Yes, this is pretty common usage of got in a casual, conversational context. It's not incorrect, but it is informal. In writing, this usage is generally only appropriate if the author is deliberately evoking an informal, conversational tone.
In general, one would instead use found, saw, or another more traditional verb.
In a scientific context, accuracy and precision are different things, and some of this difference carries over to these words' non-scientific common usage.
An accurate scientific measurement is one that is very close to the real value. A precise measurement is one that is repeatable with very small variations, whether it's close to the real value or not. An exact scientific value is one that has absolutely zero error: maybe it's a purely defined quantity like the number of centimeters in a meter.
In common usage, accurate describes being close to the correct value. Precise, on the other hand, emphasizes the small margin of error or paying attention to the smallest details. Exact means zero deviation from the correct value.
You should also know that accurate and precise can be applied to either a number or the person generating the number. Exact, on the other hand, is seldom used to describe a person and only describes the number.
Describing a number, say the number of people attending a concert:
I need an accurate count of the people at the concert. [says that you need a number that's close to the truth, but maybe 3,000 is a good enough answer even if it's not exactly correct]
I need a precise count of the people at the concert. [says that you need a number with a small margin of error, something like 2,945 give or take a few]
I need an exact count of the people at the concert. [says that you want a count of every single person with zero error]
Describing an accountant:
She is a very accurate accountant; she has not made a mistake yet. [close to the truth]
She is a very precise accountant; she tracks every cost down to the tenth of a cent. [very small margin of error]
You cannot say "she is a very exact accountant." Exact does not describe people.
There is another adjective, exacting, that is applied to people, but it means "having very high expectations [of others]."
She is a very exacting accountant; she makes me submit receipts for every transaction, no matter how small. [very high expectations]
Looking at your sample sentences:
- Could be either "precise" (emphasizing "down to the penny") or "exact" (emphasizing "this is the correct value").
- "Accurate" would be best. You might think "exact" (meaning zero error), but you don't use "exact" to describe people.
- All three are possible here. I'd probably use "accurate".
- I'd use "precision" here, in the sense of "attention to the tiniest detail."
Added examples:
- Either "precise" (meaning "down to the minute or second") or "accurate" (meaning "correct"). "Exact" is fine but sounds less natural.
- This actually brings up a different shade of meaning. I talked about these three adjectives as modifying a measurement or guess. Here you're using them to modify a real thing that that the speaker is guessing at. In that case, you cannot use the word accurate. You can use either exact (meaning absolutely correct) or precise (meaning known in detail).
- Same as 2.
- Same as 2.
- All three options are possible. "Exact" and "precise" are pretty close synonyms, meaning "identical meanings in both languages." Using "accurate" is a little looser, meaning that there's no word in the other language that comes close to the meaning in English.
Best Answer
Both expressions are very common and universally understood in mathematical English written by native speakers -- which means that that, as a practical matter they are both right. And if a dictionary claims that "denote" can't be used in this way, that is the dictionary's fault.
Your assumption that (2) requires that the symbol has already been mentioned does not match how these phrasings are used in practice. Indeed, a sentence such as (2) would commonly be found in an early section of a paper entitled "mathematical preliminaries" or something like that, where the author briefly summarizes the symbolism that he's going to use in the rest of the paper.
As a writer your wording would probably sound most natural if you use the bare statement of fact "N denotes ..." for notation that already exists, such that the sentence just reminds the reader what is already true, whereas "We denote ..." or "Let N mean ..." and so forth are for definitions that are part of the original content of what you're writing, and therefore are new to the reader.
However, as a reader you shouldn't try to extract any meaning at all from the difference between these phrasings. The trend I just sketched here is not followed consistently enough for that to work. Go by context instead.
By the way, your third sentence
sounds slightly clunky to me; simply
would be more straightforward.
(Note that actually neither of your sentences could usefully be used in a mathematical paper, since there is no real doubt that N and "the set of natural numbers" means the same, whereas differing traditions about whether zero counts as a natural number or not. Defining N to mean the set of natural numbers without revealing whether this set includes 0 for you is just a waste of ink and/or taunting the reader).