A fallacy is an incorrect result arrived at by apparently correct, though actually specious reasoning. The great Greek geometer Euclid wrote an entire book on geometric fallacies which, unfortunately, has not survived (Gardner 1984, p. ix).

The most common example of a mathematical fallacy is the "proof" that 1=2 as follows. Let a=b, then


The incorrect step is (4), in which division by zero (a-b=0) is performed, which is not an allowed algebraic operation. Similarly flawed reasoning can be used to show that 0=1, or any number equals any other number.

Ball and Coxeter (1987) give other such examples in the areas of both arithmetic and geometry.

See also

Dissection Fallacy, Division by Zero

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Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 41-45 and 76-84, 1987.Barbeau, E. J. Mathematical Fallacies, Flaws, and Flimflam. Washington, DC: Math. Assoc. Amer., 1999.Bogomolny, A. "Fallacies.", M. "Fallacies." Ch. 14 in The Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon and Schuster, pp. 141-150, 1959.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, 1984.Pappas, T. "Geometric Fallacy & the Fibonacci Sequence." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 191, 1989.

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Cite this as:

Weisstein, Eric W. "Fallacy." From MathWorld--A Wolfram Web Resource.

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