Why do the notation of the set of positive integers and the set of positive reals are different?

Question

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 $X^+$?

Answer 1

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.