0.999... is the repeating decimal consisting of an infinite string of nines after the decimal point, and it is equal to exactly 1, not merely close to it. Following the standard rules for interpreting decimal notation, its value is defined as the smallest number that is greater than or equal to every term in the sequence 0.9, 0.99, 0.999, and so on, and this smallest number can be shown to be exactly 1. The equality can be proved in several ways, ranging from intuitive arguments based on the properties of finite decimals to fully rigorous proofs using the Archimedean property of the real numbers or the tools of calculus, such as infinite series and limits. The same phenomenon, that a number can have two different decimal representations, is not unique to 1: every nonzero terminating decimal has an equal counterpart ending in an infinite string of the largest digit, and the same fact holds in other number bases as well, such as 0.111... equaling 1 in binary.
Facts
Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. 0.999..., Wikipedia
Introduction section
It can be proved that this number is 1; that is, 0.999 ... = 1.
Infinite series and sequences subsection, under Analytic proofs
This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.
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