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Re: Number Line
Posted:
Jan 19, 2008 7:18 AM
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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.
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