Research: 0.999… = 1?

Whenever the equality of 0.999… and 1 arises, teachers can expect a high degree of disbelief from students, and proofs may do little to abate their skepticism. This equality challenges students’ conceptions of the real line, limits, and decimal representation, but students have a strong historical and intuitive basis for their resistance. The purpose of this paper is to investigate the reasons students reject the equality and to consider the consequences of this rejection.

With this purpose in mind, we have organized the paper in the following way: We begin by outlining various arguments supporting the equality and then review some of the pedagogical struggles noted in research that explain students’ resistance. Next, we justify students’ intuitive resistance by presenting a system of hyperreal numbers in which the equality does not necessarily hold. Finally, we consider the implications of adopting such a system, which forces students to choose between conflicting properties; the example offered to the students is the conflict between the Archimedean property for real numbers and the existence of infinitesimals.

Arguments for equality

There are many arguments that support the equality of 0.999… and 1. Here are four of these arguments which I studied.

Relying on the decimal expansion for 1/3

A common argument for the equality goes as follows: If 0.333… = 1/3 then digit-wise multiplication by 3 would imply that 0.999… = 1. Of course, this argument relies on students’ acceptance of the equality of 0.333… and 1/3, ha! Research has shown that students generally accept this equality, even while rejecting the equality of 0.999… and 1 (Fischbein, 2001). Students might resolve this tension by asserting, “Well, then, maybe 0.333… doesn’t equal 1/3.”

Subtracting off the infinite sequence

The figure below outlines a more formal argument that doesn’t depend on similar equalities. Yet students might still object.image

The issue with this argument is whether x can be cancelled. Richman (1999) asserted that skeptics might reject the equality by claiming that not all numbers can be subtracted from one another! Moreover, if we consider 0.999… as the limit of the limit of the sequence 0.9, 0.99, 0.999, … then we see that the corresponding products, using the standard algorithm for multiplication of 9 by 0.999… produces a limit of 8.999…, which leads back to the same central issue that x might not be 1 after all.

Generating a contradiction

A third argument for the equality works by contradiction: If 1 and 0.999… are not equal, then we should be able to find a distinct number in between them (their average), but what could that number be other than 0.999… itself? Still, students might argue that some pairs of distinct numbers simply do not have averages; some students have even argued that there are numbers between 1 and 0.999… – namely, ones represented by a decimal expansion that begins with an infinite string of 9’s and then ends in some other number (Ely, 2010). Even when students cannot find fault with the argument, they still might not believe the result.

Defining the decimal expansion with limits

Since Balzano formalised the definition of limits in the early 19th century, Calculus has been grounded in the formal definitions of limits that is taught in Precalculus and many college-level mathematics courses. The figure below lays out Balzano’s formal ε −N definition for limits of sequences.

image

This definition amounts to a kind of choosing game: Assuming S is the limit of a sequence, {Sn}, for any positive distance, ε, you choose, I can find a natural number, N, so that whenever the sequence goes beyond the Nth term, the distance between any of those terms and S is less than ε. The definition says that if the tail of a sequence gets arbitrarily close to a number, then the number is the limit of the sequence.

We can think about the decimal representation, 0.999…, as the limit of an infinite series:

9/10 + 99/100 + 999/1000 + …

Thus, we arrive at the following conclusion:
image

The equality holds because for any real value of ε that you choose, I can find a natural number N such that 1 is within ε of
image

This means that we have devised a way to answer the question, “How close is close enough?” The answer is that we are close enough to the number 1 if, when given an ε neighbourhood extending some distance about the number 1, we can find a number N such that the terms at the tail end of the series are inside that neighbourhood. When this happens, we no longer distinguish between the terms of the series and the number 1.

Why students remain skeptical

There is a historical basis for students’ (our, as well) skepticism in accepting any of the arguments above, and researchers have found several underlying reasons for why we reject the equality—some more logical than others (Ely, 2010; Fischbein, 2001; Oehrtman, 2009, Tall & Schwarzenberger, 1978). For example, many students conceive of 0.999… dynamically rather than as a static point; we interpret the decimal expansion as representing a point that is moving closer and closer to 1 without ever reaching 1 (Tall & Schwarzenberger, 1978). Starting from 0, the point gets nine-tenths of the way to 1, then another nine-tenths of the remaining distance, and so on, but there is always some distance remaining (cf. Zeno’s paradox). This conception aligns with Aristotle’s idea of potential infinity and his rejection of an actual infinity: 0.999… is a process that never ends, producing a decimal expansion that is only potentially infinite and not actually an infinite string of 9’s (check out Dubinsky, Weller, McDonald, & Brown, 2005, for an excellent discussion of historical struggles with infinity and related paradoxes). This issue points to a confusion between number and their decimal representations: Would students be inclined to say that one-third is a process that never ends simply because its decimal expansion is 0.333…?

