0.999...
alternative decimal expansion of 1
0.999... is a repeating decimal that represents the number 1. The three dots represent an infinite list of "9" digits. Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is 1; that is,
0.999
…
=
1.
{\displaystyle 0.999\ldots =1.}
Despite common misconceptions, 0.999... is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent exactly the same number.
There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only elementary arithmetic and the Archimedean property: for each real number, there is a natural number that is greater (for example, by rounding up).
The public source identifies “0.999...” as alternative decimal expansion of 1. This brief keeps that definition visible, then builds a research path around alternative, decimal and expansion.
Why this record matters
A short description can identify a subject without explaining its stakes. For “0.999...”, the useful work is to connect “alternative decimal expansion of 1” to the records capable of establishing context and consequence.
Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Sep 10, 2026. The linked authority identifier is Q713113. None of the 0 selected statements returned an explicit reference.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “0.999...”, its source revision and the description used here.
- Expand the search: follow 0.999... primary sources, 0.999... archive and alternative research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “0.999...”?
- Which institution is responsible for the underlying evidence?
- What terminology or title could unlock a more precise catalogue search?
Search terms from this dossier
This entry incorporates text from “0.999...” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.