Axiom

Pronunciation: /ˈæksiəm/ ?

An axiom is a statement that is taken to be true without proof^[1]. An axiom is taken to be self-evidently true, or obviously true. In contrast, a conjecture and a hypothesis are believed to be true, but are not self-evident.

In math, all proofs start directly or indirectly with one or more axioms. The word postulate means the same thing as axiom. An axiomatic system is a logical system that is based on axioms.

Examples

Name	Representation	Description
Betweenness Assumption	Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 1: Betweeness Assumption Created with GeoGebra.	Given collinear points A, B, and C, C is between A and B if AC + CB = AB. Click on the blue points in manipulative 1 and drag them to change the figure. Notice that if C is between A and B that AC + CB = AB. If C is not between A and B then AC + CB > AB.
Angle Addition Postulate	Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 2: Angle addition postulate. Created with GeoGebra.	If two angles are adjacent, they can be added together to form a larger angle. The measure of the larger angle is the sum of the measures of the two smaller angles. Click on the blue points in manipulative 2 and drag them to change the figure. Notice that m∠BAC + m∠CAD = m∠BAD.
Table 1: Examples of axioms

References

1. axiom. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=axiom&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
2. Mendell, Henry. Aristotle and Mathematics, sect 2. Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, CSLI, Stanford University. (Accessed: 2009-12-16). http://plato.stanford.edu/entries/aristotle-mathematics/#2.
3. Keller, Samuel Smith. Mathematics for Engineering Students, Plane and Solid Geometry, pg 3. D. Van Nostrand Company, 1908. (Accessed: 2010-01-02). http://www.archive.org/stream/mathengineer00kellrich#page/3/mode/1up/search/axiom.

More Information

• McAdams, David. Euclidean Geometry. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem.org. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Euclidean Geometry.
• Peil, Timothy. Introduction to Axiomatic Systems. 2010-01-16. http://www.mnstate.edu/peil/geometry/C1AxiomSystem/AxiomaticSystems.htm.

Printed Resources

Cite this article as:

Axiom. 2010-01-01. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/a/axiom.html.

Translations

• Chasque para ver este artículo en español.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem.org and are licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Revision History

2010-01-01: Added "References" (McAdams, David.)
2008-10-26: Changed manipulatives from sketchpad to geogebra (McAdams, David.)
2008-10-10: Changed text to compare hypothesis and conjecture, expanded 'More Information' (McAdams, David.)
2008-07-08: Added axiomatic system (McAdams, David.)
2008-04-07: Added More Information (McAdams, David.)
2008-03-25: Added examples (McAdams, David.)
2007-09-03: Added synonym 'postulate' (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)