Infinity

An unbounded number greater than every Real Number, most often denoted as $\infty$. The symbol $\infty$ had been used as an alternative to M (1,000) in Roman Numerals until 1655, when John Wallis suggested it be used instead for infinity.

Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results which follow from Georg Cantor's treatment of Infinite Sets. Informally, $1/\infty=0$, a statement which can be made rigorous using the Limit concept,

$$\lim_{x\to\infty} {1\over x}=0.$$

Similarly,

$$\lim_{x\to 0^+} {1\over x}=\infty,$$

where the notation $0^+$ indicates that the Limit is taken from the Positive side of the Real Line.

See also Aleph, Aleph-0, Aleph-1, Cardinal Number, Continuum, Continuum Hypothesis, Hilbert Hotel, Infinite, Infinite Set, Infinitesimal, Line at Infinity, L'Hospital's Rule, Point at Infinity, Transfinite Number, Uncountably Infinite Set, Zero

References

Infinity

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 19, 1996.

Courant, R. and Robbins, H. ``The Mathematical Analysis of Infinity.'' §2.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 77-88, 1996.

Hardy, G. H. Orders of Infinity, the `infinitarcalcul' of Paul Du Bois-Reymond, 2nd ed. Cambridge, England: Cambridge University Press, 1924.

Lavine, S. Understanding the Infinite. Cambridge, MA: Harvard University Press, 1994.

Maor, E. To Infinity and Beyond: A Cultural History of the Infinite. Boston, MA: Birkhäuser, 1987.

Moore, A. W. The Infinite. New York: Routledge, 1991.

Morris, R. Achilles in the Quantum Universe: The Definitive History of Infinity. New York: Henry Holt, 1997.

Péter, R. Playing with Infinity. New York: Dover, 1976.

Smail, L. L. Elements of the Theory of Infinite Processes. New York: McGraw-Hill, 1923.

Vilenskin, N. Ya. In Search of Infinity. Boston, MA: Birkhäuser, 1995.

Wilson, A. M. The Infinite in the Finite. New York: Oxford University Press, 1996.

Zippin, L. Uses of Infinity. New York: Random House, 1962.