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I read from my lecture notes that $\mathbb{R}_+^*=\{x|x>0\}$ and in http://mathworld.wolfram.com/PositiveInteger.html that $\mathbb Z^+$ is the positive integers. Why do we have to put plus to upper corner when we deal with positive reals and to lower corner when we deal with integers? Is there any general rule to remember that if $X$ is a set of numbers then is the set of positive numbers $X_+$ or $X^+$?

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    $\begingroup$ Well in the first example there's already something taking up the upper right hand corner, but I've seen $\mathbb{R}^+$ used to denote the same set, as well as $\mathbb{R}_+$, $\mathbb{R}^{ > 0}$ and $\mathbb{R}_{ > 0}$. The notation isn't universal in any way; there's not really a general rule like that: if you're writing something and have a set $X\subseteq\mathbb{R}$ and want to talk about only the positive elements inside of $X$, feel free to use $X_+$ or $X^+$ or any variant, as long as you say what you mean. I.e., say ``Let $X^+ = \{x\in X\mid x > 0\}$." $\endgroup$
    – Stahl
    Commented Dec 12, 2014 at 20:29

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The primary reason is that $\mathbb{R}_+^*$ has a star in the superscript, and it would look silly to put a plus sign there as well.

As to why we put the star there in the first place, it is because $\mathbb{R}^*$ is common notation for the set $\{x\in\mathbb{R}:x \neq 0\}$ equipped with the operation of multiplication of real numbers. The reason to consider this is that under this operation $\mathbb{R}^*$ is a group. Moreover just taking the positive part of this set, $\mathbb{R}_+^*$ is also a group. This is also something commonly done with the rational numbers $\mathbb{Q}$ and the multiplicative group of rational numbers, $\mathbb{Q}^*$. (Also the complex numbers, but for the complex numbers you don't have a notion of positive unless you restrict to the real axis.)

Most sets of numbers don't have an interesting algebraic structure, so you probably won't see a star on them. For these sets there is no convention on where the plus sign goes when you restrict to the positive elements. So long as you are consistent in your notation, you can use a subscript or superscript.

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