In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"?
The Unicode standard lists all of them inside the Mathematical Operators Block.
- ≈ : ALMOST EQUAL TO (U+2248)
- ≃ : ASYMPTOTICALLY EQUAL TO (U+2243)
- ≅ : APPROXIMATELY EQUAL TO (U+2245)
Best Answer
The notations $\cong$ and $\simeq$ are not totally standardized. Both are usually used for "isomorphic" which means "the same in whatever context we are." For example "geometrically isomorphic" usually means "congruent," "topologically isomorphic" means "homeomorphic," et cetera: it means they're somehow the "same" for the structure you're considering, in some senses they are "equivalent," though not always "equal:" you could have two congruent triangles at different places in a plane, so they wouldn't literally be "the same" but their intrinsic properties are the same. I've seen colleagues use both for isomorphic, and some (mostly the stable homotopy theorists I hang out with) will use $\cong$ for "homeomorphic" and $\simeq$ for "up to homotopy equivalence," but then others will use the same two symbols, for the same purposes, but reversing which gets which symbol.
The $\approx$ is used mostly in terms of numerical approximations, meaning that the values in questions are "close" to each other in whatever context one is working, and often it is less precise exactly how "close." Topologists also have a tendency to use $\approx$ for homeomorphic.
The main take-away from this answer: notation is not always standardized, and it's important to make sure you understand in whatever context you're working.