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Abstract

The notion of number line was formed in XX c. We consider the generation of this conception in works by M. Stiefel (1544), Galilei (1633), Euler (1748), Lambert (1766), Bolzano (1830-1834), Meray (1869-1872), Cantor (1872), Dedekind (1872), Heine (1872) and Weierstrass (1861-1885).
On the History of Number Line
Galina Sinkevich, associate professor, Department of Mathematics,
Saint Petersburg State University of Architecture and Civil Engineering
Vtoraja Krasnoarmejskaja ul. 4, St. Petersburg, 190005, Russia
Abstract. The notion of number line was formed in XX c. We consider the
generation of this conception in works by M. Stiefel (1544), Galilei (1633),
Euler (1748), Lambert (1766), Bolzano (1830-1834), Mцray (1869-1872),
Cantor (1872), Dedekind (1872), Heine (1872) and Weierstrass (1861-1885).
A number line is an abstract notion which evolved early in 20
th
century. A
number line should be distinguished from solid and geometrical line. A solid
number line or an interval is an image which came into existence in the
ancient world. As a notion, a geometrical line or axis formed in analysis in the
period from the 16
th
to 18
th
century. The notion of a right line or a curve as a
locus emerged in the 17
th
century in the earliest works on analysis [1].
As a concept, a number line formed in works of Cantor and Dedekind,
СoаeЯer, ЭСe Эerm ТЭselП, ПТrsЭ Фnoаn Кs К “nЮmЛer sМКle” КnН ЭСereКПЭer, Кs К
“nЮmЛer lТne” СКs Лeen ЮseН sТnМe 1912μ “TСЮs, ЭСe posТЭТЯe КnН neРКЭТЯe
numbers together form a complete scale extending in both directions from
гero” Д2].
A continuum, the philosophic idea of the continuous or extended, was the
prescience of a number curve. It dates back to the ancient world (Zeno,
Aristotle), Middle Ages (Boethius), and beginning of the Modern Age
(G. Buridan, T. Bradvardin), and thereafter, Leibniz comes.
In the ancient world, numbers were presented as a set of natural numbers.
Rational positive proportions of geometric magnitudes were quantities.
Irrational quantities π КnН
2
were determined through approximants. The
reasoning was based on Eudoxus' method of exhaustion. The estimations were
assumed as greater and smaller. The approximation techniques reached the
peak in works of Archimedes and later, in works of oriental mathematicians.
Zero was not regarded as a number for a long time. Although results of
certain problems were negative (e.g., Diophantus obtained such negative
numbers), such negative numbers were not deemed to be competent; in
certain rare cases, they were interpreted as a debt. Before the modern age,
only positive routes were sought when solving equations. Irrational numbers
conventionally from Euclid were understood as non- extractable radicals.
Irrational number were called (e.g. by Newton) surdi (deaf) or false.
ImКРТnКrв nЮmЛers аСТМС ПТrsЭ КppeКreН Тn 1545 Тn CКrНКno’s аorФs аere
called sophistic numbers.
Michael Stifel (14871567) was the first to define negative numbers as
numbers that are less than zero and positive numbers as those that are greater
than zero. It was he who described zero, fractional and irrational numbers as
numbers. Stifel wrote in what way whole, rational and irrational numbers
relate to each other.
LeЭ Юs КННress SЭТПel’s ЛooФ oП 1544 “Arithmetica integra” Д3]. SЭТПel
recognized that there are infinitely many fractions and irrational numbers
between two nearest whole numbers. He considered a unit segment (2, 3) and
located infinite sequences therein
,...
7
3
2,
7
1
2,
6
5
2,
6
1
2,
5
4
2,
5
3
2,
5
2
2,
5
1
2,
4
3
2,
4
1
2,
3
2
2,
3
1
2,
2
1
2
and
,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5
333333
3
3
33
33
,...26,24,23,22,21,20,19,18,17,26,25,24,23,22,21
4
4
4
44
4444
33
3
3
33
[3, p.104]. See [4] for more detail.
In 1596, in keeping with the traditions of German Rechenhaftigkeit
(calculability) and following Archimedes whose works won prominence in
Europe in the middle of the 16
th
century, Ludolph van Ceulen (15391610)
calculated a 35-НeМТmКl nЮmЛer π. SТnМe ЭСen, nЮmЛer π аКs Фnoаn Кs
“LЮНolpСТne nЮmЛer” ЮnЭТl ЭСe enН oП ЭСe 19
th
century.
