Hi Todd! On Wed, 26 Feb 2020 12:32:57 -0800 ToddAndMargo via perl6-users <perl6-us...@perl.org> wrote:
> On 2020-02-26 12:14, Tobias Boege wrote: > > On Wed, 26 Feb 2020, ToddAndMargo via perl6-users wrote: > >>>> $ p6 'say (99/70).base-repeating();' > >>>> (1.4 142857) > >>>> > >>>> means that 142857 also repeats (it does not), but > >>>> that it is best it can figure out with the precision > >>>> it has? > >>>> > >>> > >>> What are you talking about? It does repeat. I suggest you take a piece > >>> of paper and compute the decimal digits of 99/70 by hand until you are > >>> convinced that it is 1.4 and then an endless stream of 142857 repeating. > >> > >> I used gnome calculator to 20 digits: > >> 665857/470832 > >> 1.41421356237468991063 > >> Sorry. Not seeing any repeating patterns. > >> > > > > Todd, you were asking about 99/70, so I answered you about 99/70. > > I even quoted it. Now you come with 665857/470832. For that number, > > you will see the repetition after about 580 decimal digits. > > > >> Here is NAS doing it to 1 million digits (they have too > >> much time on their hands): > >> https://apod.nasa.gov/htmltest/gifcity/sqrt2.1mil > >> No repeats. > >> > >> So why does base-repeating tell me there is a repeating > >> pattern when there is not? > >> > > > > Sigh. We already settled that Rat and Num in Raku are rational numbers > > and that √2 is an irrational number. So what can you definitely not apply > > the base-repeating method to in Raku? -- The honest value of √2. > > > > NASA apparently computed the first million digits of √2 and you see no > > repeated digits. Good. In fact a real number is irrational if and only > > if its decimal expansion has no repeating pattern. You can view this as > > a reason for why they are not super easy to deal with on a computer. > > > > But that has nothing to do with the numbers we looked at above. Those were > > obviously rational numbers. They were obviously different from √2 because > > that number is not rational -- and we were looking at rational numbers. > > We were looking at rational numbers close to √2. What makes you think that > > NASA computing digits of the number √2 has any bearing on the correctness > > of `(99/70).base-repeating`? Because √2 and 99/70 are obviously not the > > same number. > > > > We are not working out the decimal expansion of √2. We are working out > > decimal expansions of rational numbers close to, but different from, √2. > > Even though they are close, structural properties of the expansions, > > like the existence of a repeating pattern, are radically different. > > > >> Ah ha, 99/70 does have a repeat: > >> 1.4142857 142857 142857 1 > >> > >> Maybe 665857/470832 I just do not go out enough digits to > >> see a repeat. > >> > >> √2 does not repeat though, but I am thinking that > >> I am stretching the poor machines abilities a bit too far > >> and that is where the repeat comes from > >> > >> $ p6 'say sqrt(2).Rat.base-repeating();' > >>>> (1.4 > >>>> 14213197969543147208121827411167512690355329949238578680203045685279187817258883248730964467005076) > >>>> > >> > >> So, with the technology on hand, the approximation of √2 > >> does have a repeating pattern of > >> > >> 14213197969543147208121827411167512690355329949238578680203045685279187817258883248730964467005076 > >> > >> (And in engineering terms, is meaningless) > >> > > > > Yes, √2 has no repeating decimal pattern and the repetition returned to > > you is the one of the rational approximation to √2 that Raku computed > > when you asked for `sqrt(2).Rat`. > > > > (Somehow it seems like you understood halfway through writing your > > response but decided to keep the first half anyway. I don't know why.) > > > >>>> And what are the unboxing rules for (665857/470832)? > >>>> > >>> > >>> No idea what you mean. > >> > >> When is <665857/470832> unboxed to a Real number to > >> operate on it? Or is it never unboxed? > >> > > > > I'm no boxing expert, but I know that Rat has high-level arithmetic > > defined for it and there is no native rational type to unbox to. > > That would have to go through Num, I suppose. So I see neither a need > > nor a target for unboxing a Rat. But I still have no idea what kind > > of operations you have in mind. That said, Rat is not a NativeCall- > > related type. > > > > Thank you! > > The more I learn about Raku, the more fascinating I find it. > > All trivia aside, sqrt(2) has way more precision for any > real world application I throw at it. > > Speaking of trivia, and off topic, did you know that > √2 caused a major religious upheaval when the result > of a 1,1,√2 triangle came out? The poor Pythagoreans: > all numbers had to be rational. Hippasus even > got murdered for blowing the whistle on √2. > See https://en.wikipedia.org/wiki/Hippasus for what we more accurately know about that. > Now-a-days, we just torture our computers with it. > > :-) > > -T -- Shlomi Fish https://www.shlomifish.org/ List of Portability Libraries - https://shlom.in/port-libs The Knights Who Say “Ni” once said “Ni” to Chuck Norris. They are now no longer The Knights Who Say “Ni”. — http://www.shlomifish.org/humour/bits/facts/Chuck-Norris/ Please reply to list if it's a mailing list post - http://shlom.in/reply .
