nonsimplifiable

Nonsimplifiable is an adjective used to describe an object that cannot be simplified under a given set of rules or operations. In mathematics and related fields, it denotes a state in which the form has been reduced as far as the applicable simplification process allows.

In fractions, a fraction is nonsimplifiable when the numerator and denominator share no common factor other than one. Equivalently, the numerator and denominator are coprime, and the fraction is in lowest terms. For example, 7/12 is nonsimplifiable, while 8/12 can be simplified to 2/3.

In algebra and symbolic manipulation, an expression or form may be described as nonsimplifiable if no further allowed transformations reduce it to a simpler appearance or size under the chosen rules. This usage is context-dependent, because the concept of “simplification” depends on the allowed operations. For instance, expanding a product can sometimes be viewed as making a form more complex rather than simpler, so not all algebraic transformations count as simplifications.

Terminology aside, many mathematicians prefer more standardized terms such as irreducible, in lowest terms, or in simplest form, depending on the object being considered. Nonsimplifiable is thus a descriptive label that can be ambiguous unless the rules of simplification are explicitly stated.

See also: irreducible, lowest terms, simplest form, simplification, greatest common divisor, irreducible polynomial.