Unlike Stifel, G. Galilei (1564 1642) felt that mathematics and physics
are linked with each other; all his reasonings were accompanied by examples
from optics, mechanics, etc. A line was understood as a result of movement.
In 1633, Тn СТs ЛooФ enЭТЭleН “TСe DТsМoЮrses and Mathematical
Demonstrations Relating to Tаo Neа SМТenМes”, Galilei discoursed of the
МoЮnЭТnР НТsЭrТЛЮЭТonμ “SIMP. If I should ask further how many squares there
are one might reply truly that there are as many as the corresponding number
of roots, since every square has its own root and every root its own square,
while no square has more than one root and no root more than one square.
But if I inquire how many roots there are, it cannot be denied that there are
as many as there are numbers because every number is a root of some square.
This being granted we must say that there are as many squares as there are
numbers because they are just as numerous as their roots, and all the numbers
are roots. Yet at the outset we said there are many more numbers than
squares, since the larger portion of them are not squares. Not only so, but the
proportionate number of squares diminishes as we pass to larger numbers.
Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of
all the numbers; up to 10000, we find only 1/100 part to be squares; and up to
a million only 1/1000 part; on the other hand in an infinite number, if one
could conceive of such a thing, he would be forced to admit that there are as
many squares as there are numbers all taken together.
So far as I see we can only infer that the totality of all numbers is infinite,
that the number of squares is infinite, and that the number of their roots is
infinite; neither is the number of squares less than the totality of all numbers,
nor the latter greater than the former; and finally the attributes "equal,"
greater," and "less," are not applicable to infinite, but only to finite,
quantities.” Д5, “FТrsЭ НКв”].
Numbers considered before the 18
th
century were natural, rational and
irrational numbers (as non- extractable radicals Kestner [6]), that is to say,
КlРeЛrКТМ ТrrКЭТonКl ones. TСe КssЮmpЭТon ЭСКЭ nЮmЛer π Тs ТrrКЭТonКl аКs
voiced by Arab scientists starting from the 11
th
century [7, p. 176].
In 1748, speaking of tangents of angles less than 30
0
Тn СТs “InЭroНЮМЭТon Эo
AnКlвsТs oП InПТnТЭelв SmКll”, EЮler ЮseН ЭСe ЭermЭoo ТrrКЭТonКl” (nТmТs sЮnЭ
irrationals) [8, p. 120]. In the same work, he made an assumption that in
addition to irrational algebraic numbers (i.e. products of algebraic equation
with rational coefficients), there are transcendental irrational numbers as well
which are obtained as a result of transcendental calculations, e.g. taking
loРКrТЭСms. Аe аoЮlН noЭe ЭСКЭ sвmЛols π КnН e esЭКЛlТsСeН КПЭer EЮler’s
“InЭroНЮМЭТon Эo AnКlвsТs oП InПТnТЭelв SmКll” аКs pЮЛlТsСeН, КlЭСoЮРС π
could be found in works of various mathematicians starting from the
ЛeРТnnТnР oП ЭСe 18ЭС МenЭЮrв. TСe sвmЛol π Рoes ЛКМФ Эo GreeФ περι-
КroЮnН, КЛoЮЭν or περι-ετρο – perimeter, circumferenМeν or περι-φέρεια –
circumference, arc.
In his work of 1766 which was published in 1770 [9] translated into
Russian [10, p. 121143], Johann Heinrich Lambert (17281777) proved
ТrrКЭТonКlТЭв oП nЮmЛers π КnН e. His reasoning was in 1794 supplemented by
LeРenНre (LeРenНre’s lemmК) Д10, p. 145–155, 11]. Fourier studied this issue
as well (1815).
In 1767, in its SЮr lК resolЮЭТon Нes цqЮКЭТons nЮmцrТqЮes” (BerlТn, 1769),
Lagrange defined irrational numbers as those determined by a nonterminating
continued fraction.
In 1821, A. L. Cauchy defined irrational numbers as limits of converging
sequences. However, he did not define procedures or operations therewith
[12].
In 1830s, Тn СТs mКnЮsМrТpЭ enЭТЭleН “TСeorв oП VКlЮes” (GrößenleСre),
Bernard Bolzano made an attempt to develop a theory of a real number.
