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?

• @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.

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

• "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.

• 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

This is a math question.

The answer is yes, periodicity isn't confined to being in one dimension.

Euler's formula sums up the idea of the sum e^{jt} = cos(t) + jsin(t) travelling in a circle in the complex plane as t goes from 0 to 2pi. See also phasors.

Another word for this type of variable is modular.

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7+8=15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below, 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 is congruent to 0 modulo 12.

You can also refer, as suggested, to a "modular variable."