A description of an object by properties that are different from those mentioned in its definition, but are equivalent to them. The following list gives a number of examples.
1. A rational number is defined as the quotient of two integers, but it can be characterized as a number admitting a finite or repeating decimal expansion.
2. An equilateral triangle is defined as a triangle having three equal sides, but it can be characterized as a triangle having
two angles of 
.
3. A real square matrix is nonsingular, by definition, if it admits a matrix inverse, but it can be characterized by the condition that its determinant be nonzero.
Of course, a characterization should not merely be a rephrasing of the definition, but should give an entirely new description, which is useful because it contains a simpler formulation, can be verified more easily, is interesting because it places the object in another context, or unveils unexpected links between different areas.
A special type of characterization is classification, which translates an abstract property into a complete list of examples and models.
