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PROOF THAT 0.999... EQUALS 1

The recurring decimal 0.999… equals 1, not approximately but exactly. More precisely, the standard real number represented by 0.999… (where the 9s repeat forever) is exactly equal to the standard real number 1.

Proofs of this equality vary depending on the level of rigor demanded and on what results are assumed to be already known. All proofs rely on properties of the standard real numbers. There are other so called "non-standard" real numbers,[1] for which the equality does not hold. In many of these number systems, 0.999... is not well-defined.

Contents

Background

In arithmetic with decimal fractions, a simple division of integers like

1⁄3

becomes a recurring decimal,

0.3333…,

in which digits repeat without end. There also exist numbers that are not quotients of integers, such as the square root of two (1.41421356…) and pi (3.14159265…) with an endless number of digits that do not repeat. A benefit of the decimal notation is that most calculations — addition, subtraction, multiplication, division, comparison — use manipulations that are much the same as for integers. And like integers, in most cases a different series of digits means a different number (ignoring trailing zeros as in 0.250 and 0.2500). The one notable class of exceptions is numbers with trailing repeating 9s.

It should be no surprise that a notation allows a single number to be written in different ways. For example,

1⁄2 = 3⁄6.

The 9s case does surprise, perhaps because any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1. But, if the 9s do not eventually stop, if they continue infinitely, then 0.999... is no longer less than 1, but exactly equal to it. Thus infinity, a sometimes mysterious concept, plays an important role behind the scenes. (See the "In popular culture" section below).

Digit manipulation

Elementary proofs assume that manipulations at the digit level are well-defined and meaningful, even in the presence of infinite repetition.

Fraction proof

The standard method used to convert the fraction 1⁄3 to decimal form is long division, and the well-known result is 0.3333…, with the digit 3 repeating. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…; but 3 × 1⁄3 equals 1, so it must be the case that 0.9999… = 1.[2]

Algebra proof

Another kind of proof adapts to any repeating decimal. When a fraction in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. To see this, consider that subtracting 0.9999... from 9.9999… can proceed digit by digit; the result is 9 − 9, which is 0, in each of the digits after the decimal separator. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.9999…, be called c. Then 10c − c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1.[2]

Real analysis

Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. Rigorous proofs are generally not studied before the university level.

Geometric series proof

In real analysis, a standard development of decimal expansions is to define them as sums of infinite series. For 0.999... one can apply the powerful convergence theorem concerning infinite geometric series:[3]

Theorem. If | r | < 1 then ar+ar^2+ar^3+\cdots = \textstyle\frac{ar}{1-r}.

Since 0.999… is such a sum with a common ratio r=\textstyle\frac{1}{10}, the theorem makes short work of the question:

0.999\ldots = 9({\textstyle\frac{1}{10}}) + 9({\textstyle\frac{1}{10}})^2 + 9({\textstyle\frac{1}{10}})^3 + \cdots = \frac{9({\textstyle\frac{1}{10}})}{1-{\textstyle\frac{1}{10}}} = 1.\,

Among mathematicians, this geometric series does not require such fanfare; one textbook relegates it to a single line in an appendix and performs the summation without comment.[4]

The sum of a geometric series is a centuries-old result. Its earliest derivations, which preceded the modern understanding of series as limits, used a term-by-term manipulation that is similar in spirit to the algebra proof given above.[citation needed] Today, such manipulations are known to be generally invalid without special justification. The standard, rigorous proof of the theorem works with limits in a manner similar to the limit proof given below; it can be found in any proof-based introduction to calculus or analysis.[5]

Nested intervals proof

The series definition above is a simple way to define 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(s) that are to name it.

If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine subdividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number x must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labelled by an infinite sequence of digits b0, b1, b2, b3, ..., and one writes x = b0.b1b2b3...

Under this definition, it may not be clear how a sequence of digits then determines a real number. Here one applies 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 b0.b1b2b3... is the unique number contained within all the intervals [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], and so on.

0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1. In this formalism, the fact that the real number 1 has two decimal representations reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits.

