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What is the formal mathematical notation for representing a decimal number using variables as it's digits?

I am honestly surprised that this has not been asked yet on MSE. Let me clarify a little using an example. Consider the $3$ digit decimal number $123$. Say I have $3$ variables for each of the decimal places $a,b,c$ such that $$a=1$$ $$b=2$$ $$c=3$$ I know that I can represent the number using $$a\times 100+b\times 10 + c$$ But this is too tedious. All I want to know is if there is any shorthand notation developed specifically for this purpose.
I have faint memory of using some notation for this exactly ONCE in my life but I think my memory is playing tricks on me. Does anybody actually remember some notation?
Thanks

Note

Not to be misunderstood as asking for notations of summations, et cetera. I am asking for specific purpose notation only.

David K
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AvZ
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  • Seems like a duplicate of my extremely misunderstood question. – Git Gud Mar 09 '15 at 15:01
  • @GitGud I expected all such questions to have the tag [tag:notation] – AvZ Mar 09 '15 at 15:06
  • I agree. I don't know why I didn't add it, perhaps I didn't know it existed. – Git Gud Mar 09 '15 at 15:08
  • I didn't vote to close as a duplicate because I wasn't sure I understood your question and now I confirmed this. The way I see it, the notation is exactly $123$ or $abc$. – Git Gud Mar 09 '15 at 15:12
  • @GitGud That is commonly used for $a\cdot b\cdot c$ so that is not correct formally. – AvZ Mar 09 '15 at 15:19
  • There are ambiguous notations everywhere, concatenation of symbols is a common representation of finite sequences (which is what this is). Context should rid the reader of any possible ambiguity, but formally, there's nothing wrong with $abc$. In any case, a common way to remove ambiguity is to use $(abc)_{10}$ to specify base $10$ or $\langle abc\rangle$ to make it clear it is a finite sequence. – Git Gud Mar 09 '15 at 15:21
  • As another vote for $(abc){10}$, look at the description of a base-$b$ integer on the number-systems wiki (that's supposed to be a subscript $b$ at the end of ${\left(d_m d{m-1} \cdots d_2 d_1 d_0\right)}_b$). – David K Mar 10 '15 at 00:50

3 Answers3

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I have seen $\overline {abc}$ used by some people on this site. Others will just write $abc$ and state that it means concatenating the digits instead of multiplication. Usually it is clear what problems want it that way.

Ross Millikan
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  • I can work with $\overline{abc}$ I guess... – AvZ Mar 09 '15 at 15:28
  • Sometimes an overline is used to indicate repeating digits in the fractional part of a decimal number. But if you don't have to write about repeating decimals then that is not a conflict. – David K Mar 10 '15 at 00:52
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For integers, it is $$n = \sum_{m=0}^k a_m10^m $$

where $n$ is the whole number, $k$ is the number of digits and $a_m$ gives the value of the digit at position $m$ (as it varies from $0$ to $k-1$).

For real numbers, there are subtleties having to do with duplicate representations of numbers like $0.5 = 0.4999999\ldots$ but the idea is the same.

ghosts_in_the_code
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Mark Fischler
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  • You misunderatood my question. I am not asking shorthand notation for summations. – AvZ Mar 09 '15 at 15:01
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Alternatively, I have found another less convenient but formally correct symbol
$$100a+10b+c=a\|b\|c$$ This of course is also usable on numbers $$123=1\|2\|3$$ The "$\|$" (represented by \| in $L^AT_EX$) operation is called concatenation (as mentioned by @GitGud and @RossMillikan).
More can be read about this operation here

AvZ
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    Doesn't the use of this notation tie you into identifying "the" natural numbers with the particular representation of them as finite sequences of decimal digits? Not that this is a strictly mathematically incorrect thing to do - the set of decimal numerals, under a recursively definable successor operation, will surely do as well as anything else as a model of the Peano axioms - but it's philosophically worrying, and mathematics normally tends to steer clear of such inessential commitments, preferring to remain "basis-free" as far as possible. – Calum Gilhooley Mar 09 '15 at 16:20
  • @CalumGilhooley I haven't even the slightest idea how Peano Arithmetic works, so I basically understood precisely $0$ of your comment. As the old saying goes "Monkey see, Monkey do", I just used what I found in the link included with this answer. I think Wolfram MathWorld is a citeable resource IMHO, but my interpretations of their data can ofcourse, be incorrect. – AvZ Mar 09 '15 at 16:27
  • Any obscurity is my fault and not Peano's! :) I didn't mean anything technical (my own knowledge of logic is nearly 0). I only meant that writing "$100a+10b+c=a|b|c$", in which the expression on the RHS, by the definition of "$|$", denotes a string of symbols, implies that the expression on the LHS must also denote a string of symbols, whereas I doubt if you really mean to imply that any expression denoting a natural number necessarily denotes a string of symbols. (This may be a defensible position, but it is surely not one that all users of the natural numbers should be required to adopt.) – Calum Gilhooley Mar 09 '15 at 16:39
  • @CalumGilhooley Would this fix it? $$100a+10b+c=(a|b|c)_{10}$$ – AvZ Mar 09 '15 at 16:45
  • Yes, I think so, if it is understood that numbers 0-9 may be identified (according to context) with the decimal digits that denote them (also, that symbols are identified with strings of length 1!). All of this seems to me a permissible abuse of language. Alternatively, one might use a notation involving something like "Quine corner quotes", a.k.a. "quasi-quotation" ... but that sort of thing gets so finicky that it would drive me mad (probably others too). I'd prefer to stick with something like $abc$, or $[a, b, c]$, with or without additional notation to make explicit the use of base 10. – Calum Gilhooley Mar 09 '15 at 17:02