Pronunciation: /ˈfæl ə si/ ?
Plural - fallacies

A fallacy is a logical argument that appears true but contains errors in logic. One common fallacy in mathematics is dividing by zero. The logic in proof 1 seems to prove that 1 = 2. This is done by dividing by zero.

Proof 1: 1 = 2
10 × 1 = 0
0 × 2 = 0
These are the criteria of the proof.
20 × 1 = 0 × 2Apply the transitive property of equality.
3( 0 × 1 ) / 0 = ( 0 × 2 ) / 0Divide both sides by the same quantity. The fallacy is in this step. Division by zero is undefined. In reality both sides of the equation are undefined. Since an undefined is involved, no conclusion can be made.
41 = 2Cancel the zeros.


  1. fallacy. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=fallacy&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
  2. Carroll, Lewis. The Game of Logic, pop 32-36. Macmillan and Company, 1887. (Accessed: 2010-02-04). http://www.archive.org/stream/gameoflogic00carruoft#page/32/mode/1up.
  3. True, Charles K. The elements of logic, adapted to the capacity of younger students, and designed for academies and the higher classes of common schools, revised edition, pp 95-106. Carlton & Porter, 1861. (Accessed: 2010-02-04). http://www.archive.org/stream/elementsoflogica00trueuoft#page/95/mode/1up/search/fallacy.
  4. Nathaniel Bluedorn, Hans Bluedorn, and Johannah Bluedorn. The Fallacy Detective: Thirty-Six Lessons on How to Recognize Bad Reasoning, 2nd edition. Christian Logic, June 30, 2002.

More Information

  • Classic Fallacies. University of Toronto Mathematics Network. 2009-03-12. http://www.math.toronto.edu/mathnet/plain/falseProofs/fallacies.html.

Printed Resources

Cite this article as:

Fallacy. 2010-02-04. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/fallacy.html.


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Revision History

2010-02-04: Added "References" (McAdams, David.)
2008-12-29: Added vocabulary link to 'transitive property of equality' (McAdams, David.)
2008-08-01: Added vocabulary links 'apply', 'cancel' and 'undefined' (McAdams, David.)
2008-06-05: Initial version (McAdams, David.)

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