Tall and Schwarzenberger (1978) analysed student reasons for accepting or rejecting the equality and found that they generally fit into the following categories:

  • Sameness by proximity: The values are the same because a student might think, “The difference between them is infinitely small,” or “At infinity it comes so close to 1 it can be considered the same.”
  • Infinitesimal Difference: The values are different because a student might think “0.999… is the nearest you can get to 1 without actually saying it is 1,” or “The difference between them is infinitely small”.”

Oehtrman (2009) went on to suggest that there is potential power in the approximation metaphor because this type of thinking closely resembles arguments for the formal definition of a limit. In fact, early definitions of limit by mathematicians such as D’Alembert included the language of approximation:

One magnitude is said to be the limit of another magnitude when the second may approach the first within any magnitude however small, though the first magnitude may never exceed the magnitude it approaches” (Burton, 2007, p. 603).

Consequences of accepting infinitesimals and rejecting equality

Consider the argument for equality that used limits outlined in the previous section. What if you were allowed to choose ε to be infinitely small? Then the game is up; one cannot possible hope to bring the sequence within such an intolerant tolerance! However, you should beware that, in order to win (i.e. choosing a value for ε that makes the limit argument fail, thus proving 0.999… does not equal 1), you have violated the Archimedean property.

The Archimedean property states that, for any positive real number, r, we can choose a natural number, N, large enough so that their product is greater than 1. That means any real number is farther from 0 than 1/N for some N. No matter how close r is to 0, if we zoom in on 0 enough, there is no number “next to 0,” or infinitely close to 0. If r were allowed to be an infinitesimal, this would not be the case; r would be less than 1/N for all N, or stay perpetually next to 0, which violates the Archimedean property. Thus, the only way to maintain this intuitive property of the real line is to reject infinitesimals, as we have done in the historical development of the real line (standard analysis).

Conclusions and implications

The Archimedean property captures one of the most intuitive ideas about the real line (Brouwer, 1998). Starting from that property, we can use the definition of limits to show that equality of 0.999… and 1 must hold. Thus, we can see that the Archimedean property and the formal definition of limits imply the equality. The only was to reject the equality is to reject the property or to reject our definition of limits.

In the history of mathematics, the development of calculus prompted speculation about the existence of infinitesimals, while motivating the construction of limits (Burton, 2007). Even the Archimedean property arose from a pre-calculus concept—namely Archimedes’ method of exhaustion. If history is any guide for motivating and developing ideas in the classroom, then Katz and Katz (2010 draw a natural conclusion in suggesting that we delay the discusion of irrational numbers and infinite decimal expansions until after limits are formally addressed.

Consideration of this equality might generate meaningful discussion about students’ intuitive concepts. Imagine a Precalculus classroom full of students who have studied decimal expansions but have never studied irrational numbers except to prove that some numbers (such as the square root of 2) cannot be expressed as a ratio of two integers. Some students might have wondered, but non had formally studied whether this property is related to repeating or terminating decimal expansions. On the first day of a unit on limits, the teachers could ask whether 0.999… equals 1. This article outlines potential connections students might make through arguments about this equality—connections between decimal expansions, the real number system, and limits. It seems that this kind of discussion does not typically happen because we ask some questions too early and others not at all.

References

  1. Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematics Education in Science and Technology, 32, 487-500.
  2. Brouwer, L. E. J. (1998). The structure of the continuum. In P. Mancosu (Ed.), From Brouwer to Hilbert (pp. 54-63). Oxford, England: Oxford University Press.
  3. Burton, D. M. (2007). The history of mathematics: An introduction (6th ed.). New York, NY: McGraw Hill.
  4. National Governors’ Association and Council of Chief State School Officers. (2010). Common sore state standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
  5. Dubinsky, E., Weller, K., McDonald, M., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part I, Educational Studies in Mathematics, 58, 335-359.
  6. Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41, 117-146.
  7. Fischbein, E. (2001). Infinity: The never-ending struggle. Educational Studies in Mathematics, 48, 309-329.
  8. Katz, K. U., & Kats, M. G. (2010). When is .999… less than 1? The Montana Mathematics Enthusiast, 7, 309-329.
  9. Keisler, H. J. (1976). Foundations of infinitesimal calculus. Boston, MA: Prindle, Weber & Schmidt.
  10. Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 40, 396-426.
  11. Richman, F. (1999). Is 0.999… = 1? Mathematics Magazine, 72, 396-400.
  12. Sierpinska, A. (1994). Understanding in mathematics (Studies in Mathematics Education Series: 2). Bristol, PA: Falmer Press.
  13. Tall, D. O., & Schwarzenberger, R. L. E. (1978). Conflicts in the learning of real numbers and limits. Mathematical Teaching, 82, 44-49.
  14. Weisstein, E. W. (2011). Convergent Sequence. In Wolfram MathWorld. Retrieved from http://mathworld.wolfram.com/ConvergentSequence.html
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