ A < a
1
Next suppose
in either case.
case.
a
ao < b
o
If firstly
+ i
a2n
<
b
b
n > O,
b2n’
then
+ B
1
B
in the alternating-Sylvester
The growth condition
+ i), i
ai(
a/+ I
> i,
implies that
1
An a2-n
since
a. > 1
for
a2-t
__<
1
+
i
i
a2n+l
(i-
a2n+2
i
a
2 n+2
+i
)+
i
a2n+1
and by observing Convention 1.
a2n+l
U2n
i
+
a2 n+i
a2n
1__(i_
a2n+ l
a2n
i > 1,
1
A’2n
a2n+2
i
(i-
>
1
+
a2n+ 1
Furthermore,
1
a2n+2
(i
i__!___)
a2n+l +I
i
i
(i
a2n+3 +I
U2n+3
1
<
a2n
Thus
An
>
a2n+i
It now follows from A
that
A < B
b2 n
ao +
2n
a-l a- l
+
A’2n,
In the alternating-Engel case, from
B
ao+
ai+ I
a- l l_ _a2
+
>_-- a. + i, i > i,
B’2n
A. KNOPFMACHER AND J. KNOPFMACHER
608
A’2n
>__
1
1
2n
a2na2n+l
1___ (1-
a2na2n+la2n+2
1
a2n
a2n+l
> 1__ 1
1
a2n
1
+
Thus again
.
<
1
1
a2n
a2na2n+l
1
1
a2n
a2 ha2 n+ 1
a2n+2 +1
i
a2n+l
as in the alternating-SyIvester case.
A’2n
(i-
a2na2n+la2n+ 2
)_
a2n+l
1
+
Also
1
+
a2na2n+la2n+2
(1-
1
a2n+ 1 + 1
1
a2n
>
>
a2n+ l
1__>= B’
2n
b2n
A < B now follows from
and the result
ao + a 1I
A
1
ala2
1
ao + 1_
a
ala2
B
1
+..._
ala2 a2n_l
..._
+
An
1
ala2..a2n_1
BI,’n
Note that if
0 and the result remains valid in this case
m then
result is proved in a similar fashion if (ii) holds.
B’2n
b2n
3.
The
CONSTRUCTIONS AND ORDER PROPERTIES
In the constructions below, standard facts about the ordered field Q of all
rational numbers are taken as understood. With the results of Section 1 as initial
and order relations on them as follows"
motivation, we now define two sets E* and
*
Let E* be the set of all formal infinite sequences A
>_--a.+r
(ao,al,a2
for i>_--1,a11.
integers
Also, let
i such that ai+ 1
all formal infinite sequences A
of
integers
))
((ao,al,a 2
al>_-- 1 and ai+l>--_ai(ai+l) for i 1.
of
the set of
* suchbe that
x
Finite sequences (rational numbers) are included in our sets E* and
use of the property all sequences in E*
We will frequently
mae
Convention 2.
using
and
satisfy
a.
implies
a.
j > i
we shall use corresponding lower-case letters
*
"digits" of the elements of the respective sets, and we define
In both the sets E* and
to denote the
for all
CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES
A < B
609
if and only if
a2n < b2n or
a2n+l > b2n+l
(i)
(ii)
where
2n or
i
i
2n+1
>_-- 0
is the first index
such that
a.
b.
In both cases, < is a "total ord%ing" 6ation, i.e., it is traitive
and satisfies the trichotomy law.
LEMMA 3.1.
We use the same argument in both cases.
PROOF.
Next let A < B and B < C. Suppose ar
for r <
b
r or
and
for r <
c
If i < j then a
r
r
a. > b.
c. (i odd).
Firstly, trichotomy is
b
obvious.
,b
and
(i)
If i
(ii)
a
then
j
> bz >
ci
c
a
r
r
( odd).
c
,
r < j
for
a
and
for
r
<
b c
(i even) or
ai < bi < ci (i even) or
a b < c ( even) or
for r < j, and
If Z > j then a
r cr
b. > c. ( odd). Thus A < C in each case.
(iii)
a.
