Yes, Jacob Klein himself was, and his book is, well known to me.
-----Original Message----- From: Humberto Bortolossi [mailto:email@example.com] Sent: Wed 2/25/2009 9:06 AM To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG 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 subject.
In the early modern period a crucial transformation occurred in the classical 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 multitude. Numbers were merely collections of discrete units that measured some multple. Magnitude, or the other hand, was usually described as being continuous, or being divisible int parts that are infinitely divisible. A distinction was made between 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 the traditional Greek notion of numbers was wrong: he believed that numbers wre continuous rather discrete(3). Stevin also developed a system for indefinite decimal expansions of number that implicitly contained the idea of a numerical continuum(4). Moreoever, François Viète (1540-1603) introduced an improved form of symbolism: unknowns where distinguished from given magnitudes by being labelled with capital vowels while the given magnitudes were denoted by captial consonants. This powerful new symbolism was used to denote both unknown magnitudes and numbers; it indicated 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 modern period in several ways. Stevin belonged to a tradition of computational mathematics that had a practical, problem-solving orientation. These practitioners were independent of the more theoretically oriented mathematics of the humanists(6). The demands mande by bookkeeping, accounting, measuing, navigation, and war had created dissatisrfaction 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 many examples of commom business problems as well as tables of interest, hnows that 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 easily 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 implicit idea of a numerical continuum) did not explicit become a part of the theoretical 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.
------------------------------------------------------------------------------------------------------------------------------------- http://mathforum.org/kb/message.jspa?messageID=1186247&tstart=0 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 ISRAEL firstname.lastname@example.org & Kibbutz Revivim D. N. Halutsa 85515 ISRAEL email@example.com
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 ternary.
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.)
D A C B |...|...|...|---|---|---|---|---|
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 <http://www.pballew.net/wallispg.pdf>.
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 <Samuel.Kutler@sjca.edu> Date: Mon, Feb 23, 2009 at 9:00 AM Subject: Re: Number line To: MATH-HISTORY-LIST@enterprise.maa.org
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 <JJJRLandau@netscape.com> [mailto: JJJRLandau@netscape.com] Sent: Sun 2/22/2009 6:16 PM To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG Subject: Re: Number line
> --- Samuel.Kutler@sjca.edu wrote: > > From: "Kutler, Samuel" <Samuel.Kutler@sjca.edu> > To: <MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG> > 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 Dedekind Translated from the German by Wooster Woodruff Beman Chicago: Open Court Publishing Company, 1901
<quote> 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. </quote>
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:
<quote> 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. </quote>
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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