Pronunciation: /ˌkɒn trəˈdɪk ʃən/ Explain

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. Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs. 3rd edition. pp 22-24. Academic Press. August 15, 2005. Last Accessed 8/6/2018. Buy the book
  2. Larry W. Cusick. Proofs by Contradiction. California State University, Fresno. Last Accessed 8/6/2018.

Cite this article as:

McAdams, David E. Contradiction. 6/29/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/5/2010: Added "References". (McAdams, David E.)
6/16/2008: Added text on 'indirect proof' and 'more information'. (McAdams, David E.)
9/17/2007: Initial version. (McAdams, David E.)

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