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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.

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  • $\begingroup$ Seems like a duplicate of my extremely misunderstood question. $\endgroup$ – Git Gud Mar 9 '15 at 15:01
  • $\begingroup$ @GitGud I expected all such questions to have the tag notation $\endgroup$ – AvZ Mar 9 '15 at 15:06
  • $\begingroup$ I agree. I don't know why I didn't add it, perhaps I didn't know it existed. $\endgroup$ – Git Gud Mar 9 '15 at 15:08
  • $\begingroup$ 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 $abc$. $\endgroup$ – Git Gud Mar 9 '15 at 15:12
  • $\begingroup$ @GitGud That is commonly used for $a\cdot b\cdot c$ so that is not correct formally. $\endgroup$ – AvZ Mar 9 '15 at 15:19
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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.

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  • $\begingroup$ I can work with $\overline{abc}$ I guess... $\endgroup$ – AvZ Mar 9 '15 at 15:28
  • $\begingroup$ 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. $\endgroup$ – David K Mar 10 '15 at 0:52
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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.

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  • $\begingroup$ You misunderatood my question. I am not asking shorthand notation for summations. $\endgroup$ – AvZ Mar 9 '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

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    $\begingroup$ 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. $\endgroup$ – Calum Gilhooley Mar 9 '15 at 16:20
  • $\begingroup$ @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. $\endgroup$ – AvZ Mar 9 '15 at 16:27
  • $\begingroup$ 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.) $\endgroup$ – Calum Gilhooley Mar 9 '15 at 16:39
  • $\begingroup$ @CalumGilhooley Would this fix it? $$100a+10b+c=(a\|b\|c)_{10}$$ $\endgroup$ – AvZ Mar 9 '15 at 16:45
  • $\begingroup$ 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. $\endgroup$ – Calum Gilhooley Mar 9 '15 at 17:02
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I won't define the decimal notation for all real numbers but will give one for just the natural numbers. First I will give a definition of natural number addition and multiplication. According to one definition, addition is defined as follows

  • $\forall x \in \mathbb{N}x + 0 = x$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}x + S(y) = S(x + y)$

where S is the successor function. Multiplication is defined in terms of addition as follows

  • $\forall x \in \mathbb{N}x \times 0 = 0$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}x \times S(y) = (x \times y) + x$

From those definitions, we can show that the following statements are all true

  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x + y) + z = x + (y + z)$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N} x + y = y + x$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} (x \times y) \times z = x \times (y \times z)$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N} x \times y = y \times x$
  • $\forall x \in \mathbb{N}\forall y \in \mathbb{N}\forall z \in \mathbb{N} x \times (y + z) = (x \times y) + (x \times z)$

We also recognize that fact that $\forall z \in \mathbb{N}$, there is exactly one ordered pair of natural numbers $x$ and $y$ such that $z = (10 \times x) + y$ and $y < 10$. We can define the decimal notation of every natural number as follows. We define a notation for all the natural numbers from 0 to 9 as the single digit we learned is its notation. For any natural number $z$ greater than 9, if you break it down into an expression of the form $(10 \times x) + y$, then the notation for $z$ is defined to be the notation for $x$ followed by the digit that represents $y$. We can then literally define 122 to mean $(10 \times ((10 \times 1) + 2)) + 2$. It's not that hard to show that also $122 = (100 \times 1) + (10 \times 2) + (1 \times 2)$.

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