Bolzano used the exhaustions method and the notions he stated in 1817
regarding the exact least upper bound, and the sequence convergence criterion
[BolгКno, B. ReТn КnКlвЭТsМСer BeаeТs Нes LeСrsКЭгes, НКß гwischen zwey
АerЭСen, НТe eТn enЭРeРenРeseЭгЭes ResЮlЭКЭ РeатСren, аenТРsЭens eТne reelle
Wurzel der Gleichung liege. Prag: Gottlieb Haase. 1817. 60 s.]
1
. He
ТnЭroНЮМeН ЭСe noЭТon oП К meКsЮrКЛle nЮmЛer, relКЭТons НesМrТЛeН Кs “eqЮКl
Эo”, “РreКЭer ЭСКn”, “less ЭСКn”ν asserted density (pantachisch) of a set of real
numbers. Bolzano introduced infinitely great and infinitely small numbers. If
his manuscript were published and recognized by contemporaries, we would
have probably dealt with another kind of analysis, a nonstandard one.
Remember what Putnam said about the emergence of the epsilon-delta
language: IП ЭСe epsТlon-delta methods had not been discovered, then
ТnПТnТЭesТmКls аoЮlН СКЯe Лeen posЭЮlКЭeН enЭТЭТes (УЮsЭ Кs ‘ТmКРТnКrв’
1
This criterion was called after Cauchy, КlЭСoЮРС Тn BolгКno’s аorФs, ТЭ аКs stated in 1817, аСТle Тn CКЮМСв’s
works it was stated in 1821.
numbers were for a long time). Indeed, this approach to the calculus enlarging
the real number systemis just as consistent as the standard approach, as we
know today from the work of Abraham Robinson. If the calculus had not been
‘УЮsЭТПТeН’ АeТersЭrКss sЭвle, ТЭ аoЮlН СКЯe Лeen ‘УЮsЭТПТeН’ КnваКв” ДPЮЭnКm,
1974]. [13]. The treatment of numbers as variable quantities might have
sЭКrЭeН аТЭС LeТЛnТЭг’ КsserЭТon Эo ЭСe eППeМЭ ЭСКЭ
dx
dy
x
y
lim
. Along with
МonsЭКnЭ nЮmЛers, ЭСere Кre ЯКrТКЛle nЮmЛers Тn BolгКno’s аorФs, ЛoЭС
measurable and nonmeasurable. The limit or boundary of such numbers can
ЯКrТКЛle Кs аell. “IП К ЯКrТКЛle, СoаeЯer, meКsЮrКЛle nЮmЛer Y constantly
remains greater than a variable, however, measurable X, and moreover, Y has
no least value, and X has no greatest one, then there is at least one measurable
number A which constantly lies between the boundaries of X and Y.
If, further, the difference
cannot infinitely decrease, then there are
infinitely many such numbers lying between X and Y.
However, if this difference infinitely decreases, then there is one and only
one such number. And, finally, if the difference
XY
infinitely decreases and
either X has the largest or Y has the least value, then there is no measurable
number constantly lying between X and Y [14, p. 525]. There is already a
prescience of the notion of a cut here, which is going to appear in 1872 in
DeНeФТnН’s аorФs. HoаeЯer, Тn 1830, noЛoНв аКs КаКre oП ЭСe eбТsЭenМe oП
transcendental numbers. Bolzano was more of a philosopher; his ingenious
mathematical wisdom was not based on his professional activities, although
his philosophical approach to understanding of the continuity formed the line
of development of the concept of a number and continuity. The compelled
ban from teaching, the lack of academic intercourse and scientific literature
prevented his ideas from taking shape of an independent theory. However, his
ideas became part of the theory of functions and the theory of sets.
In 1840, G. Liouville started developing the notion and theory of
transcendental numbers. In 1844, he published a small article in Comptes
Rendus where he said that an algebraic number cannot be approximated by a
rational fraction [15]. His further research constituted the theory of
transcendental numbers. In 1873, Hermite proved the transcendence of
number e Д16]ν Тn 1882, LТnНemКn proЯeН ЭrКnsМenНenМe oП nЮmЛer π Д17]ν Тn
1885, his proof was simplified by Weierstrass [18].
However, the theory of a real number had not been created as yet. The
Эerms ‘гero’, ‘РreКЭer ЭСКn’, ‘less ЭСКn’, or ‘eqЮКl Эo гero’ МoЮlН noЭ Лe
rigorously defined. Therefore, in his lectures of 1861 in differential calculus,
АeТersЭrКss proЯeН ЭСe ЭСeorem Кs Пolloаsμ “A continuous function where a
derivative that is within certain intervals of arguments is always equal to zero
amounts to a constant” Д19, p. 118].
