In this series, I discuss some ways of convincing students that 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.

Methods #2 and #3 are indirect methods. We start with a decimal representation that we know and end with .

**Method #2**. This technique should be accessible to any student who can do long division. With long division, we know full well that

Multiply both sides by :

Though not logically necessary, this method could be reinforced for students by also considering

**Method #3**. With long division, we know full well that

and

Add them together:

Though not logically necessary, this method could be reinforced for students by also considering any (or all) of the following:

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