Decimalis

Decimalis is a term encountered in some educational texts and speculative mathematics to encapsulate the study of decimal representations of real numbers. In this sense, decimalis provides a framework for discussing how numbers are written in base-10, how those representations behave, and what information the digits convey about the number.

Definition and core ideas: A real number x is expressed as its decimal expansion x = d0.d1d2d3..., where di denote digits 0–9. Decimalis analyzes properties such as terminating representations, recurring decimals, and the structure of the non-repeating part (preperiod) and the repeating cycle (period). It highlights the correspondence between decimal forms and algebraic properties, such as the fact that a terminating decimal corresponds to a rational number whose simplified denominator has only the prime factors 2 and 5.

Examples: Terminating decimals include 0.5 and 0.125. Non-terminating repeating decimals include 0.333... and 0.142857142857.... Non-terminating non-repeating decimals include irrational numbers such as sqrt(2) ≈ 1.41421356..., whose decimal expansion never settles into a repeating pattern.

History and usage: The name derives from Latin decimalis, related to ten or decimal. In formal mathematics, decimalis is not a standard branch but appears in teaching contexts to describe decimal-related topics, sometimes in theoretical discussions or in fiction to label a decimal-centric framework.

See also: Decimal expansion; Repeating decimal; Rational number; Irrational number.