0.999...


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,
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.
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. Other proofs generally involve basic properties of real numbers and methods of calculus, such as series and limits. Why some people reject this equality is a question studied in mathematics education.
In [|other number systems], 0.999... can have the same meaning, a different definition, or be undefined. Every non-zero terminating decimal has two equal representations. Having values with multiple representations is a feature of all positional numeral systems that represent the real numbers.

Elementary proof

It is possible to prove the equation using just the mathematical tools of comparison and addition of decimal numbers, without any reference to more advanced topics. The proof given below is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line, there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so.

Intuitive explanation

If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. For any number that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than. So, it does not make sense to identify 0.999... with any number smaller than 1.
Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines. Therefore, 0.999... cannot be identified with any number larger than 1, either.
Because 0.999... cannot be bigger than 1 or smaller than 1, it must equal 1 if it is to be any real number at all.

Rigorous proof

Denote by 0. the number 0.999...9, with nines after the decimal point. Thus,,, and so on. One has,, and so on; that is, for every natural number.
Let be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is,, for every. By subtracting these inequalities from 1, one gets.
The end of the proof requires that there is no positive number that is less than for all. This follows from the Archimedean property, which can be expressed as, "for every real number, there is a natural number that is greater". By computing the reciprocal, this implies that for every positive real number, there are natural numbers whose reciprocals are smaller. Therefore, for any positive real number, there must be some such that is smaller. This property implies that if for all, then can only be equal to 0. So, and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is,, as claimed.
This proof relies on the Archimedean property of rational and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers and infinitely large numbers. When using such systems, the notation 0.999... is generally not used, as there is no smallest number among the numbers larger than all 0..

Least upper bounds and completeness

Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: the smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers has a least upper bound and that this least upper bound is equal to one.

Algebraic arguments

Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. discusses the argument that, in elementary school, one is taught that, so, ignoring all essential subtleties, "multiplying" this identity by 3 gives. He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.
Byers also presents the following argument.
Students who did not accept the first argument sometimes accept the second argument, but, in Byers's opinion, still have not resolved the ambiguity, and therefore do not understand the representation of infinite decimals., presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness., citing, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature. concurs, arguing that knowing one can multiply 0.999... by 10 by shifting the decimal point presumes an answer to the deeper question of how one gives a meaning to the expression 0.999... at all. The same argument is also given by, who notes that skeptics may question whether is cancellable that is, whether it makes sense to subtract from both sides. similarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.

Analytic proofs

is the study of the logical underpinnings of calculus, including the behavior of sequences and series of real numbers. The proofs in this section establish using techniques familiar from real analysis.

Infinite series and sequences

A common development of decimal expansions is to define them as infinite series. In general
For 0.999... one can apply the convergence theorem concerning geometric series, stating that if, then
Since 0.999... is such a sum with and common ratio, the theorem makes short work of the question:
This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [|algebraic proof] given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999.... A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in several proof-based introductions to calculus or analysis.
A sequence has the value as its limit if the distance becomes arbitrarily small as increases. The statement that can itself be interpreted and proven as a limit:
The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 10 approaches 0 as approaches infinity, is often justified by the Archimedean property of the real numbers. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "when a large number of 9s is taken, the difference between 1 and.99999... becomes inconceivably small". Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.

Nested intervals and least upper bounds

The series definition above defines the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion to name it.
If a real number is known to lie in the closed interval , one can imagine dividing that interval into ten pieces that overlap only at their endpoints:,,, and so on up to. The number must belong to one of these; if it belongs to, then one records the digit "2" and subdivides that interval into,,...,,. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits,,,..., and one writes
In this formalism, the identities and reflect, respectively, the fact that 1 lies in both. and, so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.
One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So,,,... is defined to be the unique number contained within all the intervals,, and so on. 0.999... is then the unique real number that lies in all of the intervals,,, and for every finite string of 9s. Since 1 is an element of each of these intervals,.
The nested intervals theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define... to be the least upper bound of the set of approximants,,, .... One can then show that this definition is consistent with the subdivision procedure, implying again. Tom Apostol concludes, "the fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."