what is the difference between ≈ and ≃?

Question

Can somebody please explain what is the different between ≈ and ≃ ? in case they are the same why are there 2 different symbols for the same thing? can somebody also say what is the name of the symbols? both are supposed to be read "almost equal to"?

Answer 1

There is no universally accepted notation, but $\approx$ and $\simeq$ both generally mean "approximately equal to." However $\simeq$ is sometimes used to denote "is similar," for instance that two triangles are similar ($\triangle ABC \simeq \triangle DEF$). You would rarely, if ever, use $\approx$ in that situation.

Answer 2

Symbols such as $\sim,\approx,\simeq,\approxeq,=,\equiv,\fallingdotseq,\risingdotseq,\doteqdot,\dots$ where lines are generally parallel or squiggles generally represent equivalence relations. Some specific equivalence relations may have standard choices for which symbol to use (such as how our usual equality is almost always represented by $=$). On the other hand, some equivalence relations do not have a universally designated symbol to use, so any from that list (or similar to those in that list) may be used and is largely author preference. On the other hand, most if not all symbols in that list can be used for multiple different situations, such as how we use $=$ to represent equality between real numbers, equality between matrices, equality between sets, etc...

There are some situations where those symbols which have squiggles may be used to represent relations which are not true equivalence relations, but act similarly to equivalence relations. For example $\approx$ might represent the relation "is close in value to" where we say for example $a\approx b$ iff $|a-b|<0.5$. Here we would have something like $10.47\approx 10$ and also $9.82\approx 10$. (Note that this example is not an equivalence relation since the relation is not transitive., $9.82\not\approx 10.47$).

As for what the symbols are named, I am in the habit of either referring to them by the name of the relation they are being used to represent, or referring to them by their $\LaTeX$ designation ($\simeq$ being called "simeq" for example)

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Similarly to how there are many symbols for equivalence relations (or equivalence-like relations) in use, there are many different symbols for orders and partial orders, such as $<,\leq,\prec,\preceq,\subset,\subseteq\dots$, again with some orders exclusively using one symbol over another but symbols being used for multiple things. These symbols more commonly have one end closed with the other end open, suggesting which side is "bigger."