3
$\begingroup$

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:

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

$\endgroup$
4
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.