0.999...
Sign in to saveupright=1.35|class=skin-invert-image|thumb|alt=Stylistic impression of the repeating decimal 0.9999..., representing the digit 9 repeating infinitely 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 is1; that is, 0.999\ldots = 1.
AI overview
0.999... is a decimal number where the digit 9 repeats infinitely, and mathematically it equals exactly 1. This matters because it demonstrates how infinite decimal representations work in mathematics—the sequence 0.9, 0.99, 0.999, and so on gets closer and closer to 1, and the infinite repetition reaches that value precisely.
AI-generated from the Wikipedia summary — may contain errors.
Article
28 sectionsContents
- Elementary proof
- Intuitive explanation
- Rigorous proof
- Least upper bounds and completeness
- Algebraic arguments <span class="anchor" id="Proofs"></span><span class="anchor" id="Algebraic"></span>
- Analytic proofs <span class="anchor" id="Analytic"></span>
- Infinite series and sequences
- Nested intervals and least upper bounds
- Proofs from the construction of the real numbers <span class="anchor" id="Based on the construction of the real numbers"></span>
- Dedekind cuts
- Cauchy sequences
- Infinite decimal representation
- Dense order
- Generalizations
- Applications
- Skepticism among students
- Cultural phenomenon
- In alternative number systems <span class="anchor" id="Alternative number systems"></span>
- Infinitesimals
- Hackenbush
- Revisiting subtraction
- ''p''-adic numbers
- See also
- Notes
- References
- Sources
- Further reading
- External links
upright=1.35|class=skin-invert-image|thumb|alt=Stylistic impression of the repeating decimal 0.9999..., representing the digit 9 repeating infinitely 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 is1; that is, 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 the same number.