# What is the name of the function f(x,y) = ((x-1) mod y)+1 ?

Does the function $f(x,y) = ((x-1) \mod y)+1$ have an existing name?

f(1,5) = 1
f(2,5) = 2
f(3,5) = 3
f(4,5) = 4
f(5,5) = 5
f(6,5) = 1
f(7,5) = 2

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tagged as linear algebra? o_O –  darij grinberg Jun 4 '10 at 16:02
Hardy seems worthy of a name. It's more or less mod. –  supercooldave Jun 4 '10 at 16:08
but since we traditionally label elements {1,2,...,y} instead of the better {0,1,...,y-1}, it is actually a good question, I'd say. –  Martin Rubey Jun 4 '10 at 16:11
I know its not the most interesting function in the world. I just want to check if it has a well known existing name to avoid confusing my readers. –  Niall Murphy Jun 4 '10 at 16:46
In that case, what kind of readers do you expect? –  Will Jagy Jun 4 '10 at 17:49

In math, as opposed to in computer science, when you apply "mod y" you land in the integers modulo y, denoted Z/yZ, not back in the integers. This means that mod 7, the symbols 1 and 8 denote the same thing, i.e. the equivalence class {...,-13,-6,1,8,15,...}.

A more computer-y way to say this is that for mathematicians, "integer mod 7" is a different kind of data class than "integer."

All this is just a long way of saying that this is probably the wrong place to ask your question.

If I were defining this function in a math paper I'd say something like "Let f(x,y) denote the unique number in {1,2,...,y} which is congruent to x modulo y" or "By the division algorithm there exists a unique number in {1,2,...,y} which is congruent to x modulo y, we denote this number f(x,y)."

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Thats pretty interesting, thanks for the detail. So I think from this information it is probably better to just name the function something like "mmod". –  Niall Murphy Jun 4 '10 at 18:48

Well, it's the quotient map from $\mathbb{Z}$ into $\mathbb{Z}/5\mathbb{Z}$, with the slightly unconventional relabelling of 0 as 5. Nothing much more interesting than that.

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You could conceivably describe this as the "least positive residue" modulo y (as opposed to the "least non-negative residue" modulo y).

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"least positive residue" is a name for normal modulo in some textbooks. Some people reference the "least positive (non-zero) residue" but i think it still doesn't capture that the number y is in the range. –  Niall Murphy Jun 5 '10 at 19:00

I think that my yesterday silly question implicitly gives a name to your function -- the feng shui function. To be more specific, for applying some knowledge from the feng shui system, one needs to start with his house number, say $n$, compute its sum of decimal digits, then do this again with the resulted number, and so on. The final result is a decimal digit in the range $\lbrace 1,2,\dots,9\rbrace$, namely $f(n,9)$, as you may check. If one does a similar computation in base $b$, the result is always $f(n,b-1)$. Isn't this a good reason for the name?!

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Could be. In "Concrete Mathematics" Graham, Knuth, and Patashnik, define a kind of reverse modulo they call "murmur". They then joke that since Mod was a fashion in the 60s, the murmur function, defined in the 70s, should be called Punk! So what should the feng-shui function be dubbed? Hip? GaGa? ...? –  Niall Murphy Jun 5 '10 at 19:08
As I am far from what is in fashion right now, I could refer to my son's understanding of that: the feng-shui function is Ska. –  Wadim Zudilin Jun 6 '10 at 0:44