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Although I fully accept that $0.999...=1$, I have always wondered if this means that they always have identical properties. For example:

  1. Does $3-2=0.999...$?
  2. Can $0.999...$ be used to describe the cardinality of a set (even if this is contrived and would not be used by convention)?
Mauro ALLEGRANZA
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Joe
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    In your experience, what do you take the equal sign ($=$) to mean? – Andrew Chin Apr 14 '20 at 07:19
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    Not to mention, what do you take $0.999\ldots$ to mean? – Arthur Apr 14 '20 at 07:20
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    Two ways to denote a fixed real number. This number does not depend on how it is denoted and in all cases you can choose your own favourite notation. – drhab Apr 14 '20 at 07:23
  • @AndrewChin Informally, I tend to think of equal sign as meaning 'makes'. E.g. $5+7$ makes $12$. However, I understand that the equal sign should really be taken to mean that the LHS has the same value as the RHS. So $5+7$ makes $12$ only because $12=12$. – Joe Apr 14 '20 at 07:23
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    An equal sign is used to branch the equality of two expressions. All properties of one side must be identical to all the properties of the other side in order for the use of the equal sign to be valid. – Andrew Chin Apr 14 '20 at 07:26
  • @Arthur I understand $0.999...$ to be $9/10 + 9/100 + 9/1000 + \cdots$. Infinite series are defined to equal the limit of adding more and more terms within the series, so $0.999...=1$. Do I understand this correctly? – Joe Apr 14 '20 at 07:31
  • A more fundamental answer can be given thanks to Nonstandard Analysis : https://en.wikipedia.org/wiki/Nonstandard_analysis . Less seriously an humorous approach (paper in French, section 6 ) : https://fr.scribd.com/document/14755203/Une-querelle-des-Anciens-et-des-Modernes . – JJacquelin Apr 14 '20 at 08:39

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If $x=y$, any truth about x is a truth about y, and vice versa. So yes, $3-2$, and the cardinality of the set of even prime numbers, are both not only $1$, but also $0.\dot{9}$.

J.G.
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  • I broadly accept your answer, and I feel stupid for not realising it was simply a matter of notation. However, isn't it the case that $0.\overline9$ is a non-terminating decimal, whereas $1$ is a terminating decimal? Again, this distinction might be one about the representation of the number, rather than the number itself, but I thought it would be sensible to clarify. – Joe Apr 26 '20 at 10:35
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    @Joe Correct, it is just a difference in representation rather than the number, just as $10$ is the binary representation of a number we normally denote $2$. – J.G. Apr 26 '20 at 10:43
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Consider that an object can (in a Platonic sense) be referred to or defined in different terms. My "mother's sister" and my "cousin's mother" are clearly different strings with different ways of defining a certain person. But they are the exact same person, so anything I say about the first must be true about the second. Another example is with the "two" objects defined as $(4+1)$ and $(2+3)$. Again, they are different strings but they refer to the same object, so anything we say about $5=(4+1)$ must be true about $5=(2+3)$. The strings $1$ and $0.\bar{9}$ are just different names for the same thing. A rose by any other name would smell as sweet, and wouldn't have any different intrinsic properties.

Jam
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What do you mean by nontrivial? Would an infinitesimal error count as trivial? If it does, then there is no trivial difference. If it does not, then of course there is.

Allawonder
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