Is $.000\dots001$ equal to $0$?

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I am not particularly formally educated in maths. I have just learned that $.999\dots=1$, so it feels somewhat natural to me that $.000\dots1 = 0$. Is this the case?

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Nice question. In fact, there is no such real number as $.000 \ldots 1$.

The digits in a decimal expansion -- even an infinite one -- are ordered by the natural numbers, so it makes sense to speak of the first digit, the second digit, and so on, after the decimal point. The string of symbols $.000 \ldots 1$ is not a decimal expansion, because there is no answer to "which digit is $1$"?