Talk:Quote notation

10-adic number

It is strange that the connection with the p-adic number representation is not mentioned. In fact is the quote notation another notation for 10-adic numbers. As usual a sequence

123'45.6 has the meaning ...12312312345.6

with the extended interpretation of the sum of a geometric series.

...123123 = 123(1+1000+1000x1000+ ...) = 123x(1/{1-1000}) = -123/999

Hence 123'45.6 = 45.6 + 100x(...123123)=45.6 - 100x123/999

Madyno (talk) 14:51, 13 March 2018 (UTC)

@Madyno:

As far as I see the connection with the p-adic number representation IS mentioned: "The construction of this system follows the approach of Kurt Hensel's p-adic numbers."

Moreover, it can also be used with decimal (NOT 10-adic) numbers. Then:

123'45.6 = 12345.6 = 12345.645 = 12345 + 645/999

The difference is that with decimal notation the repeating group is to the right of the quote, whereas with 10-adic it is to the left.

This is NOT mentioned and IMHO should be mentioned in the article. --Nomen4Omen (talk) 16:55, 13 March 2018 (UTC)

Yes , I know. I've seen that the connection just was mentioned, but not applied. So I added some text. Madyno (talk) 17:08, 13 March 2018 (UTC)

average vs. worst case length

It remains unclear, how Special:Contributions/70.29.73.25 on 10 April 2019‎ at 20:42 got his so-called "experimental results". He states

"The 180,000 shortest numerator-denominator representations require 15.65 bits on average, and those same numbers in quote notation require 39.48 bits on average."

As I found out he quotes the first "Appendix added later" of the source. But they do not give some way of calculation. Nor consider that the quote notation depends additionally on the base chosen.

Also, it is not appropriate "Taking the shortest numerator-denominator numbers ...". This way of thought is like putting a wet finger into the air. Especially, if they think it results in a biased comparison in favor of numerator-denominator.

Since it is good mathematical practice to look at the worst case figures first, I replaced his section on space complexity by a look at the Carmichael function.

The subsequent considerations on time complexity are not really better. But, since they are restricted by the space complexity, they cannot harm very much.

As a conclusion, the source

Hehner, E.C.R.; Horspool, R.N.S. (May 1979), A new representation of the rational numbers for fast easy arithmetic (PDF), SIAM J. Comput. 8 no.2 pp.124-134

is not a qualified one. It cannot be, because the drawbacks are too severe. As a consequence, there is no computer algebra software using their approach.

--Nomen4Omen (talk) 16:09, 12 April 2019 (UTC)

1. The approximation of the Carmichael function#Average value given by Paul Erdős and the other numbers given in that article, both diametrically contradict the estimates given by Hehner & Horspool. The Erdős-approximation proves and the other numbers suggest that almost all (i.e. all but a subset of measure zero) of the λ-values are exponential in the length $l:=\log _{2}(n)$ of the input $n$ , namely $c^{l}$ with some $c>1$ .
2. If the user has a workspace of size $2^{l}$ bytes, which is equal to $2^{3+l}$ bits, s/he is able to safely use “quote notation” for numbers $\leq 2^{3+l}$. E.g. if s/he has 1 Megabyte s/he should stay with his denominators below $2^{23}\approx 8,000,000$. But, two such denominators will very probably blow up the workspace. --Nomen4Omen (talk) 07:50, 30 May 2019 (UTC)

invented by?

Is somebody able to tell what part of the subject may have been invented by Eric Hehner and Nigel Horspool?

As far as I can see, all algorithms, i.e. addition, subtraction, negation, multiplication, division, conversion, are very well known in the literature as well as the net. The only thing I am not absolutely sure of is the "sign determination" which, of course, is really a strange thing in the p-adics, because their is no sign in the p-adics nor are there "negative numbers" nor is there an order relation. But, of course, there are rational numbers which can be imbedded, but when mapped back, they are ordered and can be negative. So, Hehner's and Horspool's criterion is kind of surprise which may have not been detected by other people.

Certainly, the "quote" and the "exclamation mark" is a great invention. Before that, there was only the overbar or what they call "overscore" notation. --92.196.39.72 (talk) 12:07, 15 May 2019 (UTC)

Btw, before Hehner and Horspool there have been at least three notations (numerator/denominator and overbar in the reals plus overbar in the p-adics). OK, the last one they call quote notation. But they admittedly need numerator/denominator and overbar in the reals, e.g. for arithmetic if the results blow up storage space or when rounding or sorting is needed.

O I see! If you're right the invention is some kind of baptizing (giving a new name to old stuff). --Nomen4Omen (talk) 08:56, 21 May 2019 (UTC)