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987654321 / 123456789
91–100 of 120 posts
Re: 987654321 / 123456789
#92This was by far the most interesting part to me. I've never considered that code and proofs can be so complementary. It would be great if someone did this for all math proofs! "Why include a script rather than a proof? One reason is that the proof is straight-forward but tedious and the script is compact. A more general reason that I give computational demonstrations of theorems is that programs are complementary to…
I came here to quote that entire section as well I’m glad I checked the comments first.
I’ve never seen a more succinct explanation of the value of coding up scripts to demonstrate proofs.
I think I’ll tighten it up to “proofs have bugs” in the future.
Re: 987654321 / 123456789
#93This was by far the most interesting part to me. I've never considered that code and proofs can be so complementary. It would be great if someone did this for all math proofs! "Why include a script rather than a proof? One reason is that the proof is straight-forward but tedious and the script is compact. A more general reason that I give computational demonstrations of theorems is that programs are complementary to…
This is misleading in that the (Curry–Howard) correspondence is between proofs and the static typing of programs. A bug in a proof therefore corresponds to a bug in the static typing of a program (or to the type system of the programming language being unsound), not to any other program bug. (Also: complementary != complimentary.)
The point is to not be so tight, leaning on the correspondence. The fact that you’re coming at the problem differently (even that it’s a different problem, “for some” versus “for all”) is actually helpful. You’re less likely to make the same mistake in both.
There’s a technique for unit testing where you write the code in two languages. If you just used a compiler and were more confident about correspondence, that would miss the point. The point is to be of a different mind and using different tools.
Re: 987654321 / 123456789
#94Earlier quoted context omitted.
The decimal digits clearly have a conspiracy going on.
That would work in any base, I even think we would find way more interesting coincidences in base 12 (as Sumerians preferred), because it's divisible by 2,3,4,6. It's unfortunate that we have 5 fingers.
If you count the sections of your four fingers with your thumb, you can count up to 12 on one hand!
Re: 987654321 / 123456789
#95Earlier quoted context omitted.
That would work in any base, I even think we would find way more interesting coincidences in base 12 (as Sumerians preferred), because it's divisible by 2,3,4,6. It's unfortunate that we have 5 fingers.
If you count the sections of your four fingers with your thumb, you can count up to 12 on one hand!
If you count with each finger as a binary digit you can count up to 15 on one hand!
255 if you use both hands!
More like 1023 if you also use thumbs but I prefer to use them as carry, overflow bits.
Re: 987654321 / 123456789
#96I like calculator quirks like this. I remember as a kid playing with the number pad and noticing a geometric center of mass in number sequences ┌───┬───┬───┐ │ 7 │ 8 │ 9 │ ├───┼───┼───┤ │ 4 │ 5 │ 6 │ ├───┼───┼───┤ │ 1 │ 2 │ 3 │ ├───┼───┼───┤ │ 0 │ . │ │ └───┴───┴───┘ I remember seeing that (14787 + 36989) / 2 would produce 25888, in that the mean of geometric shape traced by the two sequences would average out in the…
how did you submit this table in HN??
Re: 987654321 / 123456789
#97TIL 0x denotes hexadecimal E.g. > 0xFEDCBA987654321 / 0x123456789ABCDEF (somehow I'd seen the denotation for years yet never actually known what it was).
Numbers with a leading 0 are octal (base 8) in many programming languages too.
Re: 987654321 / 123456789
#98Earlier quoted context omitted.
If you count the sections of your four fingers with your thumb, you can count up to 12 on one hand!
If you count with each finger as a binary digit you can count up to 15 on one hand! 255 if you use both hands! More like 1023 if you also use thumbs but I prefer to use them as carry, overflow bits.
I trained myself to do this by default a very long time ago and I can't imagine counting any other way.
It's so natural, useful and lends well to certain numerical tricks. We should explicitly be teaching binary to children earlier.
Re: 987654321 / 123456789
#99Math can be so fun when you find patterns like this, it’s like hidden magic in numbers.
Re: 987654321 / 123456789
#100Sounds interesting, but site is behind the great Cloudflare firewall. Cannot see it on this phone.