I have been wondering for a long time whether there is a unequivocal way to define and use the symbols commonly adopted for an approximate equality between two quantities. I am a physicist, and I often see them used interchangeably and more or less only accordingly to the taste of the author or lecturer.

But what is a reasonable and clear way to differentiate between these symbols?

I refer to:

$$\approx\ \simeq\ \sim\ $$

  • $\begingroup$ None of those have precise meanings until you specify that meaning (the middle one is even sometimes used for isomorphisms). $\endgroup$ – Tobias Kildetoft Apr 12 '16 at 10:17
  • $\begingroup$ What would be a possible choice of the different meaning that makes use of all three? Thanks! $\endgroup$ – usumdelphini Apr 12 '16 at 10:20
  • $\begingroup$ I can think of probably at least 10 different things I might use these to denote, so I don't see any obvious choices. $\endgroup$ – Tobias Kildetoft Apr 12 '16 at 10:21
  • $\begingroup$ Assuming you are only using these for numbers, I would use $\sim$ as "approximately", $\approx$ as "approximately equal" and never use $\simeq$. For example "The table is $\sim 4$ feet in length" or "$\pi\approx 3.1415$". So, when saying two numbers are almost the same $\approx$ when saying one number is almost the desired quantity $\sim$. $\endgroup$ – Sean English Apr 12 '16 at 10:29

The symbol $\sim$ is usually used for asymptotic equivalence for functions.

One says that $f\sim_a g$ if $$\lim_{x \to a} \frac{f(x)}{g(x)} =1.$$ If no $a$ is precised (which is the most usual case), the limit is taken at infinity.

On the other side, $\approx$ and $\simeq$ are used for decimal approximation for numbers. Example $$\pi \approx 3,14.$$ We prefer $\approx$ to $\simeq$ since $\simeq$ denotes often an isomorphism (for example between two groups, two rings).

In conclusion, the difference between these symbols is made thanks to the mathematical objects for which there are defined.

  • Functions : $\sim$
  • Numbers : $\approx$
  • Groups, rings (in general : category theory) : $\simeq $

NB : These are the conventions that I often read in the literature, no doubt that can be others.

  • $\begingroup$ Thank you very much! What about Taylor series and approximate values of variables. Let's say,respectively: $f(x)\simeq f(x_0)+f'(x_0)(x-x_0)$; and $r=aB/2\simeq 2$? $\endgroup$ – usumdelphini Apr 12 '16 at 10:26
  • $\begingroup$ For each $x$, $f(x)$ is a number so I recommend to use the number approximation $f(x) \approx f(x_0)+f'(x_0)(x-x_0)$. $\endgroup$ – C. Dubussy Apr 12 '16 at 11:02

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