Re: rationals, I mean there's an infinite number of rationals available arbitrarily near any other rational, that has to mean they are good enough for all practical purposes, right?
For practical purposes, they’re bad. Denominators tend to explode when you do a few operations (for example 11/123 + 3/17 = 556/2091), and it’s not easy to spot whether you can simplify results. 12/123 + 3/17 = 191/697, for example.
You can counteract things by ‘rounding’ to fractions with denominators below a given limit (say 1000) but then, you likely are better of with reckoning with a fixed denominator that you then do not have to store with each number, allowing you to increase the maximal denominator.
For example (https://en.wikipedia.org/wiki/Farey_sequence), there are 965 rational fractions in [0,1] with denominator at most 10 (https://oeis.org/A005728/list), so storing one requires just under 10 bits. If you use the fractions n/964 for 0 ≤ n ≤ 964 as your representable numbers, arithmetic becomes easier.
They're paying for the water. It's not like they're getting it for free. Sure, the municipality could just not sell outside the municipality, but most utilities are forbidden from suddenly cutting off service due to health and safety concerns.
And that doesn’t mean water bought ‘on demand’ would have to be equally costly as what people who ‘subscribed’ to water pay.
It essentially allows the user to check if a class implements an interface, without explicitly inheriting ABC or Protocol. It’s up to the user to ensure the body of the case doesn’t reference any methods or attributes not guaranteed by the subclass hook, but that’s not necessarily bad, just less safe.
All things have a place and time.
There, all instructions are 32 bits and D4D4 is only 16 bits.
We can't be 100% certain that Fermat didn't have a proof, but it's very unlikely (someone else would almost surely have found it by now).
“Fermat usually did not write down proofs of his claims, and he did not provide a proof of this statement. The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755)
[…]
Zagier presented a non-constructive one-sentence proof in 1990“
(https://www.quora.com/What-s-the-closest-thing-to-magic-that... shows that proof was a bit dense, but experts in the field will be able to fill in the details in that proof)
all integers have a square, while not all integers are prime.
in any given span, you'll see more primes than squares, however.
more dense?
That’s true, but I don’t see how that’s an argument. “All integers have a prime ‘the nth prime’, while not all integers are squares” similarly is true, but not an argument as to which set is denser.
Also, how does “plenty” compare to the millions (50 million or so, it seems from a quick search) x64s that Intel sells per year? Do they even sell 1% of that?