I have two questions regarding 'approximation' (or a better word, if any, please see Q.2):
$1-$ I usually see terms like $\pi \approx 3.14159$ or $\pi \simeq 3.14159$ or $\pi \sim 3.14159$, etc. Which of the symbols $\approx$, $\simeq$, ... is the universally accepted or "standard notation" to use in Mathematics?
$2-$ What does actually approximation of a real number mean? $3$ and $5$ are the nearest numbers to $4$ but why there is no sign of e.g. $3 \approx 4$ in Mathematics? In other words, when defining approximation (i.e. "asymptotic equality of functions") in Analytic Number Theory we use the evaluation of of $\lim_{x \to \infty } (f(x)/g(x)) =1 \ \text{or} \ 0$ for "big or small O-notation" respectively. But what about numbers in lieu of functions?
PS, Sorry! Many subquestions in Q.2 are for better explanation of one question regarding understanding better the concept of the approximation.
The correct notation for an approximation (in the physicists' sense: this is not a mathematical concept) is $\;\color{red}\approx$. , as indicated by the name of the LaTeX command:
\approx. $\color{red}\simeq$ is mostly used for ‘isomorphic to’, and $\color{red}\sim$ for ‘equivalence’.In mathematics, one only speaks a ‘approximation
at a certain degree of precision’. This supposes the existence of a metric. If the metric is discrete, like in $\mathbf Z$, this is totally uninteresting. But you have the $p$-adic valuation, for instance, which is non-trivial.