I am writing a blog post about cryptography and I need to show some examples that involve sums modulo N.

I would like (if possible) to use a different "plus" symbol for these sums. The kind of examples I am writing are tabular look like this

example sum

which is a base 36 sum modulo 36. If it was in line I could just write $( A + 4 )_{mod\ 36} = E$, but that's not a possibility in the context.

Is it appropriate the circled plus ($\oplus$) in this case?

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    $\begingroup$ How is this related to mathematics education? $\endgroup$ – Rory Daulton Sep 26 '15 at 11:45
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    $\begingroup$ I thought it was appropriate given that you have a notation tag and it's about explaining something to a public. Feel free to close if it is off topic. It's not about school, or teaching, but certainly explaining something is education. $\endgroup$ – Sklivvz Sep 26 '15 at 12:30
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    $\begingroup$ I'm voting to close this question as off-topic because it is about communication of mathematics and has no link to mathematics education. $\endgroup$ – Benoît Kloeckner Sep 26 '15 at 14:32
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    $\begingroup$ If the question would say that OP is preparing lecture notes for their class or their course for next week, and wants some pedagogical input to decide on the notation to use, would this be off-topic too? If not, what is the difference? (I do agree the question is a bit borderline, but in my mind rather within the border, but I am willing to be convinced otherwise.) $\endgroup$ – quid Sep 26 '15 at 14:49
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    $\begingroup$ This comment is just to indicate that I agree with @quid's assessment of the question (though I see the four votes to close). $\endgroup$ – Benjamin Dickman Sep 27 '15 at 0:34

To use a special plus-symbol like $\oplus$ can be appropriate in this context, depending on what you want to high-light. It is quite common at least at the start of presenting some new structures to use "special" symbols for the operations and often they are derived from the usual symbols for addition and multiplication. This serves well as a reminder to the reader that this is not good-old addition but something else, yet still some kind of addition. Later on, one will often fall-back to the standard symbols on the grounds that "it is clear from context" what is meant.

The specific symbol you propose can be a good choice, as it is a standard symbol visually close to $+$, provided you do not want to discuss direct sums or related.

A common use of $\oplus$ is to denote the direct sum of (commutative) groups. For example, it is common to write things like $\mathbb{Z}/\mathbb{7}Z \oplus \mathbb{Z}/49\mathbb{Z}$.

There should be no actual risk of confusion though.

Alternative ideas include:

  • $\hat{+}$
  • $+_{n}$ with $n$ the modulus.

Depending on your approach, you could also use $(A+4) \% 36 = E$, with $\%$ the binary operator $A\%B$ being the reduction of $A$ modulo $B$. At least this is a very common programming-language symbol.

In contrast, although some introductory sources use $\oplus$ or $+$ with funny marks on it, these are absolutely not standard, and I have witnessed people believe implicitly that all mathematical notation is universal and "global scope", so they have trouble recovering from thinking that $\oplus$ is funny kind of operation on funny numbers (as opposed to direct sum of suitable objects...).

For that matter, my experience teaching basic crypto to computer science and engineering people suggests to me that it is almost pointlessly difficult to get across the idea of integers-mod-$N$ as equivalence classes, and the operation of addition therein as something in its own right. Instead, for most basic purposes I found that the $n\% N$ operation was more palatable, and could be explained quickly.

And then if you're addressing more sophisticated people, very possibly you can use plain-old $+$ and just say "in $\mathbb Z/36$", rather than have non-standard symbols.

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    $\begingroup$ Just be careful with $\%$ for negative operands, programming languages have some weird definitions. $\endgroup$ – vonbrand Sep 27 '15 at 0:39

In brief: it's your article, you can do what you want. You also reap the reward.

If the idea is to just mention arithmetic in base 36, without any other context, you can use your own algebraic symbols. That there is a relation to base 10 addition might suggest your using a symbol with a plus shape, but by itself it may not matter.

If the idea is to do arithmetic in base 36 and in other bases, it might make sense to use different universes and different symbols. So your example might be instead box{A} box{+} box{4} box{=} box{E} . I use box{} to indicate that for each symbol that normally represents an A or a 4, I am using a different symbol that is related to the symbol I normally use, so that I have some familiar handle to use to grapple with these unfamiliar objects and strange new operations and relations. For each different base, use a different box.

If the idea is to do arithmetic in many bases, AND SHOW HOW THEY ARE RELATED TO WHAT YOU ARE DOING, then I recommend a different approach. You should use a + for something familiar, another symbol for an operation in an unfamilar base, and a procedure or mapping or way of translating one problem to another. You should say that normally you might do a base 10 computation like 10+4 to get 14, but because it is base 36, you have to "box" everything and so the computation, while on some level is the same, has a different representation: box{10+4} gives box{10} box{+} box{4} which is A_ +_ 4_ , which is E_ in our new representation. Hey, box{14} is also E_! Maybe this new world isn't so strange after all!

It really matters what connections you are trying to establish. I think people should know about the mapping as well, so that they can make the translation themselves. Hide that mapping, and the reader is left with wondering why bother with a different base.

Gerhard "Notation Is Important; Intent Importanter" Paseman, 2015.09.27


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