01-05-2023, 01:53 PM
In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence. This number is equal to 1. In other words, "0.999..." is not "almost exactly" 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 technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined.
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.
There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise given by Stillwell (1994, p. 42), 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 0.999... = 1.
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.
More precisely, the distance from 0.9 to 1 is 0.1 = 1/10, the distance from 0.99 to 1 is 0.01 = 1/102, and so on. The distance to 1 from the nth point (the one with n 9s after the decimal point) is 1/10n.
Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than 1/10n for every integer n. In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than 1/10n for all n. This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so 1 = 0.999....
Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: a 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 in fact has a least upper bound, and that this least upper bound is equal to one.
There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined.
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.
There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise given by Stillwell (1994, p. 42), 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 0.999... = 1.
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.
More precisely, the distance from 0.9 to 1 is 0.1 = 1/10, the distance from 0.99 to 1 is 0.01 = 1/102, and so on. The distance to 1 from the nth point (the one with n 9s after the decimal point) is 1/10n.
Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than 1/10n for every integer n. In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than 1/10n for all n. This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so 1 = 0.999....
Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: a 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 in fact has a least upper bound, and that this least upper bound is equal to one.
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