# Word to describe a mathematical variable that repeats, like an angle or time

In mathematics a variable can be said to be constrained if it must lie within certain bounds, for example:

``````0 < x < 1  (the variable x is constrained: it lies between zero and one)
``````

However some kinds of variables (such as an angle, or time of day) can be constrained with an additional property that their values form a closed loop: if I increase the value past the upper bound it reappears at the lower bound. For example:

``````11 o'clock + 2hours = 1 o'clock
``````

Is there an adjective or name to describe this kind of 'circular' variable?

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@Bogdan: I guess you might know better than I would since I'm not a frequent contributor to math.SE, but by analogy, if this came up on the physics site, where I'm a moderator, there's a chance it would be considered off topic since it's arguably about language, not physics. –  David Z Aug 24 '11 at 18:30

What you've described is referred to as "wrapping" and a variable that wraps around is called a wrapped variable.

As jimreed rightly points out, the cause of wrapping is due to modular arithmetic and it is very uncommon to explicitly mention that a variable is "wrapped" because it is assumed to be understood. The only time I've seen mathematical/scientific articles use the term wrapped variable is when they also talk about "unwrapping" the variable. So in your clock example, you would add 12 after each time it completes a full circle to get an unwrapped variable.

I don't have a wiki link to the definition (because, as I said, it's not used commonly). However, the article on wrapped distribution uses the term wrapped variable.

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You could use periodic if the repetition follows any kind of function.

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"Periodic" is the first thing that comes to my mind (though I come from a physics perspective, not pure math). –  David Z Aug 24 '11 at 18:42
I have extensive classroom mathematics experience and `periodic` is the term we almost always used for a function that repeats. The "length" of each repetition is called the `period` of the function. –  Rice Flour Cookies Sep 6 '11 at 14:40
Yeah, its periodic. See complex phasors for the basic idea. –  bobobobo Nov 6 '11 at 23:35

You might say that the variable is cyclic although that term more correctly applies to the values the variable can have than the variable itself.

Instead of talking about the variable, I would say that the operations on that variable use modular arithmetic.

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+1 for 'cyclic' –  Richard Inglis Aug 24 '11 at 16:35
it is not "cyclic", since Z (the set of integers) is cyclic but unbounded. Also, it is not "modular arithmetic" because time is a continuous variable, unless we restrict to the whole hours: 0,1,...,24. –  Theta30 Aug 24 '11 at 16:40