What does $a\approx b$ actually mean?

Question

If we are to use things, they should be defined. I've never seen a mathematical definition of $\approx$ though, something that any given computer could precisely understand and execute, regardless of human input and customs.

If I say $\frac{\sin(x)}{x}\to1$ as $x\to0$, I can actually give meaning to this outside of saying "look at the graph, it gets close to 1".

$$\lim\limits_{x\to0}\frac{\sin(x)}{x}=1\iff\forall\epsilon>0\exists\delta>0:0<|x|<\delta\implies\left|\frac{\sin(x)}{x}-1\right|<\epsilon$$

We can also say $\frac{\sin(x)}{x}-1\in\mathcal{O}(x^{2})$ as $x\to0$ meaningfully without talk of mystical "small values".

$$\frac{\sin(x)}{x}-1\in\mathcal{O}(x^{2})\iff\exists\delta>0\exists M>0:0<|x|<\delta\implies\left|\frac{\sin(x)}{x}-1\right|\leq Mx^{2}$$

But can we rigourously say $\frac{\sin(x)}{x}\approx1$ as $x\approx0$ or as $x\to0$? Let's look at it as any other binary relation.

$a\approx b\implies b\approx a$ should hold if it measures some kind of closeness, so it's likely symmetric.

$a\approx a$ I'm not sure about. If you said $0.\dot{9}\approx 1$ you'd expect to be chastised. "They're equal, not approximately equal" some would say, as if it can only be one. So I'm not sure what the consensus would be on whether it's reflexive. I'd say probably.

$a\approx b\land b\approx c\implies a\approx c$ should probably not hold. Some would agree that $3.3\approx3.4$ and that $3.4\approx3.5$, but would they say $3.3\approx3.5$? Perhaps, so let's keep going. $3.5\approx3.6$, $3.6\approx3.7$, ... ${}^{10000}10000\approx{}^{10000}10000+0.1$ $3.3\approx{}^{10000}10000+0.1$ would likely not be seen as true, so it's not transitive. And if it does have reason to be true, we've got no shortage of numbers of increasing size (and we could also go backwards to the negatives for one of the sides, for extra security).

Looking at potential properties still doesn't define it though. Surely there must be some non-arbitrary $\epsilon$ action somewhere, for this sort of topic. We could simply state that $\forall\epsilon>0, a\approx_{\epsilon}b\iff |a-b|<\epsilon$, but that isn't a single binary relation. If we select one of those $\epsilon$ to have $\approx_{\epsilon}$ mean $\approx$, well, which? Every single $\epsilon$ is an arbitrary choice.

I notice that $a\gg b$ and $a\ll b$ suffer the same pitfalls, but they're used much less than $\approx$. It's currently more useful to get the answer to the question: does $\pi\approx\frac{22}{7}$, and precisely why or why not?

Answer 1

There are old (Leibniz, Cauchy) and newer (Laugwitz, Keisler, Robinson, Nelson) contexts where infinitesimals are used consistently (means, avoiding the obvious pitfalls), and where $a\approx b$ actually means that $a-b$ is infinitesimally small. This is also a transitive relation over finite, limited numbers of steps.

This notation of a softened equality sign found then its way into more engineering contexts in the sense of a "difference too small to care about", which will necessarily occur in calculations in finite space and time. For instance $\pi$ to 12 digits is enough for all practical calculations inside the solar system, the error due to this truncation is small against the common measurement uncertainties.

As you already noticed, exploring theoretical consequences of the controversial infinitesimals and connected notations deeper either requires restrictions on what "finite" or "limited" means, or leads to contradictions. There are several frameworks for non-standard analysis that make this overall consistent (but require lots of caveats or special specifications like $st(x)$ - "$x$ is standard").

In all these contexts, $x\approx 0 \implies f(x)\approx f(0)$ just expresses the continuity of $f$ in $x=0$ (if $f$ is not defined using non-standard means). In that sense this notation is a less dynamic looking variant of $x\to 0\implies f(x)\to f(0)$.