In 1869, a French mathematician Charles MцrКв
(1835-1911) developed
the theory of a real number [20]. Being based on converging sequences and
СКЯТnР ТnЭroНЮМeН К relКЭТon oП eqЮТЯКlenМe ЛeЭаeen ЭСem, MцrКв ТnЭroНЮМeН
the notion of a nonmeasurable number as a fictive limitμ “An ТnЯКrТКnЭ
converges to a certain fictive nonmeasurable limit if it converges to a point
which does not allow for a precise definition. If nonmeasurable limits of two
МonЯerРТnР ЯКrТКnЭs Кre eqЮКl, ЭСese ЯКrТКnЭs аТll Лe oП eqЮКl ЯКlЮe” Д21, p. 2].
MцrКв НeПТneН ЭСe relКЭТon oП МompКrТson oП КnН operКЭТon аТЭС
nonmeasurable numbers. However, his sophisticated language, clumsy terms;
СТs remoЭeness Пrom mКЭСemКЭТМКl lТПe oП PКrТs (MцrКв lТЯeН Тn К ЯТllКРe КnН
dealt in winegrowing for many years; thereafter, he gave lectures at Dijon
university), his utter antagonism to or lack of knowledge of achievements in
mathematics after Lagrange, put restrictions on his theory. He believed that
“one аТll neЯer Мome КМross НТsМonЭТnЮoЮs ПЮnМЭТons аСТМС СКЯe no
derivatives and are non-integrable, so no worries about them. There is no
point in addressing the Laplace equation, Dirichlet principle, because
derivatives are defined, calculated, stirred into differential expressions the
аКв LКРrКnРe аКnЭeН ЭСem Эo Нo, ЭСКЭ Тs, аТЭС ЭСe Сelp oП sТmple operКЭТons”
[22]. Therefore his contemporaries did not accept his theory, although now
ЭСeв МКll ЭСe МonМepЭ oП К reКl nЮmЛer ЭСe MцrКв–Cantor concept.
German mathematicians took the initiative of developing the concept of a
reКl nЮmЛer. In 1872, аorФs oП E. HeТne “LeМЭЮres on ЭСe TСeorв
FЮnМЭТons” Д23], G. CКnЭor “EбЭensТon of One Theorem from the Theory of
TrТРonomeЭrТМ SerТes” Д24, p. 9–17] КnН R. DeНeФТnН “ConЭТnЮТЭв КnН
IrrКЭТonКl NЮmЛers” Д25] аere pЮЛlТsСeН.
A professional mathematician and teacher, E. Heine explained the theory of
a number in terms of fundamental sequences, having introduced a relation of
eqЮТЯКlenМe КnН orНer. HТs eбplКnКЭТon СКН mЮМС Тn Мommon аТЭС CКnЭor’s
theory as it was developed in the course of joint discussions with him. He was
ahead of his colleagues methodically. Hitherto, the notion of a limit in
analysis had often been provided in terms of countable sequences [26, 27].
R. Dedekind approached to the definition of a number as an algebraist,
having provided an arithmetic definition of a number. Dedekind considered
properties of equality, ordering, density of the set of rational numbers R,
(numerical corpus, К Эerm ТnЭroНЮМeН Лв DeНeФТnН Тn sМСeНЮles Эo DТrТМСleЭ’s
lectures he published). In doing so, he tried to avoid geometric
represenЭКЭТons. HКЯТnР НeПТneН ЭСe relКЭТon oП ‘РreКЭer ЭСКn’ КnН ‘less ЭСКn’,
Dedekind proved its transitivity, existence of infinitely many other numbers
between two different numbers, and that for any number it is possible to make
a cut of a set of rational numbers into two classes, so that numbers of either
class are less than this number and numbers of another class are greater than
this one, in which event the very number which makes such cut may be
attributed to as one class or to the other, in which event it would be either the
greatest one for the first class or the least for the second one.
Subsequently, Dedekind considered points on a straight line and
established the same properties for them as he had just found for rational
numbers, thus stating that a point on a straight line corresponds to each
rational number. “EКМС poТnЭ p of a line separates the line into two parts, so
that each point of one part is located to the left of each point of the other part.
