Impropers

Impropers is a term used in mathematics to describe objects that do not meet a context-specific criterion of “properness.” Rather than denoting a single defined class, impropers refer to several related ideas that arise when a standard condition is not satisfied, and the exact meaning depends on the mathematical setting.

One common use is in fractions. An improper fraction is a rational number in which the numerator is greater than or equal to the denominator. For example, 7/4 and 5/5 are improper fractions. They can be converted to mixed numbers, such as 7/4 = 1 and 3/4. Proper fractions, by contrast, have a numerator smaller than the denominator. Improper fractions are often convenient for arithmetic, while mixed numbers are sometimes preferred for interpretation.

Another familiar context is the improper integral. An improper integral is an integral where either the interval of integration is unbounded or the integrand has a nonremovable infinite discontinuity on the interval. Such integrals are defined as limits, for example ∫ from a to ∞ f(x) dx = lim t→∞ ∫ from a to t f(x) dx, or ∫ from a to b f(x) dx with a vertical asymptote at c, defined as a limit that excludes the singular point. Convergence or divergence of these limits determines the value of the improper integral.

In probability and statistics, the term appears in the notion of improper priors. An improper prior is a prior distribution that does not integrate to 1 but can be used in formal Bayesian analysis if the resulting posterior distribution is proper. Overall, impropers describe cases where a standard, well-behaved condition does not hold, with the specific interpretation depending on the mathematical framework.