# Numeral systems diversity

## Contents

## Alternative titles

- the hexadecimal numeral system (first/old/previous title until 2010-07-11)
- Numeral systems and early mathematics and engineering learning (environments)
- Lukas Girtanner struggling with numbers
- Lukas Girtanner's struggle with numbers
- in an ideal case, the word "still" could be added. or alternatively "(still?)", but the ideal case will not happen
- numeral systems diversity (new/current title since 2010-07-11: The title "numeral systems diversity" might more appropriately reflect the idea of using as many diverse numeral systems in mathematical or engineering contexts where numbers are being used than the old/previous title "the hexadecimal numeral system".)
- numeral system diversity (without the s)
- diversity of numeral systems
- diversity in the learning of numeral systems
- learning as many numeral systems as possible
- learning as many diverse numeral systems as possible
- learning as many as diverse numeral systems as possible

## General remarks

- Should the title of this page here be changed because it is a general discussion about numbers and numeral systems in learning environments (especially given the fact what mathematics actually is - not about numbers at all)?

- The (constant - at least in the long- and medium-term) discussion/change/optimization of a particular/specific numeral system - especially in the context of didactics, education and learning environments - as a compromise with reality in order to ultimately reach mathematization? Isn't actually ninety-nine (0x63) percent (or even more; or even one hundred (0x64) percent? - but in case of one hundred (0x64) percent, I would be disappointed) of this whole page here a constant compromise with reality? The page compromise with reality might become a really large page. Actually, aren't ninety-nine (0x64) percent (see the considerations before) of this website here a compromise with reality? Therefore, actually, hasn't the page compromise with reality already been written respectively it is constantly being written?

- How problematic is it or does it make sense (is it justifiable / to which extent / under what kind of circumstances) to "endorse" a particular numeral system as a "pure mathematician" and/or a person with mathematization goal in mind?
- the hexadecimal system would also have to have name that is not based on "... decimal" because isn't the term "hexadecimal" a little bit long especially when it is the numeral system being the most commonly used daily and casually?
- for mathematics - the choice of a particular numeral system does not matter at all
- the choice of a particular numeral system is relatively important in terms of engineering
- the choice of a particular numeral system is an issue for people with high but not very high intelligence (and therefore the majority of intelligent people)

- aren't numbers just as unmathematical as for example a real world object? aren't numbers just some real world objects that only indirectly have to do something with mathematics?

There might still be disadvantages of the hexadecimal numeral system:

- The hexadecimal numeral system has a "poverty" in terms of prime number base: the prime number two (0x2) is the one and only base (in contrast to for example ten=two*five (0xA=0x2*0x5) or six=two*three (0x6=0x2*0x3) respectively twelve=two*two*three (0xB=0x2*0x2*0x3) or twelve=four*three (0xB=0x4*0x3))
- In case of a numeral system with a base that is larger than ten (0xA), hand counting would be difficult for human babies, toddlers and children since they have only two*five=ten (0x2*0x5=0xA) fingers. But to which extent does hand counting matter in case of (very) intelligent children? Don't they quickly progress beyond this development stage? And isn't it anyway possible to hand count also up to sixteen (0x10), not as elegantly as in case of ten (0xA), but there are several possibilities: 5+5+5+1 or 5+3+5+3 or 4+4+4+4 (the version 4+4+4+4 being the way that would most closely resemble the structure of the hexadecimal system).

But there are a lot of advantages of sixteen (0x10) as a base number:

- especially the children with high (but not very high) intelligence might be motivated more to do robotics and more general electrical engineering if the numeral system that they know best is also the one that they use most/mainly in robotics and more general electrical engineering in order to minimize their effort when using numbers during their engineering process. (For the mathematically very talented children, the contrary might be the case, that they feel underchallenged by the ease of application of usual numeral system for engineering or they might be disappointed by the use of a numeral system like the hexadecimal system because it has only one single prime number as its base despite being a comparably large number. For these mathematically very talented children, the use of as many different numeral systems as possible is recommended and the UMIS should be able to switch between as many different numeral systems as possible - maybe with the exception of very large numbers as its base. However, mathematically even more intelligent children might not or hardly do engineering at all and also hardly use any number(s) in their pure mathematics effort and in their case, the considerations on this page here might not be applicable anyway. So, this page here is actually only valid for all those children who do not do pure mathematics exclusively from birth (probably the vast majority of children, at least at the moment)).
- increased "visual address space" (a more dense digit space)
- full binary compliance
- no need for new fonts (an indirect engineering issue, especially in the present-day engineering world)
- even the largest binary-based number space with no new need for new fonts (see also the point above)
- square of square of 2
- compatibility with bytes
- nature/evolution (more specific: natural biological evolution on the planet Earth) chose 4 as the base for its lifeforms (the 4 DNA proteins). This might be an indicator that either 2, 4, 8 or sixteen (0x10) might be a good choice. And not 2 * 5 = 10 (0xA)
- similarly to nature, (electrical) engineering and computer science also evolved it respectively found it most useful

