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If we have $a \equiv b \pmod{n}$ then $a$ and $b$ are congruent to each other modulo $n$, correct?

What do we "call" $a$ and $b$? Because sometimes these numbers can be negative. Would they be remainders? Residues? Do we say these remainders (residues?) belong to the same "congruence class mod $n$"? Are remainders and residues synonymous?

Trying to get the terminology / lingo right here. When do we use which words? How do we describe what all this notation is representing / saying?

Ethan Bolker
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Sean Hill
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    Yes; see congruence relation : "two numbers $a$ and $b$ are said to be congruent modulo $n$". $a$ and $b$ are numbers and $n$ is called the modulus of the congruence. – Mauro ALLEGRANZA Oct 31 '18 at 13:35
  • Are $a$ and $b$ remainders? Residues? – Sean Hill Oct 31 '18 at 13:37
  • "Like any congruence relation, congruence modulo $n$ is an equivalence relation, and the equivalence class of the integer $a$, denoted by $a_n$, is the set ${ \ldots, a − 2n, a − n, a, a + n, a + 2n, \ldots }$. This set, consisting of the integers congruent to a modulo $n$, is called the congruence class or residue class or simply residue of the integer $a$, modulo $n$." – Mauro ALLEGRANZA Oct 31 '18 at 13:38
  • Thus, residue of $a$ modulo $n$ is the set of all integers congruent to $a$ modulo $n$. – Mauro ALLEGRANZA Oct 31 '18 at 13:40
  • Huh for some reason I thought residue meant like remainder or a single number and not a set of integers, interesting. Is there a word for a given integer taken from the residue of $a$ mod $n$? – Sean Hill Oct 31 '18 at 13:44
  • reminder is pertinent because $a$ and $b$ are congruent modulo $n$ exactly when they have the same reminder when divided by $n$. Thus, the conguence class (or residue) contains all numbers that have the same reminder. – Mauro ALLEGRANZA Oct 31 '18 at 13:50
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    @Sean "residue" is an overloaded (ancient) term. It can mean either a representative of a congruence class or the entire class. It can also mean "congruent", e.g. older texts often write "$b$ is a residue of $a$ modulo $n$" for $a\equiv b\pmod{n}.,$ It predates modern (abstract) algebra so one can't expect it to be completely consistent with modern syntax and semantics. – Bill Dubuque Oct 31 '18 at 14:02
  • @MauroALLEGRANZA Does it make sense to say that a negative number can have a positive/negative remainder? For example $-10$ divided by $7$ might have remainder $-3$ or remainder $4$? – Sean Hill Oct 31 '18 at 14:06
  • @Sean remainders typically denote normal / canonical reps, e.g. least nonnegative or least magnitude (signed). reps, e.g. $\bmod 5$ they are ${0, 1, 2, 3, 4}$ or ${ -2, -1, 0, 1, 2 }. $ More generally we can use any complete systems of incongruent reps – Bill Dubuque Oct 31 '18 at 14:08
  • We have that $-10$ divided by $7$ gives a quotient $q$ and a reminder $r$. The rule is : $7 \times q+r=-10$. – Mauro ALLEGRANZA Oct 31 '18 at 14:10
  • See the post : negative number divided by positive number, what would be remainder? – Mauro ALLEGRANZA Oct 31 '18 at 14:11
  • And see modulo operation. – Mauro ALLEGRANZA Oct 31 '18 at 14:12

1 Answers1

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In a congruence are the numbers remainders or residues?

If you want to give it a name, the word should be "residues" (at least according to some books). More precisely, we have the following

Definition: Let $a,m\in\mathbb Z$, $m\geq 1$. An integer $b$ such that $a\equiv b \pmod{m}$ is called a residue of $\boldsymbol a$ modulo $\boldsymbol m$.

Sources: See the references in MathWorld, Encyclopedia of Mathematics and Proofwiki (but be careful with the pages as pointed out by @BillDubuque in the comments). Explicitly, see p. 55 of Shanks, p. 49 of Hardy & Wright, and p. 54 of Andrews.

Addendum: If you want a more modern reference to verify that the term is still in use in this way today (at least by some authors), see p. xi of Deza or p. 38 of Fine.

Addendum 2: the "remainder" is called the smallest non-negative residue (Deza, p. xi) or the least residue (Koshy, p. 214). It is not the same as "residue".

Addendum 3: If you want a historical reference, the aforementioned use of the term "residue" was also used by Gauss when the modern theory of modular arithmetic was being born (see Disquisitiones Arithmeticae , p. 1).

