There is a string link to Farey Numbers. See:

https://en.m.wikipedia.org/wiki/Farey_sequence

See in particular:

If ⁠a/b⁠ and ⁠c/d⁠ are neighbours in a Farey sequence, with ⁠a/b⁠ < ⁠c/d⁠, then their difference ⁠c/d⁠ − ⁠a/b⁠ is equal to ⁠1/bd⁠. Since

this is equivalent to saying that 

This provides a very elegant way to check the numbers in these series. Since it uses integers, it is much more accurate.
Note, for any non rational numbers, there are always two closest approximations of interest. The one less than and the one above. They are Farey numbers and obey the above calculation.

Cool bro.

Rishi

why is it that $2pi i$ looks “correct”, but the other five permutations 2ipi, i2pi, ipi 2, pi 2i, pi i2 look “incorrect” despite formally describing the exact same quantity? Similarly for z=x+iy versus z=x+yi, e^{itheta} versus e^{theta i}, etc.

and

The symbol in only is a viable solution in a portion of the use cases. For instance, an assertion such as O(n) cdot O(n) = O(n^2) would not be correctly describable as O(n) cdot O(n) in O(n^2). Perhaps O(n) cdot O(n) subset O(n^2) would be defensible, but now one has to devote a non-trivial amount of thought into deciding which of the connectives =, in, ni, subset, supset to use in a given context. For instance the assertion “Since sin(y) = sin(x) + O(|y-x|), we have sin(x+O(1/n)) = sin(x) + O(1/n)” would now become “Since sin(y) in sin(x) + O(|y-x|), we have sin(x+O(1/n)) subset sin(x) + O(1/n)“. Using the equality sign for all of these use cases instead is more intuitive and corresponds more closely to how the verb “is” (“to be”) is actually used in mathematical English.

[Also, the use of set notation such as in or subset gets particularly confusing when the objects one is manipulating are themselves sets. For instance, the fact that a continuous function f: {bf R}^d to {bf R}^d maps bounded sets to bounded sets can be described in my notation as f(B(0,O(1))) subset B(0,O(1)), or equivalently that f(x) in B(0,O(1)) whenever x in B(0,O(1)). These relationships become significantly more difficult to describe if one is already using in, subset as connectives, as described above.]

followed by:

It’s a question of mereology. The orthodox foundations of mathematics are based on set theory and first-order logic, and so mereological assertions such as “3 is a prime” and “primes larger than 2 are odd” are translated either into set-theoretic assertions such as 3 in { mathrm{primes}} and {mathrm{primes} } cap { mathrm{greater than } 2} subset {mathrm{odd}} or into first-order logic sentences such as mathrm{IsPrime}(3) and forall n: mathrm{IsPrime}(n) & (n > 2) implies mathrm{IsOdd}(n). With years of training, we all get used to these translations, and even internalize them to the point where we don’t notice anymore that a translation has actually occurred (except when we have to teach confused undergraduates who have not yet had experience with such translation). Nevertheless most of us still often think in mereological terms rather than set-theoretic or first-order terms (unless when actually dealing with sets as geometric or other structured spaces, or when focusing on logical, computational, or game-theoretic aspects of one’s assertions, in which case the set-theoretic or first-order formalism becomes advantageous). One mathematician can say to another, “By Heine-Borel, compactness in a Euclidean space is the same as being closed and bounded” and understand and use it immediately without requiring translation to set theory or first order logic; indeed, such a translation would only serve to slow that mathematician down as he or she would usually have translate it back into mereological form in order to wield it effectively. Because of this, I think it is worth adjusting our notational conventions to more closely align with our actual thought processes, especially if it can be done without sacrificing any precision or rigour. For instance, the two statements discussed above would now be translated as “3 = hbox{prime}” and “mathrm{prime} & mathrm{greater than } 2 = mathrm{odd}“, thus having almost identical grammatical structure to mathematical English.