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Topic: Re: Number line
Replies: 2   Last Post: Feb 27, 2009 6:29 AM

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Kutler, Samuel

Posts: 5
Registered: 4/2/07
Re: Number line
Posted: Feb 27, 2009 6:29 AM
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Yes, Jacob Klein himself was, and his book is, well known to me.

-----Original Message-----
From: Humberto Bortolossi []
Sent: Wed 2/25/2009 9:06 AM
Subject: Number line


I've also tried to find out who is the father of "number line" idea.
Apparently, there are two candiates: Stevin and Wallis. Below are some
excerpts of books and articles about the subject. It seems that classical
books on history of mathematics (Boyer, Cajori, Eves) don't discuss the

Best regards, Humberto.

*Simon Stevin*,M1
Chapter 1

Transformation of The Number Concept

In the early modern period a crucial transformation occurred in the
conception of number and magnitude(1). For the Greeks, the unit, or one, was
not a
number; it was the begnning of number and it was used to measure a
Numbers were merely collections of discrete units that measured some
Magnitude, or the other hand, was usually described as being continuous, or
divisible int parts that are infinitely divisible. A distinction was made
arithmetic and geometry; arithmetic dealt with discrete or unextended
quantity while
geometry dealt with continuous or extended quantity(2). In the early modern
period a
transformation occurred in this classical conception of number and
magnitude. For
instance, Simon Stevin (1548-1620) insisted in this 1585 Arithmétique that
traditional Greek notion of numbers was wrong: he believed that numbers wre
continuous rather discrete(3). Stevin also developed a system for indefinite
expansions of number that implicitly contained the idea of a numerical
Moreoever, François Viète (1540-1603) introduced an improved form of
unknowns where distinguished from given magnitudes by being labelled with
vowels while the given magnitudes were denoted by captial consonants. This
new symbolism was used to denote both unknown magnitudes and numbers; it
that numbers and magnitudes where, in a sense, interchangeable(5). This
association of
numbers with magnitudes encouraged the notion that numbers could also be
treated as
though they were continuous in the Aristotelian sense. Thus the traditional
idea of
discrete numbers versus continuous magnitude was challenged in the early
period in several ways. Stevin belonged to a tradition of computational
that had a practical, problem-solving orientation. These practitioners were
of the more theoretically oriented mathematics of the humanists(6). The
demands mande
by bookkeeping, accounting, measuing, navigation, and war had created
with traditional methods. Stevin began this careers as a bookkeeper; his
first work,
publish in 1582, was titled Tables of Interest. This work, which includes
examples of commom business problems as well as tables of interest, hnows
mercantile computations were becoming quite complex(7). Stevin's 1585
edition of The
Disme, dedicated to the wish that "any man's business may be performed
thereby," was motivated in part by a desire to ease practical compuattions
(8). Although
Stevin's development of an indefinite decimal expansion of number (with its
idea of a numerical continuum) did not explicit become a part of the
discussions about mathematical continua in Europe in this period, it
certainly advanced
the notion of numbers being continuous rather than discrete.

(4) Meaning in this case the Stevin noted that numbers can be pictured along
a number line.

Dear Swapna,
The development of the number line is really the development, in the 16th
and 17th centuries, of the idea of number itself . In this connection,
certainly Descartes stands out (see not only La Geometrie but also the
Regulae, esp. reg.xviii). But perhaps a more important and earlier figure
in this story is Simon Stevin. Unlike the Greeks who saw the unit as the
basis of all numbers (as in Euclid's definition), Stevin took "0 as the true
and natural beginning" for number. Thus, for Stevin, the principle for
number was likened to the principle for lines or other continuous
magnitudes--and Stevin says this explicitly: "As to a continuous body of
water corresponds a continuous wetness, so to a continuous magnitude
corresponds a continuous number."

You might look at J. Klein's Greek Mathematical Thought and the Origin of
Algebra for a more detailed and very deep discussion of these issues.

Dr. Michael N. Fried
Graduate Program for Science and Technology Education
Ben Gurion University of the Negev
P.O. Box 653, Beer Sheva
Kibbutz Revivim
D. N. Halutsa 85515

This flow approach seems similar to our number line, which may have begun
with Flemish mathematician Simon Stevin 1548-1620.

