Why does 0.999… = 1? (Part 3)

In this series, I discuss some ways of convincing students that 0.999\dots = 1 and that, more generally, a real number may have more than one decimal representation even though a decimal representation corresponds to only one real number. This can be a major conceptual barrier for even bright students to overcome. I have met a few math majors within a semester of graduating — that is, they weren’t dummies — who could recite all of these ways and were perhaps logically convinced but remained psychologically unconvinced.

Method #4. This is a direct method using the formula for an infinite geometric series… and hence will only be convincing to students if they’re comfortable with using this formula. By definition,

0.999\dots = \displaystyle \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \dots

This is an infinite geometric series. Its first term is \displaystyle \frac{9}{10}, and the common ratio needed to go from one term to the next term is \displaystyle \frac{1}{10}. Therefore,

0.999\dots = \displaystyle \frac{ \displaystyle \frac{9}{10}}{ \quad \displaystyle 1 - \frac{1}{10} \quad}

0.999\dots = \displaystyle \frac{ \displaystyle \frac{9}{10}}{ \quad \displaystyle \frac{9}{10} \quad}

0.999\dots = 1

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  1. Formula for an infinite geometric series (Part 11) | Mean Green Math
  2. Why Does 0.999… = 1? (Index) | Mean Green Math

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