This development of decimal representations can be found in an informal digression within Bartle and Sherbert's Introduction to real analysis, a university-level text.[6] Apostol's Mathematical analysis takes a two-part approach: first Apostol considers half-open intervals like [0, 1), then notes that using intervals like (0, 1] results in slightly different decimals; 1 is written 1.000... in the first definition and 0.999... in the second.[7]

From first principles

Proofs at a more advanced level draw on the axiomatic foundations of mathematics. They use careful and sound definitions of integers, fractions, real numbers, infinity, limits, and equality. The validity of manipulations at the elementary level is a logical consequence of these foundations.

One requirement is to characterize numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a positional notation, so that the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5. Without sign the value is determined as follows:

a_n \cdots a_2 a_1 a_0 . b_1 b_2 \cdots = \,\! a_n \times 10^n + \cdots + a_2 \times 10^2 + a_1 \times 10^1 + a_0 \times 10^0
{} + b_1 / 10^1 + b_2 / 10^2 + \cdots .

A minus sign negates the value. For purposes of this discussion the integer part can be summarized as b0. To proceed further, to give any meaning to the sum of the bk terms, requires a theoretical exploration of numbers, especially real numbers.

The natural numbers — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. Peano axioms are the usual formal definition, and these in turn draw upon axiomatic set theory. There is little difficulty, conceptual or formal, in extending natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found less than, greater than, or equal.

Order proof

The step from rationals to reals is a huge extension, and order is an essential part of any construction. In the Dedekind cut approach, each real number x is the infinite set of all rational numbers that are less than x.[8] In particular, the real number 1 is the set of all rational numbers that are less than 1.[9]

Every positive decimal expansion then easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form 1 − (1⁄10)n.[10]

The equation "0.999... = 1" means that these two Dedekind cuts are the same set, each containing the same rational numbers. If the equation were untrue, there would have to be some rational number r such that r < 1, but r > 1 − (1⁄10)n for every positive integer n. Defining q to be the rational number 1 / (1 − r), one would have q > 10n for every n, which is impossible. (There is no infinite rational number; the rational numbers are Archimedean.)[11] So 0.999... = 1.

This approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in Mathematics Magazine, which is targeted at undergraduate mathematicians.[12] Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."[13] A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "Other number systems" below.

Limit proof

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where |z| is the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that are Cauchy using this distance. That is, in the sequence (x0, x1, x2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xm − xn| ≤ δ for all m, n > N. (The distance between terms becomes arbitrarily small.)[14]

A sequence (x0, x1, x2, ...) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. Now if (xn) and (yn) are two Cauchy sequences, taken to be real numbers, then they are defined to be equal as real numbers if the sequence (xn − yn) has the limit 0. Truncations of the decimal number b0.b1b2b3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.[15] Thus in this formalism the task is to show that the sequence

\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right) = \left(1, {1 \over 10}, {1 \over 100}, \dots \right)

has the limit 0. Considering the nth term of the sequence, for n=0,1,2,..., it must therefore be shown that

\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.

This limit is an elementary result, and so again 0.9999… = 1.[16]

This approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.[17]

Generalizations and applications

Proofs that 0.999... = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a doppelgänger with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. Second, a comparable theorem applies in each radix or base; for example, in the base 2 version 0.111… equals 1, and in the base 3 version 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.[18]

These facts are applied in a characterization of the middle-thirds Cantor set:

The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.[19]

Repeating nines also turn up in another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.[20] A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.[21]

Other number systems

Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999..." as naming a real number is ultimately a convention, and, as Timothy Gowers argues in Mathematics: A Very Short Introduction, the identity 0.999... = 1 is a convention as well:

"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."[22]

One can place constraints on hypothetical number systems where 0.999... ≠ 1, with their new objects and/or unfamiliar rules, by reinterpreting the above proofs. As Fred Richman puts it, "one man's proof is another man's reductio ad absurdum."[23] If 0.999... is to be different from 1, then at least one of the assumptions built into the proofs must break down.

Infinitesimals

These proofs rely, explicitly or implicitly, on properties of the standard real numbers, including the Archimedean property that there are no nonzero infinitesimals. There are mathematically coherent ordered algebras, including various alternatives to standard reals, which are non-Archimedean; but it is difficult to discuss decimal expansions in them, because:

  • They may have multiple elements with the same decimal expansion to an infinite number of places.
  • Dividing through by an infinitesimal, when defined, would result in elements larger than every integer, which therefore cannot be expressed by decimals in the usual fashion at all.