We may now introduce symbols <--_, > and
and (9reotest) lome bound/, in the usual way.
LEMMA 3.2.
Evg non-emptw ube;t
has a ast uppe boun (supremum).
o *
B
by a sequence
which is bounded above
a isb.an
Now A <B
b
AX with a
since otherwise there is nothing to prove.
such that every
AEX, and there is a largest index
We may assume
a
bI
I
for elements of
X of E*,
(bo,b I
Be.
Assume
and define (east) upp bound
(respeetive!W, *) mich is
First consider a non-empty subset
PROOF.
(o,l,m,m
,
b
a
upper bound for
X,
AX
for some
where
o
for every
has
since otherwise
is the maximum value of
X
be the least
If + 1 is odd let
We now define c o
bo
1
+ 1 is even
b o. If
of any AX with a
possible value for the digit
1
of any AX with
let
be the greatest possible value for the digit
+1
1
In either case
has no largest value.
a
m if
b o, where we take
1
+1
). Otherwise we continue
if
m we are done, and put C
(Co,C 1
1
to define
as the least possible value or greatest possible value depending on
2
of all elements of
+ 2 is odd or even, respectively, for the digit
whether
2
m we are done and
in X. Again if
the form (Co,C
C+l,a+2,a+ 3
2
I
as the
define
to
inductively,
Continue
put C
).
(Co,C 1
1
C+l,m,m
for
the
value
(+i+l even)
least possible value (++i odd) or.greatest possible
).
of an element of X of the form (Co,C
digit
c+Z,a++l,+Z+ 2
a+
c+
c+
c b
c+
c+
c,m,m
c+
a+
c+
a++
c++
1
1
The
m.
If, when ++1 is even, a+i+ 1 has no largest value, we take c+Z+
m. We then take
process terminates if at any stage
1
process constructs a non-terminating
this
Otherwise
C
).
(Co,C 1
c+,m,m
c++
A. KNOPFMACHER AND J. KNOPFMACHER
6 i0
In either case we have CE* since Cr+l>--_c +i for
C
A then C >A since either
i I c
> a. (i even)
or e. < a. (xL odd) for the first index
a.
>k such that c.
C
sequence
I
(c o Cl,
i. Also if
c
(by the definition of the sequence
C
Lastly,
c-,O
d
r.
< i
< m,dm
A
(eo,C
of the form
since otherwise
sup)<
em
If
is odd then
m
D< C. Then
every element
Hence
> Cm
satisfies D < A --<_ D
has an upper bound
)<
in
Cm,am+l,am+ 2
dm
X
In the cases m 0 or
contradiction. If m is even we have d < c
(e
e < m (m > O) every element of the form A
1,....cm,
m
satisfies D < A <_-- D
In the case em m am m for every A we can choose
)X with arbitrarily large am. For am > dm we
A
(c e I,
1
choose
have
m for some A
m and a
< A <_-- D. Finally, if c
o,c
am_ am,am+
A
(Co,C
Cm_l,
))
((Co,Cl,C 2
saisfy
4.
and
m,
The argument for
C
,
S*
D < A
I)
once more.
is almost identical to the above, except that the sequence
defined inductively via suitable elements of
ci+ >_--ci(i+l),Cl
S*
will now
>_-- 1.
EMBEDDING AND DENSITY OF RATIONALS
The following proposition justifies our use of Convention 1.
PROPOSITION 4.1
The tntZng-EngZ and ntng-Sylvt ogoCtm
dne 1 1 ord-prving maps
PE*
(I
whose images oe dense in E*
E*
and
and
*
PS*
(I
*
respectively.
-
It is an immediate consequence of the results quoted earlier that the
By
1 maps
two algorithms define 1
pE (1-- E* and PS*
are then
maps
these
Proposition 2.3 and the definition of order in E* and
PROOF.
,
order-preserving.
