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When reading mathematical texts I encounter phrases such as "the decimal number 2," the "binary number 10," etc., so much that I begin to imagine there really is such a thing as a "base-n number," whereas the reality (at least as far as I understand it) is that numbers are distinct from their representations. In other words, it seems to me that our minds tend to conflate numbers with number words, so that, e.g., the phrase "the decimal number 70" invites ambiguity of meaning between "the number represented by 70 in the decimal numeral system," on the one hand, and "the string '70' as a word in the decimal language," on the other. Nevertheless the two are of course distinct, and failing to observe the distinction can lead to confusion. For example, it can be easily proven that any real number has exactly one additive inverse such the the sum of the two is 0. However, in the decimal system, all of the members of the set $S = \{-70, -070, -0070,-00070, \ldots\}\cup\{-70., -70.0, -70.00, -70.000, \ldots\}$ are distinct words, yet as "decimal numbers" any of them could be substituted for $x$ in the equation $70 + x = 0$, thereby appearing to violate the uniqueness of the additive inverse.

I am writing a paper where maintaining the distinction between numbers and representations is crucial, but I find it both tedious and unconducive to readability to repeatedly call out that distinction using phrases like "the number represented by $x$ in the base-$n$ numeral system" and "the string $x$ as a word in the base-$n$ language" at every turn. Any suggestions on how to make the writing less awkward?

JustAsking
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    A common notation is $(x)_b$ to indicate that $x$ is represented in base $b$: e.g., $(101)_2 = 5$. – Théophile Nov 17 '20 at 18:57
  • @Théophile: even without parenthesis. The suffix $b$ can be used for binary, $d$ for decimal (because $10$ would be ambiguous), $h$ for hexadecimal, $o$ for octal. –  Nov 17 '20 at 18:58
  • Sure, if the context is clear, then without parentheses is fine too. There may be situations where $x_2$, for example could be confusing, but if not, then keep it simple. – Théophile Nov 17 '20 at 19:00
  • The solution of $70+x=0$ does not involve any representation of $x$, there is no ambiguity. –  Nov 17 '20 at 19:02
  • @Théophile: in general $x_2$ will not be used as a variable is base-independent, except in the rare cases that $x$ denotes a string of digits susceptible to be interpreted in different bases. –  Nov 17 '20 at 19:05
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    @YvesDaoust Right, understood. I'm just thinking of the OP's phrases like "the number represented by $x$ in the base-$n$ numeral system", which would amount to $x_n$. – Théophile Nov 17 '20 at 19:07
  • How about "number" versus "name", as in "the number with decimal name 30 is equal to the number with decimal name 30.0" and so on. – kimchi lover Nov 17 '20 at 19:20

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