Readit NewsTexting is unironically a better use of time than reading infinite jest, or gravities rainbow, etc.
"let {x_n} be a sequence"
As the author points out, a sequence is a function. The statement {x_n} is the set of terms of the sequence, its range. A function and its range are two different things. And also sets have no ordering. It might seem like a minor thing, but I thought we were trying to be precise?
A second example: at the high school level, I'm pretty sure a lot of textbooks don't carefully distinguish between a function and the formula defining the function very well.
The author of this web page has a section on what he calls "double duty definitions". Personally, I don't find anything wrong with the language "let G=(V,E) be a graph". G is the graph and we're simultaneously defining/naming its structure. So, some of this is a matter of taste. And, to some extent, you just have to get used to the way mathematicians write.
In most cases it is not as much abusing notation as overloading it. If you think of the context of a formula (say, adjacent paragraphs) as its implicit arguments (think lambda captures in c++), then it is natural that curly braces can denote both a set and a sequence, depending on this implicit input.
Such context dependent use of symbols is actually rather convenient with a little practice.
1{hidden bit} + (1-2^-52){mantissa with all ones}
the relative accuracy — corresponding to the absolute accuracy of a single bit in mantissa — is about 2^-53. The hidden bit is easy to forget about...
> the largest integer value that can be represented exactly is 2^53
— I am confused as to why it not 2^52, given that there are 52 bits of mantissa, so relative accuracy is 2^-52, which translates to absolute accuracy larger than 1 after 2^52. Compare this to the table there saying "Next value after 1 = 1 + 2^-52".
It is conjectured that in higher dimensions, the densest packing is always non-lattice. The rationale being that there is just not enough symmetry in such spaces.
Are there things we could have learned from them that are lost to time? Well, yeah, and that itself is bad, but preservation is simply not feasible, same as we don't store every single piece of information nowadays, we can't store all of language for the same reason it's interesting in the first place: It's alive.
It's also worth noting, there's a whole, whole lot that is bound to be uninteresting beyond historical knowledge and that deserves no more respect than, say, food.
Plus, it is all the more exciting to think about what caused some languages to exist and thrive for so long, and the information about the past they retained.
I have found Beyond Compare to be very good on Linux, even on large files/directories.