Pronunciation: /ˌkɒn trəˈdɪk ʃən/ ?

A contradiction is a statement that is necessarily false.[1] For example, if a proof starts with the assumption that line segment AB is less than line segment CD, and later concludes that AB = CD, both statements can not be true at the same time. One statement contradicts the other.

A proof by contradiction starts with a statement that is to be disproved. It then proceeds to show the statement false by arriving at a contradiction. An example of a proof by contradiction is Euclid's proof that if two angles are equal, then the sides opposite the equal angles are also equal. A proof by contradiction can also be called an indirect proof, or reductio ad absurdum (Latin for "reduction to the absurd").


  1. contradiction. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. contradict. WordNet. Princeton University. (Accessed: 2011-01-08).
  3. Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs, 3rd edition, pp 22-24. Academic Press. (Accessed: 2010-01-11).
  4. Larry W. Cusick. Proofs by Contradiction. California State University, Fresno. (Accessed: 2010-01-20).
  5. Henry Cohn. Dangers of proof by contradiction (advice for amateurs or beginners). Microsoft Research New England. (Accessed: 2010-01-20).

Printed Resources

Cite this article as:

Contradiction. 2010-01-05. All Math Words Encyclopedia. Life is a Story Problem LLC.


Image Credits

Revision History

2010-01-05: Added "References" (McAdams, David.)
2008-06-16: Added text on 'indirect proof' and 'more information' (McAdams, David.)
2007-09-17: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License