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In the context of modular arithmetic, I'm confused about the notation: "a (mod m)"

I understand what "$13 \equiv 1 \pmod{ 12}$" means. I understand the equivalence class "$[a]_m$". I understand the remainder operation "$a \ mod \ m$".

But I'm not sure what "a (mod m)" means. I think it's a set of equivalence classes, but I'm not 100% sure. I saw this passage in a book:

Usage of "a (mod m)" in book

I've been reading the Congruence classes section on Wikipedia, but it seems to me that section is using the notation "(a mod m)", which is yet another notation, to mean 2 different things:

  • "It is called the congruence class or residue class of a modulo m, and may be denoted (a mod m) or as [a] ..."
  • "Consequently, (a mod m) denotes generally the unique integer r such that..."

The related notations I'm aware of are:

  • $a \equiv b \ (mod\ m)$
  • a modulus m
  • $a\ (mod\ m)$
  • $a\ mod\ m$
  • $(a\ mod\ m)$
  • $[a]_m$

Regards

Bill Dubuque
  • 282,220
Dess
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  • I’m not sure I understand your question. You are aware that $\mod m$ may either return a number or a set of numbers; it returns either the remainder or the whole congruence class. It is sometimes ambiguous, but usually not important. – Malady Jun 09 '25 at 02:41
  • It is confusing because $\pmod m$ doesn't apply to $a,$ it applies to $\equiv.$ There is also a binary operation, $a\bmod m,$ but the parentheses are only used to modify the equivalence. – Thomas Andrews Jun 09 '25 at 02:41
  • I've seen some cases using $a\equiv_m b$ rather than $a\equiv b\pmod m.$ That is often useful when the expressions on either side are complex, and there is a long gap. But it is rarely used. – Thomas Andrews Jun 09 '25 at 02:45
  • But it looks like the book is using $1\pmod m$ to mean the same thing as $[1]_m.$ that seems confusing to me, but it might be an old notation? – Thomas Andrews Jun 09 '25 at 02:52
  • What is not clear? In the first excerpt they explicitly say that $,a\pmod{! m},$ is a congruence equivalence class so $,a\pmod{! m},$ means $, a + m\Bbb Z.,$ In the Wiki excerpt $,(a\bmod m),$ denotes the result of the binary mod operation, i.e. the remainder of $,a\div m,$ (the paren's are not needed), see the linked dupe. $\ \ $ – Bill Dubuque Jun 09 '25 at 02:53
  • From the linked post: Beware that some authors write "a(mod n)" to denote "a mod n", while others use it to denote the entire congruence class "$[a]_n=a+n Z$". Keep in mind that such notation overloading might lead to confusion for beginners, – Dess Jun 09 '25 at 03:06
  • @BillDubuque You mentioned that, in the wiki, it means the remainder. I agree they use it like that, but it also says this: "It is called the congruence class or residue class of a modulo m, and may be denoted (a mod m)" -- Do you mean that, in that sentence, it also means remainder? – Dess Jun 09 '25 at 03:09
  • @ThomasAndrews It seems the book used "a (mod m)" as a set of equivalence classes. But, I did wonder if it was also used to mean a specific equivalence class. And, further, since Wikipedia seems to be using yet another notation for this, if there's been a mistake somewhere. That "mod" notation seems to be very overloaded. – Dess Jun 09 '25 at 03:14
  • Btw. Check this out. I asked what I felt was a genuine question. I searched the site before I asked, but I didn't find a specific answer. I looked online, I did my research; I put effort into the question. But, despite this, the question was closed and now I can't asked any more question. This site makes it excessively difficult to ask a question such that it won't get tanked. – Dess Jun 09 '25 at 03:19
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    I have never seen a mathematician use the notation $,(a\bmod m),$ to denote a congruence class (that would seriousy conflict with notation for ideals). It also conflicts with Wiki's later use of it to denote the remainder. Wikipedia should not be considered an authoritative source on such matters. The linked dupe describes standard math notation. Best to ignore the Wiki nonsense. Note Wiki's notation $,a\pmod{!n} = a+n\Bbb Z,$ denotes a single equivalence class. It is $,\Bbb Z/m\Bbb Z,$ that is the set of all such classes (i.e. the set of all such classes for all $,a\in\Bbb Z).,$ – Bill Dubuque Jun 09 '25 at 03:24
  • Btw, the Wiki edit, 21 Jan 2023 that added the notation $,(a\bmod n),$ for $,a+m\Bbb Z,$ was done by D. Lazard, who is in fact a mathematician. It may be that this notation is peculiar to a geographical region (e.g. France), or it may simply be a typo. You can ask Lazard where he found such notation. $\ \ $ – Bill Dubuque Jun 09 '25 at 05:57
  • I like to quote humpty dumpty in these matters "The question (of definitions) is which is to be the master". You know what equivalences classes are. You know what the remainder function is. And you know the difference. So if you see someone write imprecise $a(\mod n)$ or $a \mod n$ or $(a \mod n)$ (huh??) or, god forbid $(a\mod )n$ [never seen that but some authors will take it as a challenge]. You know your el from you assbow so you know whats going on. – fleablood Jun 09 '25 at 15:00
  • "This site makes it excessively difficult to ask a question such that it won't get tanked." And the way to deal with that is to quote the who. "Don't give a damn about your reputation". – fleablood Jun 09 '25 at 15:01
  • @fleablood "And the way to deal with that is to quote the who..." -- It's not about the reputation. It's about the fact that, since I dared asked questions which got marked as a duplicate, my questions(s) got closed, and now the site won't let me post further questions. That's a weird system: You use the site as designed and you're penalized because of it. – Dess Jun 10 '25 at 00:49

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