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A board for discussion of the number twelve as the base of numeration in mathematics and physics.


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"Lagrange was Wrong, Pascal was Right"

Phaethon
Phaethon
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"Lagrange was Wrong, Pascal was Right" Empty "Lagrange was Wrong, Pascal was Right"

Post by Phaethon Mon Jun 20, 2022 5:00 pm

BREAKING NEWS June 2022:

https://sciendo.com/it/article/10.2478/rmm-2022-0004
Recreational Mathematics Magazine, Volume 9 (2022), Issue 16 (June 2022), pages 55 - 64.
Article title: "Lagrange was Wrong, Pascal was Right"

The author George Teşeleanu concludes: "Conclusions: According  to  our  experiments  the  best  positional  base  is  the  duodecimal  system [...] Thus, Pascal was right to propose the use of the duodecimal system instead of the decimalsystem, while Lagrange was wrong to stop its adoption by the French Metric Commission."
Phaethon
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"Lagrange was Wrong, Pascal was Right" Empty Re: "Lagrange was Wrong, Pascal was Right"

Post by Phaethon Tue Jun 21, 2022 10:12 pm

I am skeptical about the procedure used in this article.

The bases were classified by a "grading system" according to divisibility properties including divisibility tests such as digit sums for each number tested as a "divisor" into the base. The "divisor" was not necessarily a factor of the base. It seemed to me to be fairly arbitrary how bonuses and penalties were applied. Computational divisibility tests are mostly not what numerical bases are used for. The main kind of divisibility test used in practice is one that does not involve a computation and is by recognising the final digits visually and using memory only. If you ask me, I would say that computational divisibility tests should not be included in the grading system; only the number of digits after the fractional point of the base should participate in the grade, as these influence the accuracy to which the base can represent the fraction of a divisor.

A "quality index" for the base was then fabricated as the sum of the grades for each tested number as a divisor up to an arbitrary number of divisor terms that could be varied.

Bases larger than twenty were not depicted in the plot: "for positional numerical systems. The results for b1⩽20 are depicted in Figure 15.  We can clearly see that bases 6, 12 and 18 are the best." And then at the end of the same paragraph there is this strange statement: "An interesting fact is that although the best systems are the sexagesimal and the trigesimal ones (see Table 2b), by using the duodecimal system we have more than 90% efficiency for a fifth and, respectively, almost a third of the symbols."

Reference:
https://dozenal.forumotion.com/t24-dozenal-fifths-better-than-decimal-thirds
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