The non-standard properties make these systems unsuitable for ordinary calculations, though they are of theoretical interest. For example, the p-adic numbers are constructed from rationals in the same way as the reals, but using different orderings (one for each prime p). Their equivalent of "decimal expansions" is of interest in number theory.

Standard reals can also be extended to become dual numbers, by including a new element ε defined to combine with other reals in the usual way, but such that its product with itself is zero. Every dual number then consists of a standard real component and an "infinitesimal" component, a+bε, either of which may be zero. However, the infinitesimals are displaced off the real line, rather than ordered between standard reals.

Another way to construct alternatives to standard reals is to use topos theory and alternative logics rather than set theory and classical logic (which is a special case). For example, smooth infinitesimal analysis has infinitesimals with no reciprocals.[24]

Game theory provides alternative reals as well, with Hackenstrings as one particularly relevant example. [25]

In popular culture

This topic provokes a great deal of interest. For example, in the newsgroup sci.math, devoted to discussion of general mathematics, statistics show over one thousand postings related to 0.999...;[citation needed] and it is one of the questions answered in its FAQ.[26]

Professor David Tall has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".[27]

Related questions

  • Zeno's paradoxes, particularly the runner paradox, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was after.[28]
  • Division by zero occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the Riemann sphere, it makes sense to define 1/0 to be infinity.[29] In fact, some prominent mathematicians argued for such a definition long before either number system was developed.[30]
  • Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.[31] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems.[32]

Notes

  1. ^ For a full treatment of non-standard numbers see for example Robinson's Non-standard Analysis.
  2. ^ a b cf. with the binary version of the same argument in Martin Gardner and Silvanus P. Thompson, Calculus made easy, St. Martin's Press, New York, 1998.
  3. ^ Rudin p.61, Theorem 3.26; Stewart p.706
  4. ^ Protter and Morrey p.504
  5. ^ For example, Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, Conway p.31
  6. ^ Bartle and Sherbert pp.60-62
  7. ^ Apostol pp.11-12
  8. ^ Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number x can be named by giving an infinite set of rationals, namely all the rationals less than x. We will in effect define x to be the set of rationals smaller than x. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."
  9. ^ Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".
  10. ^ Richman p.399
  11. ^ To be precise, Pugh (p.20) defines the Archimedean property to mean that for all x, there is an integer n greater than x. "In other words, there exist arbitrarily large integers. ... An equivalent way to state the Archimedean property is that there exist arbitrarily small reciprocals of integers." For an exploration of infinitesimals, see the section #Other number systems.
  12. ^ Mathematics Magazine:Guidelines for Authors. The Mathematical Association of America. Retrieved on 2006-08-23.
  13. ^ Richman pp.398-399
  14. ^ Griffiths & Hilton §24.2 "Sequences" p.386
  15. ^ Griffiths & Hilton pp.388, 393
  16. ^ The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) Thomas' Calculus: Early Transcendentals 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).
  17. ^ Griffiths & Hilton pp.viii, 395
  18. ^ Protter and Morrey p.503; Bartle and Sherbert p.61
  19. ^ Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) assign this description as a difficult exercise.
  20. ^ Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.
  21. ^ Rudin p.50, Pugh p.98
  22. ^ Gowers p.60
  23. ^ Richman p.396; emphasis is his. This line appears in a paragraph of the published version that is not present in the earlier preprint.
  24. ^ John L. Bell. "An Invitation to Smooth Infinitesimal Analysis" (PDF). Retrieved on 2006-06-29.
  25. ^ A. N. Walker (1999). Hackenstrings and the 0.999… ≟ 1 FAQ. Retrieved on 2006-06-29.
  26. ^ Hans de Vreught (1994). sci.math FAQ: Why is 0.9999... = 1?. Retrieved on 2006-06-29.
  27. ^ David Tall (2000). "Cognitive Development In Advanced Mathematics Using Technology". Mathematics Education Research Journal 12: 210-230.
  28. ^ Wallace p.51, Maor p.17
  29. ^ See, for example, Conway's treatment of Möbius transformations, pp.47-57
  30. ^ Maor p.54
  31. ^ Munkres p.34, Exercise 1(c)
  32. ^ Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics, 2e, W. H. Freeman. ISBN 0716710889, 462. Floating point types. MSDN C# Language Specification. Retrieved on 2006-08-29.

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