# Can approximation be exact

When something is an approximation to another thing can it be exact? Meaning can an approximation be the thing being approximated? A quote from wikipedia

An approximation is anything that is similar but not exactly equal to something else.

I know that wikipedia is not absolute source of knowledge so I wanted to be sure. Can I say that the number 2 is an approximation of the number 2?

EDIT: the meaning of exact is not what I need here, I want to know if approximation can be the same as what is being approximated. As I understand it and from the dictionary definition I find exact is something very similar or nearly identical. Still most of the definitions I find stress on the fact that an approximation can not be accurate. I guess my confusion is coming from the meaning of the word's translation in Bulgarian. For us an approximation can be exact.

EDIT2: trying to add a bit more context. I am writing about an algorithm that produces a solution that is close to the perfect solution, so I called that an "approximation". However there are application of the algorithm when it is guaranteed to find the perfect solution. Thus I got a comment that I can not call the algorithm an approximation when applied to those specific applications. I am trying to verify if this comment is adequate

• I think this is a valid question (and quite interesting) but could you please consult other sources and edit your question to quote them. For example look up 'exact' and 'approximate' in a dictionary and explain how you understand them. Also please say what conclusions you have come to so far. Sep 18, 2015 at 20:14
• Thanks for the clarifying edit. However I beg to differ. I think you do need the meaning of 'exact' as well. You can only compare 'approximate' with 'exact' if you know what exact actually means, and that is not a simple matter. Sep 18, 2015 at 20:42
• @chaslyfromUK maybe you can help on that? Seems to me accurate has about the same meaning but I may be wrong. I am no native speaker Sep 18, 2015 at 20:47
• Clearly there is a language difference because in English, 'accurate', 'exact', and 'precise' have different meanings. That is why I asked you to do some research first, using dictionaries. On Stack Exchange, the reponsibility is on the person who asks the question to show their research before others answer. In this case it is essential because otherwise you are in effect asking us to prepare mathematical and linguistic lectures on the subject. Also we need context. Are you asking about maths? Measuring cloth? I say this not as a criticism but because the question is not at all simple. Sep 18, 2015 at 20:56
• I tried to add a bit more context hoping to narrow the scope of my question. I did research but I can't find a definite answer I think. Problem is a lot of sources and specifically when talking on algorithms are not created by native speakers Sep 18, 2015 at 21:05

The third definition of approximation, as listed in the referenced link, is the one I am the most familiar with and frequently use in my field of research:

Mathematics, Physics. a result that is not necessarily exact, but is within the limits of accuracy required for a given purpose.

It sounds like that's the one that applies in your case.

Example use case: we collect data of how long neutrons take to fly from a source to our detector, and have measurements recorded a big database table like the following table.

``````+------+----------------------------+
|    # |time (milliseconds)         |
+------+----------------------------+
|    1 | 1.001299498201             |
|    2 | 0.992839823984             |
|    3 | 0.982312055471             |
|  ... | ..............             |
|89231 | 1.003934092729             |
|89232 | 0.993997584121             |
+------+----------------------------+
``````

If someone asks me whether the times are all about the same, I could find a mean and standard deviation of them and say Yes, all times are approximately one millisecond. That does not mean to say that no single entry can possibly be recorded as exactly `1.000000000000` in the database.

So I would say yes, in the context of your question and in situations like the example above, an approximation can be exact but is not necessarily exact.

• I may be missing something here but aren't you conflating 'exact' with 'precise'. Sep 18, 2015 at 21:47
• @chaslyfromUK, I don't think I am, but could you be more specific? Sep 20, 2015 at 21:31

I'd say linguistically, no, you can't, as your Wikipedia quote implies. But mathematically, yes, you can. If for example some formula being used for approximating a quantity were occasionally to give an exact result, you wouldn't take that as a failure of the formula to give a correct result.

So it's a tricky question. I think you can't say that the number 2 is an approximation of the number 2, even though it is.

• Only for very small values of 2. Sep 18, 2015 at 20:36
• @IgnacioVazquez-Abrams, Are there many values of 2 which are not very small? Sep 18, 2015 at 20:42
• Ok. I have to admit I got confused and the comments are not helping Sep 18, 2015 at 20:48
• I disagree. Linguistically you can say it, just as you can say 'The moon is made of cheese'. There is no requirement that the statement has to be true. In linguistic terms we don't have to consider the mathematical correctness. I think that the real problem is that the question itself is not yet well-formed. Sep 18, 2015 at 20:49
• @chaslyfromUK, You say there is no requirement that the statement has to be true. Of course, I agree, but I don't see the relevance of this. Sep 18, 2015 at 20:56