0.999...

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, 0.999... (also written as 0.9, 0..9, or 0.(9)) is a <a href="/facts/Repeating_decimal/yo5T4m2M">repeating decimal</a> that is an alternative way of writing the number <a href="/facts/1/m3wpSfMf">1</a>. Following the standard rules for representing <a href="/facts/Real_number/R02gw5Pb">real numbers</a> 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, .... 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 <a href="/facts/Mathematical_intuition/stJIX3kK">intuitive</a> arguments to <a href="/facts/Mathematical_rigor/v23FKfhw">mathematically rigorous</a> <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a>. The intuitive arguments are generally based on properties of <a href="/facts/Finite_decimal/HCLBt4UQ">finite decimals</a> that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only <a href="/facts/Elementary_arithmetic/aUn5tGtp">elementary arithmetic</a> and the <a href="/facts/Archimedean_property/61pC7UP6">Archimedean property</a>: for each real number, there is a <a href="/facts/Natural_number/u65og8JE">natural number</a> that is greater (for example, by rounding up). Other proofs are generally based on basic properties of real numbers and methods of <a href="/facts/Calculus/MEmUTYoz">calculus</a>, such as <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> and <a href="/facts/Limit_(mathematics)/j8JBhj4e">limits</a>. A question studied in <a href="/facts/Mathematics_education/6hXltmmF">mathematics education</a> is why some people reject this equality.
In other number systems, 0.999... can have the same meaning, a different definition, or be undefined. Every nonzero <a href="/facts/Terminating_decimal/yo5T4m2M">terminating decimal</a> has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all <a href="/facts/Positional_numeral_system/31DECoN4">positional numeral systems</a> that represent the real numbers.

0.999... open-in-new

0.999...