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. McAdams, David E.. All Math Words Dictionary, contradiction. 2nd Classroom edition 20150108-4799968. pg 45. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs. 3rd edition. pp 22-24. Academic Press. August 15, 2005. Last Accessed 6/25/2018. Buy the book
  3. Larry W. Cusick. Proofs by Contradiction. California State University, Fresno. Last Accessed 6/25/2018.

More Information

  • Euclid of Alexandria. Elements. Clark University. 9/6/2018.

Cite this article as:

McAdams, David E. Contradiction. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
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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