Wikipedia:Reference desk/Archives/Mathematics/2021 August 20

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August 20

What is the integer base with lowest radix economy, when also looking at bijetive bases, negative bases, bijetive negative bases and balanced bases?

What is the integer base with lowest radix economy, when also looking at bijetive bases, negative bases, bijetive negative bases and balanced bases?

When talking about only non bijetive, negative or balanced bases, the number base with lowest radix economy is 3 (e is the best when you also count non integer bases). But what about when you not onlycount integers but also count bijective bases, negative bases, bijective negative bases and balanced ones? 187.59.231.254 (talk) 15:23, 20 August 2021 (UTC)

Let b denote the number of digits in the base. If the length of the numeral representation of n grows ~ log_b n, then the optimal economy is reached for b = 3. This does not depend on the details of the mapping. So this applies for bijective numeration, and balanced ternary is also optimal among the balanced representations.  --Lambiam 15:24, 20 August 2021 (UTC)

This site says bijective binary has a different radix economy to binary and its even bettar than base e https://veniamin-ilmer.github.io/math/bijective_economy.html 187.59.231.254 (talk) 17:00, 20 August 2021 (UTC)

The number of digits in the representation of a positive integer n in the traditional binary format is ⌊(log₂ n) + 1⌋. For binary bijective it is ⌊log₂ (n + 1)⌋. The difference is at most 1, so asymptotically it is the same. I don't know the meaning of using base e; what is the set of digits? I do not understand how E(e, N) is defined in the section Radix economy § Comparing different bases, but this section reeks of original research. Take M = 1234567890123456789. The average value for N = 1 to M of E(2, N) for the binary bijective representation is about 116.26453; however E(e, N) is defined, it can't be worse, since e × log(M!)/M < 110.51787.  --Lambiam 19:02, 20 August 2021 (UTC)

The article claims, without citing sources, that e has the lowest redix economy, but it's hard to see how the definition is applicable for non-integer bases, and the article specifies b must be an integer in the definition. It's certainly possible to define an expansion of a number base e, or to any base >1, see Non-integer base of numeration. But in general the expansions of integers to these bases are infinite and the distribution of digits is uneven, making the given formula for radix economy invalid when b is not an integer.

The GitHub article does not seem to be counting accurately. It assumes. for example, that the only four bits are needed to store '1122', but this is only true if you know ahead of time that the representation has four digits. So if you need to store large and small numbers you must reserve enough space to store the largest possible, and so you can't assume smaller numbers use less space and this makes the average over both large and small numbers invalid. --RDBury (talk) 01:43, 21 August 2021 (UTC)

The ill-conceived concept of "average" radix economies in the GitHub article is faithfully copied, though, from the section Radix economy § Comparing different bases in the Wikipedia article. I'm going to perform a radical OR-ectomy.  --Lambiam 05:24, 21 August 2021 (UTC)