Editing Characterization (mathematics) (section)

==Examples==
* A [[rational number]], generally defined as a [[ratio]] of two integers, can be characterized as a number with finite or repeating [[decimal expansion]].<ref name=":0" />
*A [[parallelogram]] is a [[quadrilateral]] whose opposing sides are parallel. One of its characterizations is that its diagonals bisect each other. This means that the diagonals in all parallelograms bisect each other, and conversely, that any quadrilateral whose diagonals bisect each other must be a parallelogram.
* "Among [[probability distribution]]s on the interval from 0 to ∞ on the real line, [[memorylessness]] characterizes the [[exponential distribution]]s." This statement means that the exponential distributions are the only probability distributions that are memoryless, provided that the distribution is continuous as defined above (see [[Characterization of probability distributions]] for more).
* "According to [[Bohr–Mollerup theorem]], among all functions ''f'' such that ''f''(1) = 1 and ''x f''(''x'') = ''f''(''x'' + 1) for ''x'' > 0, log-convexity characterizes the [[gamma function]]."  This means that among all such functions, the gamma function is the ''only'' one that is [[log-convex]].<ref>A function ''f'' is ''log-convex'' [[Iff|if and only if]] log(''f'') is a [[convex function]]. The base of the logarithm does not matter as long as it is more than 1, but mathematicians generally take "log" with no subscript to mean the [[natural logarithm]], whose base is ''e''.</ref>
* The circle is characterized as a [[manifold]] by being one-dimensional, [[compact space|compact]] and [[connected space|connected]]; here the characterization, as a smooth manifold, is [[up to]] [[diffeomorphism]].