Pedro
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  • Random web pages should not be trusted to be authoritative. None of those pages make any sense (in fact all of them are ill defined so will only further confuse the matter). – Bill Dubuque Oct 24 '24 at 23:24
  • @BillDubuque Your opinion is not authoritative either. What about "the references therein"? And the pages are not random. They were carefully chosen. – Pedro Oct 24 '24 at 23:32
  • I have more than enough experience with number theory to be familiar with the many uses of the term. But one doesn't even need such to recognize that your linked "definitions" are ill-defined. – Bill Dubuque Oct 24 '24 at 23:35
  • @BillDubuque I really respect your experience. But those who wrote the pages and the books are experienced too. And clearly there is a consensus in the use of the term, which is reflected in the pages and supported by some books (see Shanks, p. 55). – Pedro Oct 24 '24 at 23:44
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    A problem with MW is that it says the residue when it should say a residue, and it has nonsense like "remainder of a congruence". A problem with EM & PW is that they equate "remainder" with "residue" [but remainder usually refers to a canonical ("least") residue, i.e. some normal form rep], which contradicts your answer ("they should be residues'). – Bill Dubuque Oct 25 '24 at 00:27
  • "Residue" is archaic and should be avoided in modern texts (except possibly residual uses in established terms like "quadratic residue" and "complete residue system"), e.g. this is what is done in the introductory textbooks by Silverman and Stark. – Bill Dubuque Oct 25 '24 at 01:40
  • Or like in Vinogradov's introduction a "reduced system of residues" has a special meaning. – suckling pig Oct 25 '24 at 01:47
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    @non That's what I meant by "like complete residue system" (the (poorly named) reduced system is just a subset of all units (invertibles) in a complete system). $\ \ $ – Bill Dubuque Oct 25 '24 at 01:54
  • @BillDubuque to be sure, I was basically just seconding what you said – suckling pig Oct 25 '24 at 03:57
  • @BillDubuque Modern texts (for example, Deza e Koshy) still use the term (see my edit). Also, if you accept the term "complete residue system", what is the problem with calling the elements of this set "residues"? Isn't it natural for a "class of residues" to be formed by "residues"? – Pedro Oct 25 '24 at 11:19
  • Your claim about Deza & Koshy is incorrect. Deza never uses "residue" alone. Instead they use the more modern "representative" of a residue (equivalence) class (except for one paragraph on p. xi where they use it only to convey the etymology of "least / minimal residue", which itself is only used in one other place besides p.9). Similarly Koshy never defines "residue" but only "least residue" (:= remainder). In particular neither use old-fashioned ad-hoc language like your "$b$ is a residue of $a$ modulo $m$". $\ \ $ – Bill Dubuque Oct 26 '24 at 00:20
  • @BillDubuque (1) Thanks for the correction, I should have said page xi. I meant that Koshy begins using the term without any reference to classes. (2) Saying that $b$ is a representative of the class is equivalent to saying that $a\equiv b$. So the use is occurring in the same sense that I mentioned. (3) Why the term is used, or how often in a book, or what other terms it is accompanied by, was never part of either the question or my answer (furthermore, the justification you gave for Deza's usage is exactly why I use the term when I'm teaching modular arithmetic; but, as I said, that was – Pedro Oct 27 '24 at 00:24
  • never a point). So, none of your arguments refute the fact that, nowadays, some authors call an integer $b\equiv a$ a residue of $a$. (4) Something is not old-fashioned if it is still used. For example, Fine (2023, p. 38) says that if $x ≡ y$, we say that $y$ is a residue of $x$. This is recent and printed in a book. You can't deny it. (5) According to you, no one used it and no one should use it. Then, it could be used with certain exceptions. Then, it could be used outside of these exceptions for certain purposes as long as it was not done "alone" (making references to classes as in Deza). – Pedro Oct 27 '24 at 00:25
  • Finally you can use the term without class reference as long as it is a special kind of residue (as in Koshy). I don't know what your next point will be. But this is my last comment on the subject. – Pedro Oct 27 '24 at 00:25
  • Wrong. Fine is the same as I said above for Deza - although they mention once in passing the archaic language "b is a residue of a modulo m" (for etymology / history) they never use it (it occurs only implicitly on p.101 & p.119 when they use minimal / least remainders (normal form reps) - same as in Deza). Instead they use the modern language of congruence (equivalence) classes - as do almost all modern number theorists. Re: (2) No one claimed $,a\equiv b\pmod{m},$ is archaic. – Bill Dubuque Oct 27 '24 at 01:27