There's a simple constructive proof using high-school level thinking. ... Two different reals have different decimal expansions. Go out far enough that they differ. Since this is about intuition, let's just assume the larger one is positive and irrational, and thus has an infinitely long expansion. Since the truncation has a finitely long decimal expansion, it's rational. And it's between the two original reals. Q.E.D. ... A full proof for all cases can be built similarly.
It doesn't break my brain that there's a rational between any two reals; it breaks my brain that this doesn't imply equivalent sizes between the reals and the rationals.
My first step would be to note that 13,857 is divisible by 7 if and only if 1385 is. But yes, my second step would be a lot like yours, in my case subtracting from 1400.
A standard problem with dev tools is that they're obsolete before they're ever mature.
What is mean is that by the time NewTool that fills a gap doing X really well catches up with the YZ that older tools have long experience handling, EvenNewerTool handles E better than NewTool does.
I don't have any knowledge of the specifics here, but at first guess I doubt something security-focused would be an exception to that rule.
Yeah, see almost every "configuration management" tool aside from Ansible that promptly got made irrelevant for the majority of users once containerization hit the scene.
A mathematican, biologist and engineer were so stoned that they struggled with the claim "All odd numbers are prime."
The mathematician reasoned "3 is prime. 5 is prime. 7 is prime. 9 is ... not prime. The claim is false."
The biologist reasoned "3 is prime. 5 is prime. 7 is prime. 9 is ... not prime. 11 is prime. 13 is prime. The claim looks true, within reasonable experimental error."
The engineer reasoned "3 is prime. 5 is prime. 7 is prime. 9 is prime. 11 is prime. ..."
A mathematician was working when she noticed that her trash bin is burning. She turned around and saw a fire extinguisher. “Clearly the problem has a solution”, she said to herself and went back to work.
Suspension of mathematical disbelief is hard to sustain seriously or at length.
Accordingly, the best mathematical SF/fantasy I can think of is in short stories or novellas which are meant to be humorous. In particular, those criteria characterize:
-- Most of the stories in Fantasia Mathematica and its quasi-sequel anthology Mathematical Magpie.
-- Flatland, which is novella length.
-- The Dragons of Probability, or anything else in Lem's Cyberiad.
I read the third book of the Zelazny's Amber series before I read the second, back when the series was new. I didn't do it on purpose, but I wasn't careful to avoid the risk either.
That was the first time in my life I read three novels the same day. It was 2 days before qualifying exams for my PhD. 1 day beforehand I was going to cram, but the day before that I wanted to clear my brain.
First time I read The Lord of the Rings, I was 12 or so, my dad was reading The Two Towers, my older brother was reading The Fellowship of the Ring, so I decided I should just start reading The Return of the King.
So my first memories of LotR are Pippin riding Shadowfax behind Gandalf.
When my dad finished book 2, I dropped book 3 and switched, and then to book 1 when my brother finished the first book, so that was a bit of a fragmented experience.
Maybe there's also a lesson here to read books with your kids, or at least where they can see you read. It's contagious.
My dad ended up buying the famous pirate Ace version of the Fellowship of the Ring - and it had absolutely no indication it was the first part of a combined trilogy so he thought it has a VERY abrupt weird ending until he later learned there were other volumes.
My dad had all three, my mom had only the first (of the same edition), so we had two copies of Fellowship at home. I guess I should have started in the other copy, except when I started we were on vacation and we didn't bring the duplicate Fellowship with us.
