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There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$

$$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what these statements strictly mean makes me rather uncomfortable. My question would be:

What is the definition of "approximation" in terms of calculus or algebra? Are there any constraints to whether a number can be "approximate" to another or not? If not, can we say $1 \approx 2$ just because they seem to be close? And if that is not the case, then what allows us to assert $1 \approx 1.001$?

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Wiki says "An approximation is a representation of something that is not exact, but still close enough to be useful." I guess that "what is useful" depends on the context of your problem. – Patrick Li Jan 31 '13 at 19:44
I have done my best to make it seem like a serious question; not something trivial like the current "definition" of approximation. – Parth Kohli Jan 31 '13 at 19:44
@PatrickLi Indeed, that is what I am talking about: a stricter definition of approximation. – Parth Kohli Jan 31 '13 at 19:45
up vote 16 down vote accepted

I am in Israel, so I'm approximately living around Jerusalem -- even if it would take me a two hours drive to get there. For someone coming from another galaxy, I am living approximately on the sun.

Approximations are in their nature inaccurate and relative. When we say that $x$ is an approximation for $y$ we usually mean that in our context they are "pretty close". If you deal with numbers which are much larger than $10^{100^{100}}$ then $1\approx 2$ is true, and both are pretty much zero.

In terms of a sequence if we have a sequence $x_n$ which converges to $x$ then we can say that for a large enough $n$, $x_n\approx x$. It means it's close enough. In this aspect $e\approx\left(1+\frac1n\right)^n$.

In number theory we have a notion of best rational approximation (and $\frac{22}7$ is such approximation for $\pi$) meaning that if we put some constraints on the denominator, this is really the best you can get. It's a very nice theorem that the continued fraction of $r$ can be used to generate best approximations for $r$.

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You can make an approximation $1 \approx 2$ if you want to. The question is not whether this is an approximation (it is), but whether it is a good enough approximation for whatever you want to do with it.

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There is no precise meaning to approximation. To approximate something is simply to represent it with something else that is good enough for your purpose.

Numbers are not the only things that can be approximated. One can, for example, approximate continuous functions with polynomial functions, in which case the idea is to keep the area between the original function and the approximating function small.

One can also specify the degree of approximation allowed. For example, we may want to restrict the area in the above example to a certain value. The smaller the difference between the original object and the approximating object, the better the approximation.

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Most (though not all) interesting cases of approximation in mathematics are not just "one-off" approximations of one number by another, such as your example $\pi \approx 22/7$, but rather statements that some expression can be approximated arbitrarily well by another, typically in the limit as some parameter goes to $\infty$ or $0$, such as your $e \approx (1+1/n)^n$.

In the broadest sense, as Nachbin writes in the Foreword to his "Elements of Approximation Theory"

Approximation theory is concerned with the problem of describing the elements of a topological space $E$ that may be approximated by those of a subset $X$ of $E$, that is, of characterizing the closure of $X$ in $E$.

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Professor Arnold Ross, of the Ohio State University, used to ask, "What is an approximation to $5$?" and then answer, "Any number other than $5$."

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