Hi Todd!
On Wed, 26 Feb 2020 12:32:57 -0800
ToddAndMargo via perl6-users 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
