How Many Decimal Representations Does 1 Have ? Solved:Let $ X = 0

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Here, Tao says that

any real number has at most two decimal representations.

Is this really true? I always thought $pi$ has only one decimal representation.

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$pi$ does indeed have only one decimal representation. However, some numbers have two! For example, $$1.0000…=0.9999….$$ Based on this, it”s reasonable to ask how many decimal representations a number can have. E.g., is there a number with three decimal representations?

Tao”s comment is that the answer is no: every number has either one or two decimal representations, that is, at most two.


Most real numbers, including all the irrationals like $pi$, have only one decimal expansion. The only ones that have two are rationals with terminating decimals like $frac 12$. You can write $frac 12=0.50000ldots =0.499999ldots $


any real number has at most two decimal representations.

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to dispute the truth of this, you need to find a real number with more than two decimal representations.
“All real numbers have at most two decimal representations” means that all real numbers have either zero, one, or two decimal representations.

Of course, all real numbers have at least one decimal representation, so we might have instead said:

Each real number has either one or two decimal representations— never more, never less.

We can divide all decimal expansions into three categories:

If the decimal expansion of $x$ terminates, you can get a second representation by decrementing the last digit and appending infinite 9s. For example, $0.128 equiv 0.127999overline{9}$.Conversely, if the decimal expansion of $x$ eventually results in an infinite sequence of 9s: $ldots 999overline{9}$, you can get a second representation by deleting the sequence of 9s and incrementing the final digit. For example, $0.127999overline{9} equiv 0.128$. There is only one other case: if the decimal expansion of $x$ continues forever without resulting in infinite 9s, it is the one and only representation of $x$.

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