Stevin's 1585 publication used a notation to represent decimal places, where
a Roman numeral I used as a superscript places that number in the tenths
spot, a Roman numeral II in the hundredths sport, etc. By 1653 that had
evolved into the use of a colon as the radix, separating the whole number
digits from the fractional digits of the number. Later it would evolve into
the decimal point, changing from 61:876 to 61.876, although the bible still
uses the colon to represent its chapters and passages; John 5:5. Perhaps our
use of time and goegrpahic latitude and longitude are also remnants?

The 1637 Cartesian coordinate system employs two number lines at right
angles. The continuum of numbers opens a new world of possibilities. Finding
points on number lines is more fascinating than it seems, especially in
bases other than ten. Edward works through several examples in binary and

*John Wallis*

It is worth noting here that the earliest demonstration of a number line is
usually credited to John Wallis, for a presentation in his "Treatise on
Algebra", in 1685. Wallis' intention in the chapter was to develop a
geometric algebra to deal with "negative squares and imaginary roots" and to
lay the foundation he described how negative numbers could have meaning in
representing a different "sense" than a positive number. Here are his words,
as gratiously explained to me by Professor Phillip Beeley, who works at The
Wallis Project at the University of Oxford Centre for Linguistics and
Philology. (*That meets my standard for someone in a postion to know*) .

"As for instance: Supposing a man to have advanced or moved forward, (from A
to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked,
how much he had advanced (upon the whole march) when at C? Or how many yards
he is now forwarder than when he was at A? I find (by subducting 2 from 5,)
that he is advanced 3 yards. (Because +5 -2 = +3.)


But if, having advanced 5 yards to B, he thence retreat 8 yards to D; and it
be then asked, How much is he advanced when at D, or how much forwarder than
when he was at A: I say -3 yards. (Because +5 -8 = -3.) That is to say, he
is advanced 3 yards less than nothing."

He was also kind enough to send an image of the page. I have captured the
relevant section below.

Wallis then draws a similar analogy from the line to the plane, and begins
his assault on the complex numbers. For those who wish to see the full page
and preceeding page, which Professor Beeley copied for me, the pdf file is here


For some unknown reason several historians of mathematics misunderstood
Wallis as if he claimed that negative numbers in itself were greater than
infinity. William Rouse Ball (1912, 293) writes "It is curious to note that
Wallis rejected as absurd the now usual idea of a negative number as being
less than nothing, but accepted the view that it is something greater than
infinity". Wallis did not reject at all numbers less zero. In fact, Wallis
can be considered as the inventor of the number line for negative
quantities. Morris Kline (1972; 1990, 253) possibly inspired by Ball also
completely misses the point: "Though Wallis was advanced for his times and
accepted negative numbers, he thought they were larger than infinity but not
less than zero". Some years later in his Loss of Certainty he writes (1983,
116): "Though Wallis was advanced for his times and accepted negative
numbers, he thought they were larger than infinite as well as less than 0".

We find the same misunderstanding about Euler in his Latin text De seriebus
divergentibus [E247] written in 1746, but not read to the Academy until
1754, and only published in 1760. Euler's observations are based on the
expansion posed by Leibniz in 1713 in which 1/(1-x)= 1 + x + x^2 + .... With
x = 2 you arrive at 1/(-1) = 1 + 2 + 4 + 8 .. (1) which according to Euler
is greater than infinity. He then uses an argument analogous to Wallis:
"This can be confirmed by the following example of a sequence of fractions:
1/4, 1/3, 1/2, 1/1, 1/0, 1/(-1), 1/(-2) ..." Now again the idea that
dividing a number by a negative one leads to something larger than infinity
has been systematically been misunderstood. Kline writes "Euler, the
greatest eighteenth-century mathematician believed that negative numbers are
greater than infinity" (Kline 1981, 52) and later he later repeated "Euler
concluded that ? 1 is larger than infinity" Kline (1983, 144). Sandiger
(2006, 179) "Euler is claiming that numbers greater than infinity are the
same as numbers smaller than zero" and recently William Dunham (2007, 138)
Euler "is willing to accept that 'the same quantities which are less than
zero can be considered to be greater than infinity'". Despite the last
quote, Wallis or Euler never claimed that negative numbers are greater than
infinity. The misunderstanding becomes apparent from an article by Kline
(1983) on Euler. Instead of expression (1) Kline writes that Euler obtained
-1 = 1 + 2 + 4 + 8 .. But that is taken already for granted that 1/(-1) = -1
which is precisely the identity questioned by Wallis and Euler. In fact,
Euler had no problems at all with negative numbers. In his book on
elementary algebra he writes that "we may say that negative numbers are less
than nothing" (Euler 1822, 5) and he explains so by enumerating the negative
numbers from zero "in the opposite direction, by perpetually subtracting
unity", de facto endorsing the number line.

