# Modular arithmetic – How to translate “modulus/ moduli”?

The english Wikipedia states:

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

I have doubts whether in German this word would translate to Modulus (plural Moduli) or Modul (plural Moduln).

As on a quick research I could not find neither version it also seems likely that the word is avoided altogether by only referring to the operation as in:

Es ist a kongruent zu b modulo m.

Which version is it?

• I would consider modulus as a non-translated English substantive, for which a perfect translation exists, and therefore avoid to use it in German. – guidot Apr 11 '19 at 7:15

Consider the following example expression from modular arithmetic:

x² ≡ y    (mod n)

X Quadrat ist äquivalent zu Y modulo N.

Here, modulo denotes the Modulo operation, and n is referred to as der Modulus, the plural of which is die Moduli. Actually, modulo is the ablative case of the Latin word modulus. Gauß himself used Modulus/ Moduli, not only in his Latin work Disquisitiones Arithmeticae, but also in German texts (see, for example, various passages in Carl Friedrich Gauß. Werke. Band II). Nowadays, some people refer to n as der Modul (with stress on the first syllable), the plural of which is die Moduln. For example, in his famous book „Das ist o. B. d. A. trivial!“ Beutelspacher (2009, 9th edt., p. 81) writes:

„Man nennt die Zahl n den Modul. Früher hieß dies (auf Lateinisch) der modulus; daher kommt der traditionelle Plural die Moduli; heute kann man aber auch Moduln sagen.“

Personally, I stick with Gauß and use Modulus/ Moduli, when I mean the concept in modular arithmetic. In the realm of linear algebra, I use Modul/ Moduln to denote Abelian groups that are acted upon by elements of a commutative ring.

As a last point, there is the term das Modul (with stress on the second syllable), the plural of which is die Module. Outside of mathematics, it is used to refer to any kind of modular unit.

• Modulus/Moduli is the correct English term. Are you sure it's also used in German? Because the German examples I found use Modul/Moduln, e.g. »die damit begann, dass Gauß die Bezeichnung a ≡ b mod m - gelesen: a kongruent b modulo m - einführte [...] Als "Modul" wurde zunächst die Zahl m [...]« (Page 2) – mtwde Apr 11 '19 at 9:15
• I personally like the version Modulus/Moduli better, especially to avoid any confusion with the other kind of Modul you mentioned (this one de.wikipedia.org/wiki/Modul_(Mathematik)). But probably Beutelspacher is right about the fact that it is antiquated or latinish. – Fleur Apr 11 '19 at 12:32
• Äquivalent, really? I only know it as congruence, so I would expect congruent or kongruent. And in German, I also use Modulus/Moduli. Modul/Moduln sounds wrong. – Rudy Velthuis Apr 12 '19 at 9:21
• @FloraHerzner: antiquated, perhaps, but I would call it better compatible with international terms (modulus). And it distinguishes it from the other meaning, which is called module, not modulus, in English, i.e. also differently. I think the use of Modul in German for what is called modulus in English is probably due to confusion with module and it somehow stuck. – Rudy Velthuis Apr 12 '19 at 9:26
• @RudyVelthuis Ich kenne beides, zumal die Kongruenz modulo n auf den ganzen Zahlen eine Äquivalenzrelation ist. Aber ich gebe zu: kongruent wird häufiger gesagt. – Björn Friedrich Apr 12 '19 at 9:43

The word you are looking for is indeed

Modul (pl. Moduln)

duden.de

But i have also found a script with another plural form

Modul (pl. Module)

Quoting page 7

Sei ℤm = {0,1,...,m-1}. Der ganzzahlige Rest r bei Division von a durch m

r = a mod m

ist diejenige Zahl r ∈ ℤm, für die a – r ein Vielfaches von m ist.

Die Zahl m heißt Modul.

Page 12

Der Satz gilt nur, wenn die Module teilerfremd sind. Sind sie nicht teilerfremd, so kann das System keine oder mehrere Lösungen in Zm haben.

Edit

After looking at some other scripts

Für alle a,b ∈ ℤ und jeden Modul m ∈ ℕ gilt:

Zu den wichtigsten algebraischen Konstrukten gehören [...] inbesondere die Moduln

and my own old math notes I think it's as Duden says:

Der Modul, pl: Die Moduln

• Was ist mit "Modulo" (Ez.)? – user unknown Apr 10 '19 at 22:22
• @userunknown m.W. bezeichnet Modulo nur die Rechenoperation (mod) – mtwde Apr 10 '19 at 22:30
• Die deutsche Wikipedia unterstützt auch die Variante "Modul, pl. Moduln". – IQV Apr 11 '19 at 6:28
• Thank you very much for your answer. The following statement most likely refers to the other kind of mathematical object called "Modul". "Zu den wichtigsten algebraischen Konstrukten gehören [...] inbesondere die Moduln" But as the Duden uses Modul in the modular arithmetic way, your answer is definitely correct. On the other hand I try to avoid confusion of modular arithmetic with the algebraic object, and will stick to the Gauss version. @IQV That "Modul" is also not the one from modular arithmetic, so I think my concern is justified. – Fleur Apr 11 '19 at 12:43

The German term for this is Modul. You can find it in many German books on number theory, and as an example I list here a section of a famous book by Adolf Hurwitz, which is called Die ganzen Quaternionen nach einer ungeraden Zahl als Modul.

Note that the English word modulus and German Modul have several other meanings in mathematics.

x² ≡ y modulus n

does not make grammatical sense. (Just try to explain the use of the noun modulus in that sentence.) So to make sense you have to understand something like

x² ≡ y with modulus of n

This is a bit clumsy but as modulus was Latin and all mathematicians had learnt Latin anyway the solution was simple. Just say

x² ≡ y modulo n

Here modulo is ablative which can be used without a preposition to mean "with modulus" and since numbers don't usually decline (and certainly not when written as 6 or n) we can treat n as genitive and translate as of n if we choose.

So when translating into German you either need a preposition, which is clumsy, or to stick to Latin.

I would say modulus arithmetic but modulo arithmetic makes some sense as it means "arithmetic with modulus". I don't think modulo operator makes sense. I would say modulus operator. I think computer scientists invented the modulo operator because of their well known lack of knowledge of Latin grammar as shown here.

• You are certainly right; nonetheless, this is more a comment than an answer to the OP's "question": I have doubts whether in German this word would translate to Modulus (plural Moduli) or Modul (plural Moduln). – Björn Friedrich Apr 12 '19 at 12:19