# 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 Θ(g(n)) requires that every member f(n) ϵ Θ(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.)

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Non-negative includes zero, positive does not. – Brandon Carter Jan 21 '11 at 21:06
It generally depends on the context. If your book uses both terminologies, I would guess it says "positive" when referring to "strictly positive", without zero. – Pandora Jan 21 '11 at 21:07
@kakemonsteret: this disagrees with both mathematical definition and with common usage of the word "negative." If you had no money in your bank account, you wouldn't say you had a negative balance. And your second comment willfully ignores context, which is pointless; in context, "non-negative" almost always refers to a real number. – Qiaochu Yuan Jan 21 '11 at 21:21
It also depends on local tradition. For example, the French use "positif" and "strictement positif" for what English speakers (usually) call "nonnegative" and "positive". – Hans Lundmark Jan 21 '11 at 21:37
@Hans: yep. This has lead to the gag among my friends where appending "in the French sense" turns any open condition into a closed one. – Willie Wong Jan 21 '11 at 23:44

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.

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 Thanks for clarifying. – typoknig Jan 21 '11 at 23:26

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.

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 All I know in my language are "Big", and "Small". Otherwise, people from my country simply use English for math. – muntoo Jan 22 '11 at 0:11 @muntoo: :) part of the issue is that naming conventions are just that, conventions, and not necessarily logical. The mathematical concepts are, but the labels not necessarily so; for the concept of trichotomy English goes one way, French another, other languages may have another convention. Some countries drive on ne side of the road, some on the other, and it all works out as long as you know the rules of the country you're in. – Mitch Jan 22 '11 at 19:43

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.

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As non-negative is an adjective, generally its meaning depends on what word comes after it.

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