Now let A < B
in
E
Let
k
*
be the least index for which
a
k
bk
satisfying
We show now in every possible case that we can find a rational number D
Now let A
( we take D- A+B
B
A < D < B. For A
if
we choose D
If k is even then k <
(Oo,Ol,...Ok,Ok+l+l,m,m
if
m’ i.e.,
m’ or D
ak,ak+l,k+2+l,m, m
(ao,a I
choose
we
then
>
odd
is
If instead k
Bm{,A
ak Ok;
if ak+l
m
m’ or D
D,6k+1,O+2 +1,
(o,1
D
(=o,al ak,ak+l+l,m,
in the case of
works
A similar argument
m and A{,B
if
m,m
,
Ok+l
.
Ok+l
.
ai+l
have the intuitively desirable property
m’s)
that rational numbers can be represented only by finite sequences (excluding
for
(see
series
Sylvester
and
Engel
ordinary
This des not hold in the case of
We note that sequences in E*
example [1] ).
and
x
CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES
APPROXIMATION LEMMA
A n
there exist ationols
or
n >--_ 0
A
(iii)
PROOF.
A (n)
A(2n)
A(2n+l)
Given
(rpecLve.y, *),
E*
such
A (2m+1)
A (2n+1)
A
A (2n)
(2n+1)
(2n)
inf A
sup A
A (2m)
(i)
(ii)
o
eenent A
Given anq
4.2
611
or
m < n
i
(2n+1)!
(ao,a I
A
an,,m
define the rational sequences
Next suppose that
part(i)
follows.
Then
EE*,
A (n)
by
).
(2n+1)
if m is odd,
for all n. In that case, we must have a > b
A
A< B
Then m
b
or a < b
if m is even, for the first index m such that am
m
m
m
(m)
(m)
contradiction
while m even gives the
A
odd gives the contradiction A
< B
(2n) =<C<A
(2n+l)
Similrly, suppose that A
Thus A
infA
A (m+l)
A (m+l) < B
Consider the first index m for which om# cm. If m is even we must
for all n
(m)
If m is odd we
which yields the contradiction A(m)<_-- C < A
have a > c
m
m
(m+l)
The same argument
<C < A
have a < Cm, which gives the contradiction A
m
for
and
parts
leads to
(ii)
(i)
(ao,a I
m+li
*
For part (iii) in Ex, the formula for alternating-Engel series for rationals
leads to
A(2n)
A(2n+l)
1
<__
alaz...a2n+l
1
(2n+l)!
For *, the corresponding formula for alternatinga
1,ai+ 1 >_-I
Sylvester series of rationals gives instead
a/+l
since
A(2n+l)_ A(2n)
1
a2n+l
<
1
(2n+l)!
->_-
>_--
1 and ai+ 1
ai(ai+l)
I
ALGEBRAIC OPERATIONS IN E* AND
since
5.
a
*
*
by Convention
and
as an actual subset of
1, it will simplify the discussion on algebraic operations below if we now re-deJrine
Since we alreaady regard
A (n)
for any routional
For any
A
a
(n>_-0)
A.
sup(A (2n) +
A,B( EX (or A,B*) we now define A + B
(respectively, I*) because
which exist in
sup(-A(2n+l)),
A (2n) + B (2n)
*
A (1) + B (1)
A(2n+1)
<
B(2n)),
A (0)
x
are very
At this stage we note that the formal structures of the sets E* and
similar to the set K, (based on continued fractions) used by Rieger [4] to construct
the real numbers. Thus to avoid repetition, we will refer the reader to the
corresponding result of Rieger whenever the proof of the algebraic property there is the
same.
The above operations make E (respectively, *) into an aelian group
cow.raining ((I,+) as a dense subgroup. Fuuther
LEMMA 5.1
A. KNOPFMACHER AND J. KNOPFMACHER
612
A < B
(i)
()
=>
=>
<
-
A+ C < B + C
A >
PROOF.
Obviously A + B
B + A and A + 0 A. The proof of the associative
law of addition can be found in Theorem 3.1 of [4]. Also the proof that A + (-A)
0
is found in Theorem 3.2 of [4]. The only changes that must be made are that we use
A(2n+l)
A(2n)
in place of inequality (1.4) of [4].
(2n+l)
It now follows that
E
(respectively, *) forms an abelian group with (Q,+)
A B and hence the strict monotone law
A + C B + C
(i) follows from the weak one proved below"
as a dense subgroup.