Now I perceive the essence of continuity in the opposite principle, i.e. it is as
Пolloаsμ “IП Кll poТnЭs oП К lТne ПКll ТnЭo sЮМС Эаo МlКsses ЭСКЭ eКМС poТnЭ oП ЭСe
first class lies left of each point of the second class, then there is one and only
one point which separates the line into two classes, and this is a cut of the line
into two fragments. If the system of all real numbers is separated into two
such classes that each number of the first class is less than each number of the
second class, then there is one and only one number making such cut” Д25, p.
1718].
Dedekind called this property of a line an axiom accepting which we make
the line continuous. In this case, Dedekind asserted that it was our mental act
which occurred whether the real space is continuous or discontinuous, that
such mental completion by new points did not affect the real existence of the
space.
TСe DeНeФТnН’s НeПТnТЭТon oП К nЮmЛer Тs sЭТll ЮseН in courses of analysis as
logically and categorically impeccable. However, as Cantor noted, one cannot
use the notion of a number as a cut in analysis. As soon as it concerns an
irregular set, this definition is useless.
Having introduced the notion of a number based on fundamental countable
sequences just as Heine, G. Cantor passed on Heine: he defined the notion of
a limiting point and introduced the hierarchy of limiting sets. In his work of
1874, Тn СТs КrЭТМle enЭТЭleН “оber eine Eigenschaft des Inbegriffes aller
reellen КlРeЛrКТsМen ГКСlen” [24, p. 1821], he proved the countability of a
set of algebraic irrational numbers and uncountability of a set of real numbers
and, therefore, transcendental numbers. Set-theoretic terminology had not
formed by that time, the notion of countability would appear in his works
later, so he spoke of one-to-one correspondence КnН ЮseН ЭСe Эerm ‘Inbegriff
(totality) instead of a set. Cantor postulated the one-to-one correspondence
between numbers and points on a line and asserted that this could not be
proved.
By 1878, Cantor switched from analysis of point spaces to the notion of
potency (power of set), formed a continuum hypothesis, considered
continuous mapping between sets of various dimensionalities. The more he
felt the insufficiency of the definition of continuity through the cut. In his
ЭСТrН КrЭТМle oП 1878 enЭТЭleН “EТn BeТЭrКР гЮr MКnnТРПКlЭТРФeТЭsleСre” (To the
Theory of Manifolds) [24, p. 2235], he already provided the notions of
potency and one-to-one correspondence between manifolds of various
dimension. It was in the same article that he introduced the notion of a
‘seМonН poЭenМв’, which means that the continuum hypothesis started to form.
See [28] for more detail. According to Cantor, a continuum of numbers is a
perfect connected well-ordered set.
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μ И  – 2012 . – . 180–185. Sinkevich G.
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// ,    
μ    .  18. 
 - .- . , . .. / . – . –
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28.  .И.  яя   
   / //  XI  
μ  . – μ И  – 2013 . – .
336347. Sinkevich G. Rasvitije poniatija nepreryvnosti v rabotach
Dedekinda I Cantora// Trudy XI Mieschdunarodnych Kolmogorovskich
chtenij. Jaroslavl, 2013 P. 336347.( The development of the notion of
continuity in DeНeФТnН’s КnН CКnЭor’s аorФs).
... Michael Stifel (1487-1567) was the first to define negative numbers as numbers less than zero and positive numbers as greater than zero. He was the first to describe zero, fractional and irrational numbers as numbers (Sinkevich, 2015). In his work Arithmetica Integra (1544) he has an amazing geometric awareness of the number. ...
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In modern mathematics curricula in primary and secondary education, number line is an important supervisory tool for understanding many concepts, such as different types of numbers, equations, and more. The use of the number line is supported by a large number of researchers, but there are also studies showing that students find it difficult to use. Although the concept of number line is important for teaching and there is a great deal of debate about its use, as far as we know, there are very few systematic studies that examine the epistemological development of some components regarding the concept of number line throughout history and correlate this development by learning this concept from the students. However, there are no studies that examine the concept of historical development of the number line as a whole or relate it to student behavior. In this paper, therefore, the first attempt has been made to examine the overall development of the concept of number line in the history of mathematics. We have therefore studied the historical evolution of the concept of number line and divided it into periods, according to the characteristics of this evolution. It seems that based on the slow mathematical integration of the concept of number line at the end of the 19th century, but also on some other critical points in the four historical periods that we have analyzed, some of the difficulties that students encounter when using it are likely to be epistemological obstacles.
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