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Experimental Mathematics


Experimental mathematics is a type of mathematical investigation in which computation is used to investigate mathematical structures and identify their fundamental properties and patterns. As in experimental science, experimental mathematics can be used to make mathematical predictions which can then be verified or falsified on the bases of additional computational experiments.

Borwein and Bailey (2003, pp. 2-3) use the term "experimental mathematics" to mean the methodology of doing mathematics that includes the use of computation for:

1. Gaining insight and intuition.

2. Discovering new patterns and relationships.

3. Using graphical displays to suggest underlying mathematical principles.

4. Testing and especially falsifying conjectures.

5. Exploring a possible result to see if it is worth formal proof.

6. Suggesting approaches for a formal proof.

7. Replacing lengthy hand derivations with computer-based derivations.

8. Confirming analytically derived results.

Examples of tools of experimental mathematics include computer algebra, symbolic algebra, Gröbner basis, integer relation algorithms (such as the LLL algorithm and PSLQ algorithm), arbitrary precision numerical evaluations, computer visualization, cellular automata and related structures, and databases of mathematical structures such as the Online Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences) by Neil Sloane, The Wolfram Functions Site (http://functions.wolfram.com) by Michael Trott and Oleg Marichev, and MathWorld (http://mathworld.wolfram.com) by Eric Weisstein.


See also

Arbitrary Precision, Cellular Automaton, Computational Algebra, Computer Algebra, Integer Relation, LLL Algorithm, New Kind of Science, Proof, PSLQ Algorithm, Simple Program, Symbolic Algebra, Triangle Geometry

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References

Bailey, D. H. and Borwein, J. M. "Sample Problems of Experimental Mathematics." 22 Sept. 2003. http://crd.lbl.gov/~dhbailey/expmath/expmath-probs.pdf.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M.; Borwein, P. B.; Girgensohn, R.; and Parnes, S. "Making Sense of Experimental Mathematics." Math. Intell. 18, 12-18, 1996. Reprinted in Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 243-265, 2003.Borwein, P. B. Computational Excursions in Analysis and Number Theory. New York: Springer-Verlag, 2002.Epstein, D. and Levy, S. "Experimentation and Proof in Mathematics." Not. Amer. Math. Soc. 42, 670-674, 1995.Experimental Mathematics. http://www.expmath.org/.Gibbs, W. W. "A Digital Slice of Pi. The New Way to do Pure Math: Experimentally." Sci. Amer. 288, 23-24, May 2003.Guénard, F. and Lemberg, H. La méthode expérimentale en mathématiques. Heidelberg, Germany: Springer-Verlag, 2001.Kimberling, C. "Encyclopedia of Triangle Centers." http://faculty.evansville.edu/ck6/encyclopedia/.Li, S.; Chen, F.; and Wu, Y.; and Zhang, Y. Mathematics Experiments. Singapore: World Scientific, 2003.Sloane, N. J. A. "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.Weisstein, E. W. "MathWorld." http://mathworld.wolfram.com/.Wolfram Institute. "The Wolfram Atlas of Simple Programs." http://atlas.wolfram.com/.Wolfram Research, Inc. "The Wolfram Functions Site." http://functions.wolfram.com/.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 337-342, 2002.

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Experimental Mathematics

Cite this as:

Weisstein, Eric W. "Experimental Mathematics." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExperimentalMathematics.html

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