Ball, Walter William Rouse, 1912, A Short Account of the History of
Mathematics, London, Macmillan and co. (Dover reprint, 1960)

Dunham, William, 2007, The Genius of Euler: Reflections on His Life and
Work, Mathematics Association of America, Washington.

Euler, Leonhard, 1754/55, De seriebus divergentibus, Novi Commentarii
academiae scientiarum Petropolitanae 5, (1760, p. 205-237), reprinted in
Opera Omnia I, vol. 14, p. 585-617.

Kline, Morris, 1959, Mathematics and the Physical World, New York: Crowell
Dover reprint 1981).

Kline, Morris, 1972, Mathematical Thought from Ancient to Modern Times,
Oxford: Oxford University Press, (reprinted in 3 vols. 1990).

Kline, Morris, 1980, Mathematics: The Loss of Certainty, Oxford: Oxford
University Press.

Kline, Morris, 1983, "Euler on Infinite Series", Mathematics Magazine, 56
(5), pp. 307-314.

Sandiger, Edward C. 2006, How Euler Did It, Mathematics Association of
America, Washington.

---------- Forwarded message ----------
From: Kutler, Samuel <>
Date: Mon, Feb 23, 2009 at 9:00 AM
Subject: Re: Number line

The number line, in my opinion, is our chief image
in mathematics to our so called number systems,
and did it have to wait until Dedekind to be such?
Insofar as Descartes is our father,
he sets the stage for it,
but it is unlike him to state it clearly.
Of course, Gauss and 2 others
almost simultaneously
introduced us to the complex PLANE.

-----Original Message-----
From: James A. Landau <> [mailto:]
Sent: Sun 2/22/2009 6:16 PM
Subject: Re: Number line

> --- wrote:
> From: "Kutler, Samuel" <>
> Subject: RE: Sir Andrew Wiles
> Date: Mon, 16 Feb 2009 06:23:13 -0500
> When did the term real number line appear, and when did someone say that
> there is a one-to-one correspondence between the real numbers and the
>points on the real number line?

Dedekind wrote about the number line in 1872. From the paper "Continuity
and Irrational Numbers" in _Essays on the Theory of Numbers_ by Richard
Translated from the German by Wooster Woodruff Beman
Chicago: Open Court Publishing Company, 1901

available on-line at,M1

This analogy between rational numbers and the points of a straight line, as
is well known, becomes a real correspondence when we select upon the
straight line a definite origin or zero-point o and a definite unit of
length for the measurement of segments. With the aid of the latter to every
rational number a a corresponding length can be constructed and if we lay
this off upon the straight line to the right or left of o according as a is
positive or negative, we obtain a definite end-point p, which may be
regarded as the point corresponding to the number a ; to the rational number
zero corresponds the point o. In this way to every rational number a, i. e.,
to every individual in R, corresponds one and only one point/, i. e., an
individual in L. To the two numbers a, b respectively correspond the two
points /, q, and if a~>b, then / lies to the right of q.

Dedekind in this paper did not go on to describe a one-to-one correspondence
between the real numbers and the line, but such an idea is an obvious
extension. However, Dedekind was interested not in discussing geometry but
in introducing what we call the "Dedekind cut", which he describes in terms
of cutting a line into two pieces.

Dedekind and Cantor seem to have influenced each other. Note the following
quote from the preface of the same paper:

While writing this preface (March 20, 1872), I am just in receipt of the
interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der
trigonometrischen Reihen, by G. Cantor (Math. Annalen, Vol. 5), for which I
owe the ingenious author my hearty thanks.

James A. Landau
Test Engineer
Northrop-Grumman Information Technology
8025 Black Horse Pike, Suite 300
West Atlantic City NJ 08232 USA

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