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I found a similar question asked about a decade ago in the English stack exchange, but there were no definitive answers, so I hope this is ok to ask here, where people may have more relevant experience.

When writing, what is the correct verb form for a modulo function?

For example: "function f(x) takes a logarithm of base 5, multiplies by 2, adds 3, subtracts 5, and.... modulos by 9(???)"; or "we then modulo 10 by 5"

The best I can come up with is "performs a modulo function with quotient 9", but I don't know if that's correct, and it sounds like there should be a simpler word or phrase.

T.S
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    Not sure there is a universal solution here. I tend to say something like "$f(n)$ is $n^2$ +$3(2n+n^2)^8$, all mod $9$" or something like that. – lulu Aug 08 '21 at 19:04

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I might say "reduce mod 5" or "take the remainder mod 5", and to be clearest, I'd specify the output range of the reduction operation, e.g. "reduce mod 5 into $\{0,...,4\}$".

It's a common misconception that the "mod" notation in math primarily signifies this reduction operation. Often, it just describes the equivalence relation (modular congruence) that holds between any two integers that differ by a multiple of the modulus. E.g. $28\equiv38\pmod{10}$ and $28\equiv8\pmod{10}$ are both true. (See this answer for more on the two usages.)

Karl
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    Cf. this answer – J. W. Tanner Aug 08 '21 at 20:49
  • Not true. In fact both the operational and relational use of $\bmod$ are widely used in mathematics (and many applications such as computer science). The denigration of the operational form is (alas) common, and seems to be rooted in prejudices against applied mathematics (e.g. computer science). If you disallow use of normal-form operations then you unnecessarily cripple yourself. They prove quite convenient in many mathematical contexts. – Bill Dubuque Aug 09 '21 at 01:29
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    @BillDubuque Thanks - of course it would be silly to disparage the use of a well-defined operation. I still think it's fair to say it's a misconception that the operational form (especially with some particular output range) is the only meaning of "mod". Hopefully people searching for things like "modulo function" will find both of our answers useful. – Karl Aug 10 '21 at 00:03
  • Some students learn about $!\bmod$ the operator before they learn about its relational denotation (congruences). So at that point in their education it does have a unique denotation. But that is not the issue that I address in the above comment. Your edit (adding "primarlly" before "signifies" ) changes the meaning of your claim. – Bill Dubuque Aug 10 '21 at 01:59
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There is no good English verb for this operation.

In any particular context where you need words and not formulas you must choose an appropriate expression. You might say

find the remainder when you divide by $9$

The particular example you ask about would never occur naturally. Even algebraically it makes no sense, since the logarithm to (not of) base $5$ will not usually have an integer value you can then find the remainder of.

So your hypothetical question does not have a good answer. If you ever have a concrete question like this one, ask it.

Ethan Bolker
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  • Interesting. I thought that the process of dividing one real number by another to come up with an integer quotient and a real remainder went back to antiquity. E.g., Euclid's proposition X.2 is proved by an analysis of what happens when this process is iterated. – Rob Arthan Aug 08 '21 at 22:41
  • @RobArthan Of course you can always divide by any divisor $d$ and find the remainder between $0$ and $d$. I think calling that modular arithmetic is rare, Only "modulo $1$" for the fractional part of a number and "modulo $2\pi$ when thinking about periodicity are common outside number theory. – Ethan Bolker Aug 09 '21 at 00:46