How many base 10 decimal expansions can a real number have?

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Alexis Wolf · Accepted Answer

Step 1Suppose that a number has two distinct decimal expansions δ=(δ0,δ1,δ2,…) and δ′=(δ0′,δ1′,δ2′,…) Then, we must have0=∑i≥0δi−δi′10i.Wlog, we may assume that δ0≠δ0′ There must be some smallest i for which they are unequal. By multiplying by a suitable power of 10, we may assume that it is i=0).Furthermore, we may assume that δ0&amp;gt;δ0′Thus, we have1≤δ0−δ0′=∑i≥1δi′−δi10i.Taking absolute value on all sides and using triangle inequality for series, we see that1≤δ0−δ0′≤∑i≥1|δi′−δi|10i≤∑i≥1910i=1.Thus, we have equality throughout. Note that this means that |δi′−δi|=9 for all i. In fact, we see that δi′−δi must have the same sign for all i. This sign must, of course, be positive.Thus, we have δ0=δ0′+1 and δi=0 for all i≥1, δi′=9 for all i≥1(All the manipulations above were justified as the series converged absolutely.)The above shows that if a number has two decimal expansions, it must actually just be a variant of 1=0.9―. In particular, there are at most two decimal expansions.

How many base 10 decimal expansions can a real number

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