# What’s the correct usage of “modulo”?

How would we rephrase the following sentence to be grammatical? :

10 modulo 3 equals 1.

By "grammatical", I probably mean something along the lines of standard American English.

Initially, I'd thought that its grammatical, but dictionary.com claims that modulo is an adverb and not a verb, so it seems that the above usage is wrong.

What’s the standard and accepted usage of the word modulo?

• I don’t understand the question. Modulo can be preposition or an adjective. It is neither a verb nor an adverb.
– tchrist
Jan 28, 2012 at 15:19
• AHED, Webster, Oxford, and Wordnik all list modulo as a preposition. Dictionary.com is in a distinct minority, and I wouldn't worry about phrasing something grammatically based on its possibly erroneous categorization. Jan 29, 2012 at 5:53
• If you want the mathematical sentence translated, it means "10 divided by 3 leaves 1 as a remainder". As I suspect you already know. The prepositional use of modulo outside that context is much more fraught; it's a vague nod to certain boundary conditions, and it can be way overdone, especially in Biz Biz. Jun 25, 2014 at 18:18
• Modulo is the name of a math operator. If "Two plus three equals five" is grammatical, so is 10 modulo 3 equals 1. Jun 26, 2014 at 19:23
• @Oldcat The cool kids all call it 'mod'. Jun 27, 2014 at 22:49

Mathematics would say: "10 is congruent to 1 modulo 3". The modified usage "10 modulo 3" with "modulo" as an operation (like "10 plus 3") is from computer programming.

• No, modulo as a preposition is older than computer programming is. Check the OED for mod, modulo. These are from the 19ᵗʰ century (barely): the first documented use appears to be Gauss in 1801.
– tchrist
Jan 28, 2012 at 15:10
• Yes, what I wrote is correct. It does not contradict "modulo as a preposition". Jan 28, 2012 at 19:29

The verb in your sentence is equals. The entire sentence can be parsed either as 10 modulo 3 being a noun phrase with modulo 3 operating as an adjective, or as modulo 3 equals being a verb phrase with modulo 3 operating as an adverb. Whichever analysis is preferred, the usage is perfectly standard, if slightly imprecise. The reference you cite is correct to prefer is congruent to over equals.

• 10 is a noun right? so 10 modulo 3 = [noun][adverb][noun] and that's a grammatical phrase? Jan 28, 2012 at 10:49
• @Pacerier Binary operators like 4+5 or 10%3 are grammatically prepositions. Four plus five, ten modulo three.
– tchrist
Jan 28, 2012 at 13:18

In a non-searchable and potentially ephemeral comment to the original posting, Professor Lawler kindly presented the following answer:

If you want the mathematical sentence translated, it means “10 divided by 3 leaves 1 as a remainder”. As I suspect you already know.

The prepositional use of modulo outside that context is much more fraught; it’s a vague nod to certain boundary conditions, and it can be way overdone, especially in Biz Biz.

I’ve marked this posting Community Wiki because it is John’s answer not my own, and so I deserve no reputation from it.

The most syntactically sound way I can say "10 modulo 3 equals 1" is

``````(10 % 3) == 1
``````

That holds in a great many languages, but not spoken English. I do hear it spoken often enough as a computer scientist, though, and when it's not simply, "ten mod three," it's "ten modulo three," and occasionally "ten modulus three." There are other ways you could phrase it, but these are all that I have heard spoken, and this from the lips of exceedingly well educated computer science profs.

Modulus comes from Latin; it means "little measure." "Modulus" uses the the nominative ending, which is for subjects. "Modulo" uses the ablative ending, so you can translate as "from the little measure," "with the little measure," "by the little measure," or "in the little measure," among other possibilities depending on the context. In English, "modulo" simply becomes a preposition, like "plus." You could say, "the modulus of ten with respect to three," or, "ten modulo three," but "ten modulus three" is pretty shaky. The operator "%" I have seen called both a "modulus operator" and a "modulo operator," but my texts all say "modulus operator."