Then
By the definitions of order and rational approximations we have
for n sufficiently large
A < B =) A (2n) < B (2n) A (2n+l) < B (2n+l)
+
Hence A < B= A + C
C
B
Finally, let A
B. Then A (2n+I) < B (2n+I) or -A (2n+I) >-B (2n+I)
<
for
n
-A
sup(-A
sufficiently large.
(2n+I})
Thus
sup(-B
HenCe (i i) follows,
(2n+I))
-(-X)
since
-B
-A > -B
giving
since
A
B.
X
Next, for any A,B E* (respectively, *), define
sup(A (2n) .B
(2n))
A.B
(-A). (-B)
-((-A).B)
-(A.(-B))
A >_-- O,B >_-- 0
if A
O,B =< 0
0
if A
O,B
0
if A >_-- O, B
A -1
sup((A(2n+l)) -1)
if
if
Also define
-((-A)
A > 0
if A < 0
-i
The definitions are unambiguous, since we have
A (2n)
for
B (2n)
A > O,B > 0
<
A (I) .B (1)
(A(2n+l)) -1
:<
(A(O)) -1
and the theorem of the supremum is applicable.
cases, we use the fact that
A < 0
if and only if
-A > O,
To cover all the
by Lemma 5.1(ii).
LEMMA 5.2 The above denons t.ogethe)t wfth the eov.ier operoons make
o a dense subfield,
into a feld containing
E* (Respectively,
*
A < B,C> 0
PROOF.
Clearly
C (n) > 0 for all n,
0
A.B
B.A
=,
and
A.C < B.C
A.1
A.
Also, since
C > 0
implies
we obtain easily
< A < B, C >
0:=>
A.C <--_ B.C
(with strict inequality to be shown later).
is
In order to verify that E*(X) is a field it remains only to verify that
-i
The
O.
.A
1, A
associative, and distributive relative to +, and that A
CONSTRUCTIONS OF REAL NUMBERS VIA ALTERNATING SERIES
613
associative law appears in Theorem 4.1 of [4], the distributive law in Theorem 4.2 of
-1
0 is shown in Theorem 5.1 of [4]. In each case we must
1, A
[4], and A .A
replace Rieger’s inequality (1.4} by
A(Zn+1)
A(2n)
1
(2n+l)
The strict monotone law for multiplication of positive elements now follows
B.C=
A B (for positive elements). By
from the weak one, and the law A.C
simple manipulation of the "sign" cases the result follows for all elements in
E*
(or
*).
*
form ordered fields
E* and
By standard theorems, treated for example in
form concrete
and
Chapter 5 of Cohen and Ehrlich [7], it then follows that
The models are in many ways equivalent
new models for the real number system
to that of Rieger [4], except that they arise from the (simpler) representation of
real numbers as infinite series rather than as infinite continued fractions.
The above discussion has shown that both
with the ]east upper bound property.
*
*
REFERENCES
i.
2.
3.
4.
5.
6.
7.
Irratonzahlen
Chelsea Publ. Co., 1951).
(New York
On series with alternating sign which may be connected with
two algorithms of M V Ostragradskii for the approximation of irrational
numbers. Uspe Mtem. Nauk (N S) 6,No. 5(45), 33-42(1951); Mth. Rev. IB,
444(1952).
The approximation of numbers as sums of reciprocals.
SALZER, H.E.
Ame. Mouth. Mty 54, 135-42(1947) and 55, 350-356(1948).
A new approach to the real numbers (motivated by continued
RIEGER, G.J.
fractions). AOh. Brauceig. Wis. Ge. 33, 205-217(1982).
Two new constructions of the real numbers.
KNOPFMACHER, A and KNOPFMACHER J.
Rocky Mtn. ]. Mth., 1988(in press).
The representation of real numbers by infinite series of
OPPENHEIM, A.
rationals. Acta ArXh. 21 391-398(1972).
the ReaZ Nnber System
COHEN, L.W. and EHRLICH, G. The Structure
Krieger Publ. Co., 1977).
(New York
PERRON, O.
REMEZ, E Ya.
o