Is positive the same as non-negative? I would assume the answer to my question is yes, but I want to make sure because my book uses both terminologies.  Please also indicate where zero falls into the mix.
UPDATE:
Here is an excerpt from my book:

The definition of $\Theta(g(n))$ requires
  that every member $f(n) \in \Theta(g(n))$ be
  asymptotically non-negative, that is,
  that $f(n)$ be non-negative whenever n
  is sufficiently large. (An
  asymptotically positive function is
  one that is positive for all
  sufficiently large $n$.)

 A: The real numbers can be partitioned into the positive real numbers, the negative real numbers, and zero.  A real number is one and only one of those three possibilities.  This is called "trichotomy."  Non-negative (or, correspondingly, non-positive) means not negative (not positive), so zero or positive (zero or negative).
That is, non-negative includes zero whereas positive does not.
Edit for clarity:
Non-negative means zero or positive.
Non-positive means zero or negative.
That is, non-negative includes zero whereas positive does not and vice versa.
A: If we go by your edits, about the book excerpt, it looks like the book treats non-negative as $\ge 0$, and positive as $\gt 0$.
Also, from the notation it seems like you are talking about functions whose domain is $\mathbb{N}$.
For an example of an asymptotically positive function, consider
$$ f(n) = 1$$
For an example of an asymptotically non-negative function, consider
$$f(n) = \left|\sin\left(\frac{n\pi}{2}\right)\right|$$
For sufficiently large $\displaystyle n$, we have that $\displaystyle f(n) \ge 0$. Note that this function is not asymptotically positive, because it is zero (for even $\displaystyle n$) infinitely often.
Any asymptotically positive function is also asymptotically non-negative, but not vice-versa.
For an example of a function which is neither asymptotically non-negative, nor asymptotically positive,
$$f(n) = \sin\left(\frac{n\pi}{2}\right)$$
This function takes the values $\displaystyle 1,-1 \ \text{and}\  0$ infinitely often.
A: In mathematical English, 


*

*positive is defined to be $> 0$

*negative is defined to be $< 0$


So non-negative means $\ge 0$, not the same as positive.
In mathematical French, it just happens that the word 'positif' is defined to be $\ge 0$, that is, 0 is both 'positif' and 'negatif'. 
In other languages...who knows.
A: As non-negative is an adjective, generally its meaning depends on what word comes after it.
