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.
Now-a-days, we just torture our computers with it.
:-)
-T