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In the sense of physical reality there are numbers like 1,2,3,4,... that you can easily write down and then there are numbers that you can never write down in any form of notation before the end of the universe or without collapsing the medium used into a black hole. Non-standard analysis idealizes this situation by dividing the natural integers into standard and non-standard elements. It is a question of an additional belief or axiom to allow that the Peano axioms imply the existence of natural number that are unaccessible even in the most idealized circumstances. Which includes including words like "standard" or "accessible" and "non-standard" or "inaccessible" into the mathematical language and giving them meaning according to the above.

Then infinitesimals are just the real numbers that are smaller than $1/n$ for any standard or accessible integer $n$. This leaves a gap (that is inaccessible to zooming even with the most idealized microscope) to zero that is filled with an infinity of infinitesimals.

Answer 2

There is no formal interpretation as this comparison is context-dependent.

If you are addressing a problem where the data is known to $1\%$ of relative accuracy, $a\approx b$ may mean $0.99 a\le b\le 1.01a$.

On the opposite, you may have high accuracy requirements but allow yourself to write $e^{\pi\sqrt{163}}\approx262537412640768744$, which is true to $10^{-12}$.

In yet other cases, you don't even care to be explicit about the precision.

Answer 3

Very nice question, and one that you may get very interesting answers for in Physics.SE as well. I can only offer a few points on how I interpret the $\approx$ symbol which I think reflect how it is commonly used, and I would also point out how you can even make a definition from those if you insist on doing so:

1. The relation $\approx$ is not transitive. I would also add, for the vast majority of use cases we don't use this symbol to write something like $a \approx b$ where $a$ and $b$ have predetermined and well defined numeric values. The interest in using the symbol is when the LHS is a variable that can take on some range of values, or when we have an uncertainty about its exact value.

2. So obviously, $\approx$ comes to tell us there is some error/uncertainty in the numerical value written on its RHS. Now, one can and does often quantify quite more accurately what exactly is that uncertainty by the context in which this symbol is used. However, even without a very definite context, we can invent the straightforward definition that the uncertainty cannot be larger than the precision with which the RHS is presented. In other words, we postulate that the context providing the uncertainty can be deduced by the form of the RHS. For example if we have $x \approx 3.5\cdot10^{21}$, we can define this as meaning $x \in [3\cdot 10^{21},4\cdot 10^{21})$. I think that would be very acceptable and in fact more often than not, the implied assumption in using this symbol. Accordingly, assuming we don't already have the exact numerical value for $\pi$ (as that would violate rule 1.), writing $\pi \approx \frac{22}{7}$ should be ambiguous and invalid, because the expression $\frac{22}{7}$ does not provide us with an explicit order of magnitude for reference. On the other hand, writing $\pi \approx 3.14$ is fine since, referring to our established rule we can deduce: $\pi \in [3.1,3.2)$.

It would be an interesting exercise to scan the literature for usages of $\approx$ and, while ignoring the overall context, observe how similar the actual usage of the symbol is to the (pseudo-)definition I have offered here. My suspicion is that it would be quite close, or rather $\approx$ the same ;)

Answer 4

It's not a symbol with a common definition, it depends entirely on the author. Broadly (and vaguely) speaking, if $a\approx b$ then in that context you can treat $a$ similarly to $b$.

Answer 5

I see three uses of $\approx$ in mathematical literature.

1. Giving an approximate value of a (usually explicit) numerical constant for the benefit of the reader, e.g. "Recall that $\sum_{n=1}^\infty n^{-2}=\pi^2/6\approx 1.645$." Here the degree of precision is unimportant (since the approximation will not subsequently be used for anything) and can probably be inferred.

2. Giving a conjectured value of an unknown constant, e.g. "For the square lattice, it is known that $p_c^{\text{bond}}=1/2$ and believed that $p_c^{\text{site}}\approx 0.5927$." Here the "approximation" is not known rigorously to be true, so the exact definition doesn't matter.

3. Some ad-hoc (but well-defined) notion of closeness. Here, the whole point is that there shouldn't be a standard definition of $\approx$. If you want to say two things are close in a standard way, use a standard symbol or expression with a standard meaning. But if you need to talk about two things being close in a non-standard way, you can use the all-purpose $\approx$ and define it as required.