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method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GeneralDiscreteMathematics.html">General Discrete Mathematics</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Renze.html">Renze</a> </li> </ul></nav>   <h1>Discrete Mathematics</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, <a href="/Calculus.html">calculus</a>). Whereas discrete objects can often be characterized by <a href="/Integer.html">integers</a>, continuous objects require <a href="/RealNumber.html">real numbers</a>. </p> <p> The study of how discrete objects combine with one another and the probabilities of various outcomes is known as <a href="/Combinatorics.html">combinatorics</a>. Other fields of mathematics that are considered to be part of discrete mathematics include <a href="/GraphTheory.html">graph theory</a> and the <a href="/TheoryofComputation.html">theory of computation</a>. Topics in <a href="/NumberTheory.html">number theory</a> such as <a href="/Congruence.html">congruences</a> and <a href="/RecurrenceRelation.html">recurrence relations</a> are also considered part of discrete mathematics. </p> <p> The study of topics in discrete mathematics usually includes the study of <a href="/Algorithm.html">algorithms</a>, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades. </p> <p> The related branch of mathematics known as <a href="/ConcreteMathematics.html">concrete mathematics</a>, while having some overlap with discrete mathematics, includes a quite different set of topics (Graham <i>et al. </i>1994, p. vi). </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/Algorithm.html">Algorithm</a>, <a href="/AutomataTheory.html">Automata Theory</a>, <a href="/ConcreteMathematics.html">Concrete Mathematics</a>, <a href="/Combinatorics.html">Combinatorics</a>, <a href="/Congruence.html">Congruence</a>, <a href="/DiscreteDistribution.html">Discrete Distribution</a>, <a href="/DiscreteFourierTransform.html">Discrete Fourier Transform</a>, <a href="/DiscreteGeometry.html">Discrete Geometry</a>, <a href="/DiscreteLogarithm.html">Discrete Logarithm</a>, <a href="/GeneratingFunction.html">Generating Function</a>, <a href="/GraphTheory.html">Graph Theory</a>, <a href="/Mathematics.html">Mathematics</a>, <a href="/RecurrenceRelation.html">Recurrence Relation</a>, <a href="/TheoryofComputation.html">Theory of Computation</a> <a href="/classroom/DiscreteMathematics.html" class="explore-classroom">Explore this topic in the MathWorld classroom</a>     <p class="contributor"> <i>Portions of this entry contributed by <a target="_blank" href="/topics/Renze.html">John Renze</a></i> </p>   <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=discrete+mathematics"> discrete mathematics </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=20th+Mersenne+prime">20th Mersenne prime</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=Dynamic+options">Dynamic options</a></li> </ul> </div> </div>   <h2>References</h2><cite>Balakrishnan, V. K. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486691152/ref=nosim/ericstreasuretro">Introductory Discrete Mathematics.</a></i> New York: Dover, 1997.</cite><cite>Bobrow, L. S. and Arbib, M. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0721617689/ref=nosim/ericstreasuretro">Discrete Mathematics: Applied Algebra for Computer and Information Science.</a></i> Philadelphia, PA: Saunders, 1974.</cite><cite>Dossey, J. A.; Otto, A. D.; Spence, L.; and Eynden, C. V. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0673980391/ref=nosim/ericstreasuretro">Discrete Mathematics, 3rd ed.</a></i> Reading, MA: Addison-Wesley, 1997.</cite><cite>Graham, R. L.; Knuth, D. E.; and Patashnik, O. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201558025/ref=nosim/ericstreasuretro">Concrete Mathematics: A Foundation for Computer Science, 2nd ed.</a></i> Reading, MA: Addison-Wesley, 1994.</cite><cite>Hall, C. and O'Donnell, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1852330899/ref=nosim/ericstreasuretro">Discrete Mathematics Using a Computer.</a></i> London: Springer-Verlag, 2000.</cite><cite>Lipschutz, S. and Lipson, M. L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0070380317/ref=nosim/ericstreasuretro">2000 Solved Problems in Discrete Mathematics.</a></i> New York: McGraw-Hill, 1991.</cite><cite>Lipschutz, S. and Lipson, M. L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0070380317/ref=nosim/ericstreasuretro">Schaum's Outline of Discrete Mathematics, 2nd ed.</a></i> New York: McGraw-Hill, 1997.</cite><cite>Rosen, K. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0072899050/ref=nosim/ericstreasuretro">Applications of Discrete Mathematics, 4th ed.</a></i> New York: McGraw-Hill, p. 1998.</cite><cite>Rosenstein, J. G.; Franzblau, D. S.; and Roberts, F. S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821804480/ref=nosim/ericstreasuretro">Discrete Mathematics in the Schools.</a></i> Providence, RI: Amer. Math. Soc., 1997.</cite><cite>Skiena, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201509431/ref=nosim/ericstreasuretro">Implementing Discrete Mathematics.</a></i> Reading, MA: Addison-Wesley, 1990.</cite><cite>Weisstein, E. W. "Books about Discrete Mathematics." <a href="http://www.ericweisstein.com/encyclopedias/books/DiscreteMathematics.html">http://www.ericweisstein.com/encyclopedias/books/DiscreteMathematics.html</a>.</cite><cite>Wolfram, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1579550088/ref=nosim/ericstreasuretro">A New Kind of Science.</a></i> Champaign, IL: Wolfram Media, 2002.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/discrete_mathematics/82/69/ug/" title="Discrete Mathematics" target="_blank">Discrete Mathematics</a>   <h2>Cite this as:</h2> <p> <a href="/topics/Renze.html">Renze, John</a> and <a href="/about/author.html">Weisstein, Eric W.</a> "Discrete Mathematics." 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