In English there exists the proposition modulo (see also the Wiktionary entry) or modulo the fact. Any of these, however, are not proper from English, but have their origin in mathematics, presumably from the expressionp≡q mod n. Parenthetically, the fact

6 ≡ 2 (mod 4)

is extended to more general (i.e. non-mathematical situations), where you can say:

All mammals, modulo the monotremes, give birth to live young. (Example taken from Wiktionary's link above).

Now to the actual question, which is constrained to math. In German one uses the same sign, mod or X/Y for X mod Y. While usual expressions involve only mathematical objects, one usually needs to write them down with words (here is where the question might be valid here). That is, it turns out that in concrete cases, when the involved expressions have a proper name, it doesn't sound too educated if one mentions the letters. Let me bore you with a last line specifying that:


  1. Consider F/Σn where Σn has a proper name, die symmetrische Gruppe. If you want to read out or write down that in German, then modulo should be a preposition. I'd read it as:
    "ef modulo der symmetrischen Gruppe"

  2. Consider a≡b mod n, where a,b,n are just numbers. Then after some lines it's valid to state that as: "[hier erwähnt man n]...und modulo diese(r) Zahl, ist a gleich b"

Which case should I use for modulo? (I'm going with genitive because of the answer given here, but I'd need help from an expert.)

  • 1
    Note that in German, „a gleicht b“ is not the same as „a ist gleich b“. „a gleicht b“ is more like „a is similar to b“. There is no direct correspondence to „a equals b“.
    – Carsten S
    Commented Feb 23, 2014 at 14:47
  • @CarstenSchultz "a equals b" (a=b) is exactly the same as "a ist gleich b". But the usage of = is wrong here in all cases. It's about congruence which is written using the triple bar sign: ≡ (u+2261) (according to DIN1302).
    – Toscho
    Commented Feb 23, 2014 at 15:32
  • @Toscho That would be my fault. But it actually isn't, because it's kind of hard to type in LaTeX in GLU ;) Anyway everybody would understand with a =, with a ~, with what appears in LaTeX for \simeq, etc. Thanks for finding the sign.
    – c.p.
    Commented Feb 23, 2014 at 15:34
  • @Toscho, I should have been clearer. I wanted to say that there is no translation of “a equals b” that preserves the structure of the original, i.e. is of the form „a [Verb] b“.
    – Carsten S
    Commented Feb 23, 2014 at 15:42
  • 1
    BTW: The wikipedia example is utter crap. modulo is misused for except for, which is set theoretic difference.
    – Toscho
    Commented Feb 23, 2014 at 15:53

2 Answers 2


Both algebraical constructs can be expressed using modulo.

Factorized constructs like $F/\sigma_n$ are read

  • F durch die symmetrische Gruppe or
  • F (faktorisiert) nach der symmetrischen Gruppe or
  • F modulo symmetrische Gruppe or
  • F modulo der symmetrischen Gruppe (I haven't heard anybody say F modulo die symmetrische Gruppe but I wouldn't even bother if I did.)

So in this case modulo is a preposition demanding genitive (, nominative) or a special construct used as part of an attribute.

Equality inside factorized constructs like $a\equiv b mod m$ is read

  • a ist kongruent (zu) b modulo m

So in this case modulo is a preposition as well but used as part of an adverb. If used with single variables like m, no case can be identified, but if used with grammatical constructs, it should be genitive. Example from Euclid's algorithm:

Bilde den Rest der ersten Variable modulo der zweiten Variable!

  • 2
    Interessant übrigens, dass sich bei Deinem letzten Beispiel auch die Frage stellt, ob man „Variable“ als substantiviertes Adjektiv deklinieren möchte („der Variablen“).
    – Carsten S
    Commented Feb 23, 2014 at 15:54
  • @CarstenSchultz Das ist meine ganz eigene Ignoranz.
    – Toscho
    Commented Feb 23, 2014 at 17:02

I do not know if I am an expert, but I am a German mathematician. I would use the genitive case.

  • Achso, wunderbar, dann kann ich dich vielleicht fragen, ob auch modulo etwa für $S^{n-1} \cong SO(n)/SO(n-1)$ geläufig ist. Oder sagt man dort durch?
    – c.p.
    Commented Feb 23, 2014 at 14:52
  • 1
    Gute Frage, ich bin mir ehrlich gesagt nicht ganz sicher. Ich glaube, ich würde „nach“ oder deutlicher „faktorisiert nach“ sagen, aber andere mögen das anders halten.
    – Carsten S
    Commented Feb 23, 2014 at 14:54
  • 1
    Auf jeden Fall Genitiv ;)
    – Carsten S
    Commented Feb 23, 2014 at 14:58
  • 1
    Der Genitiv ist in Ordnung, aber am geläufigsten ist mir hier der Nominativ. Dabei wird, was auch immer modulo genommen wird, als undeklinierbare Einheit betrachtet (so ähnlich manchmal wie Titel von Werken in Anführungszeichen), also z. B.: »F modulo [die symmetrische Gruppe]«, oder (Labor-Slang): »Das Ergebnis ist 2,53409 modulo Vorzeichen.«
    – Wrzlprmft
    Commented Feb 23, 2014 at 15:31
  • 2
    @Wrzlprmft, richtig, „modulo Vorzeichens“ würde ich auch nicht sagen, aber immer „modulo der Tatsache...“.
    – Carsten S
    Commented Feb 23, 2014 at 15:45

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