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I'm reading the book "Foundations of Chemical Reaction Network Theory" (it's mostly a mathematics book, and not a chemistry book), where the following quote appears:

We denote by $\mathbb{R_+}$ the positive real numbers and by $\mathbb{\bar{R}_+}$ the nonnegative real numbers.

But this is confusing: what's the difference between a positive number and a nonnegative number? Is this just a way to include $0$, as it's not strictly a positive number though also not negative (and therefore nonnegative)?

agaminon
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    The only difference is $0$. "nonnegative" is equivalent to "positive or zero". – Hayden Aug 01 '22 at 22:13
  • The term "nonnegative" is intended to mean "not negative", so it refers to the numbers that are not negative, which are the positive numbers and zero. This assumes, of course, that we're not using the Bourbaki definition of "positive" (e.g. see this answer). – Dave L. Renfro Aug 01 '22 at 22:18
  • "Nonnegative" is a common way to say $\geq 0$, since "(not) $(x < 0)$" is equivalent to $x \geq 0$. Admittedly the double-negative can be a little confusing if you haven't seen it before, but it's standard in math writing. – Jair Taylor Aug 01 '22 at 22:34
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    There's zero difference. (Sorry .... :P) – Noah Schweber Aug 01 '22 at 23:05
  • To beat a dead horse. $\mathbb R_+ = {x\in \mathbb R| x > 0} = (0,\infty)$. Bue $\overline{\mathbb R}+ = {x\in \mathbb R| x\ge 0} = [0, \infty) = \mathbb R+ \cup {0}$. – fleablood Aug 01 '22 at 23:29
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    "Is this just a way to include 0, as it's not strictly a positive number though also not negative (and therefore nonnegative)?" Yes-- that is exactly what it is. (don't you love it when that happens?) – fleablood Aug 01 '22 at 23:31

2 Answers2

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Zero or $0$ is the difference. set of nonnegative real numbers was one element extra that is $0.$

The set of positive integers is $\{1,2,3,4,\dots\}.$

The set of nonnegative integers is $\{0,1,2,3,4,\dots\}.$

For set of 'real numbers' you would need to include many other non-integral numbers too to both the above sets but still there is only one difference i.e. $0.$

Thomas Andrews
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ishandutta2007
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"Non-negative" means "positive or zero", i.e. $\overline{\mathbb{R}}_+ = \mathbb{R} \cup \{0\}$.

This is consistent with the notation of $\overline{S}$ representing the "closure" of the set $S$, which in this case means taking the positive real numbers and adding the boundary point $0$ which is important for taking limits, among other things.

ConMan
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