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For any number $x$, there exists a unique number $y$ such that the difference $y-x$ is a integral multiple of the number $2\pi$, and that $y\in[0,2\pi)$. Is there a single word or a single wording to express the number $y$? I mean, some word of the form * used as "$y$ is the * of $x$".

All of the words "radian", "angle", and "degree" can be used to denote a size not in the interval $[0,2\pi)$.

If such a terminology corresponding to the interval $(-\pi,\pi]$ is presented, I will be also appreciated.

Jack M
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kwgl
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2 Answers2

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One way to think about this would be to think this things like the domain of sine, cosine, etc., as being the space $\mathbb R/2\pi$ (read as "r mod two pi") rather than $\mathbb R$. A point in $\mathbb R/2\pi$ is an equivalence class of points of $\mathbb R$, where two points are equivalent if they differ by an integer multiple of $2\pi$. Two values of $x$ would be equivalent, i.e. would belong to the same equivalence class, if they both yield the same value of $y$ that you describe above.

Otherwise, you could use the notation $x\bmod2\pi$ and at the outset explain to the reader that each time you use that expression, you mean the concept you describe in your question.

Michael Hardy
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    Thanks for the perspective and the alternative solution! I think the former is good idea though not much suitable to my specific problem. The latter has been forbidden by an advisor of mine. :) – kwgl Aug 23 '14 at 23:08
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If the angle is being thought of as the argument of a complex number, then principal value is the term you're looking for. It's a common enough term that using the term in other contexts should seem reasonable to most readers.

Jack M
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  • Thanks for reminding the terminology "principal value"! It is defined to be in interval $[0,2\pi)$ or the interval $(-\pi,\pi]$. Thus it is neither definite. I am considering using it, by defining it to be in the interval $[0,2\pi)$. Also hope to see other solutions for my problem. – kwgl Aug 23 '14 at 23:04