Different use of approximate equality symbols
By Sebastian Wright
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:
$$\approx\ \simeq\ \sim\ $$
$\endgroup$ 41 Answer
$\begingroup$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.
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