Understanding of the reals: 0.9999…. = 1

Many people, including mathematics teachers, wrestle with the claim that 0.9999… is the same as 1. Some think of it as incorrect – the former is something just slightly less than but never “reaching” 1. Others perhaps follow the proof but think of it as mathematical “hocus-pocus” – the same as those “proofs” demonstrating that 1=2.

For school mathematics teachers: Is this fact important to know? Is it important to be able to prove it? Does it come up in classrooms? Do students care? If it is important to know, why? What is important to know about it, if anything at all? As a professor and teacher educator interested in teachers’ knowledge – particularly in ways that more advanced mathematics becomes productive for teachers – I wrestle with these kinds of questions regularly. The proof of this fact, 0.9999… = 1, can come in a variety of forms, but it draws on the notion of infinity and limits, etc. It can be proved through arguments from analysis about convergent series (sum on k of an infinite geometric series, (9/10)^k), algebraic techniques (e.g., let x=0.9999…, then 10x=9.9999…, etc.), or by computational arguments (e.g., 1/3=0.3333…, multiply both sides by 3…). But does this constitute important knowledge for teachers? And if so, why?

Simon’s (2006) notion of a Key Developmental Understanding (KDU) has become one way that I have begun to consider such questions. Simon describes a KDU as “a change in a students’ ability to think about and/or perceive particular mathematical relationships” (p. 362). In this case, the “students” being referred to are teachers. Simon emphasizes that KDUs are not missing pieces of information, but rather key understandings that foster one’s ability to think about and perceive mathematical ideas and relationships. They represent ontological shifts and transformations in teachers’ available assimilatory structures – that mathematical ideas have been re-understood, re-organized, re-structured, etc. For my own interests specifically, in what ways can mathematics that is not in a local neighborhood of the content a teacher teaches influence their understanding of and/or perceptions about the content they teach?

So what key understanding is gained from knowing that 0.9999…=1? To me, one of the understandings that is important is about the structure of the real numbers. One of the reasons that students (and teachers) like decimals – as opposed to fractions – is that it always produces the same decimal expansion. In other words, a conceptually difficult issue with fractions is that 1/4=2/8=3/12=… (an infinite number of equivalent representations) gets resolved with decimals. Type 1/4, 2/8, 3/12, etc. on the calculator and they all produce 0.25. So decimals seem easier or more consistent in some ways. But does the issue really get resolved? In fact, using decimal expansions, which necessitate infinite decimal expansions (e.g., 1/3=0.3333…) comes with its own set of conceptually difficult issues. Namely, that if we agree to an infinite decimal expansion, we have to deal with infinity. Which means we have to grapple with the odd conclusion that 0.9999… is, in fact, equal to 1. And so, in reality, decimals have equivalence classes in the same way that fractions do. And not just the seemingly trivial ones, such as 0.25 = 0.2500; but, additionally, that 0.25 must also equal 0.249999… . Indeed, any terminating decimal will have such an equivalence class; repeating decimals or irrational numbers will not. For me, this is perhaps one of the fundamental understandings that comes from knowing 0.9999…=1: that the set of real numbers has a structure that contains similarities to the set of fractions, and differences. Decimal representations, like fractions, have equivalence classes, albeit different in nature than those for the set of fractions. And this is something to be grappled with. Although decimals have some advantages in terms of understanding the relative size of fractions, they, too, come with conclusions that we must accept if we are to truly understand them.

Reference: Simon, M.A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), pp. 359-371.