I'm going to explain the symbols that are used to indicate "approximately equal" in math.
There are several different symbols that mean “approximately equal”:
≈ (U+2248) ALMOST EQUAL TO
≃ (U+2243) ASYMPTOTICALLY EQUAL TO
≅ (U+2245) APPROXIMATELY EQUAL TO
~ (U+007E) TILDE
All of these have slightly different meanings. In some contexts, you might see them all used interchangeably, but they aren't really interchangeable. So if you're writing a paper or book where precise meaning is important, it's good to know what each one means.
How the tilde differs
The tilde (~) often means “about” in informal text, but in mathematics its meaning depends on the field: it can mark similarity, asymptotic equivalence, a relation, or a statistical distribution. For example, if I write “x ~ y,” I need the surrounding definition or field to tell the reader exactly which relation I mean.
On the other hand, the symbols ≈, ≃, and ≅ are used when comparing two quantities that are approximately equal. The difference between them is subtle, but there is a slight difference in meaning. In most cases, they can be used interchangeably, but in certain contexts, one may be preferable over another.
What ≈, ≃, and ≅ mean
The ≈ symbol is the most commonly used symbol for approximately equal. It is often used in equations to indicate that two expressions are approximately equal. For example, if we have the equation “sin(x) ≈ x”, this means that the sine of x is approximately equal to x.
The ≃ symbol has the Unicode name Asymptotically Equal To and is used for asymptotic or related equivalence by convention. The exact convention should be defined in technical writing. “ln(x) ≃ x as x becomes large” is not a valid example because ln(x)/x tends to 0, not 1; a standard asymptotic example is x + 1 ≃ x as x tends to infinity, when the intended convention is clear.
The ≅ symbol has the Unicode name Approximately Equal To, but in mathematics it is also widely used for congruence or isomorphism. I would use π ≈ 3.14 for an ordinary numerical approximation and reserve ≅ for the convention used in the relevant field.
Summary
In summary, the symbols ≈, ≃, and ≅ are all used to indicate that two quantities are approximately equal. However, each symbol has a slightly different meaning. The ≈ symbol is the most commonly used symbol for approximately equal, while the ≃ and ≅ symbols are used in specific contexts. The tilde (~) is also used to indicate approximately equal, but it has a slightly different meaning.