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A limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. For example, the Point is a degenerate case of the Circle as the Radius approaches 0, and the Circle is a degenerate form of an Ellipse as the Eccentricity approaches 0. Another example is the two identical Roots of the second-order Polynomial $(x-1)^2$. Since the $n$ Roots of an $n$th degree Polynomial are usually distinct, Roots which coincide are said to be degenerate. Degenerate cases often require special treatment in numerical and analytical solutions. For example, a simple search for both Roots of the above equation would find only a single one: 1.

The word degenerate also has several very specific and technical meanings in different branches of mathematics.


Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 513-514, 1985.

© 1996-9 Eric W. Weisstein