has there maybe be a competition 1-3 billion years ago between various lifeforms that had their protein information based on different systems that were based on different (prime) numbers, for example 2, 3, 4, 5 and 6, ... and only the lifeforms with the base 4 (that ultimately evolved into the 4 stable DNA proteins) survived and extinct all other lifeforms? what might that mean for the choice of a numeral system today? and for biological life (of the future)? giving all other non-based DNA or non-4-based lifeforms an equal chance?

twelve (0xB) would incorporate the advantage of 4 and still have prime number diversity advantage because of 3

twenty (0x14) would incorporate the advantage of 4 and still have prime number diversity advantage because of 5

the main disadvantage of sixteen (0x10) is its very low prime number diversity/richness

on a learning psychology level, it matters tremendously in case of not-ultra-intelligent children because the numerical mind will be shaped like that from birth

possible to do some kind of multimodal parallel numeral system learning? writing at least several numeral systems next to each other? and in a perfect case an infinite number of numeral systems next to each other? wouldn't be that exactly the most unmathematical solution possible since mathematics and especially its (most fundamental) axioms are exactly the contrary of really existing numbers (I still remember one axiom of Zermelo-Fraenkel where there is an infinite set with an empty set and a set with an empty set it in and a set with an empty set with an empty set and a ..., this is mathematics, not numbers)? wouldn't the correct way of writing numbers be to not write any numbers at all? but what about engineering (and maybe computing)? engineering (and computing) without numbers at all? so, as part of a compromise with reality, learning and engineering still with numbers and therefore having to decide about a particular (or a priority/order of) numeral system (because/since it is impossible to introduce an infinite amount of numeral systems)?

numeral systems are not a matter of course

a compromise? maybe just write the hexadecimal system as the prime numeral system and the decimal value only for the toddlers and children who need in order to orient themselves into the rest of the world too and have to learn "the other numeral system" (the decimal numeral system) too from early age? In order to not be surprised too much what kind of numbers the rest of the world uses (the decimal numeral system)?

the ultimate justification for sixteen (0x10) simply to take the lowest prime number, namely two (0x2)? sixteen (0x10) only as a visual and "engineering" representation for 2, but not a genuine number because 2 is the genuine number? could 4=DNA on Earth also be interpreted in this way? four (0x4) just as an engineering solution by nature based also on the lowest prime number - two (0x2) in a similar way like the hexadecimal system? the lowest prime number two (0x2) as the ultimate solution? and only a limited importance if four (0x4), sixteen (0x10) and two hundred fifty-six (0x100), ...? but what about the numbers in between that are not the continuation of squares (eight (0x8), thirty-two (0x20), maybe sixty-four (0x40) and one hundred twenty-eight (0x80))?

how quickly should I begin to use the hexadecimal system on this page here too in order to serve as some kind of example? should I struggle to write every decimal number that I have written on this page here in a hexadecimal notation too or should I even write hexadecimal numbers primarily or even more strictly only the hexadecimal number just in order to serve as a first example of how to switch to a numeral system? and what would that mean from a mathematical point of view? a futile and unnecessary attempt and waste of time and energy?

proposals for new names of the hexadecimal system: hex-system, hexs (problem: still oriented to the decimal system because it is 6+10 (0x6+0xA)), simply 0x like in electric engineering?

how much should one hesitate to change such an numeral system? not much, since mathematically numeral system do not matter and in terms of engineering, the numeral system of 16 (0x10) is so prominent that one has to adapt to it anyway during youth or early adulthood if one wants to become a good electrical engineer, so why not just learning it as a prime system from birth? the transition to the new numeral system 16 (0x10) might take only a few months or years and would consist of simply only or additionally using the hexadecimal numbers. furthermore, the universal mathematical information system would also have to be adapted in a way (see next section).

the universal mathematical information system would also have to be switchable between numeral systems, in an ideal case, all numeral system would be selectable, with 16 as the default (when one "installs" or "starts") in the software/hardware. There would also have to be enough numeral systems conversion exercises (especially for the intelligent, but not very intelligent children) that would start as early in childhood as possible. Whenever mathematics is being taught and numbers are (still) used, it would be recommendable to accustom the children to using several numeral systems daily but with the hexadecimal system having priority.

as far as I am concerned, i will probably use both numbers here in the future. I have already begun to adjust this page here in terms of these numbers. another idea might also be to write out the decimal numbers in the future as words and to write the hexadecimal numbers as numbers, for example "I have eaten twelve (0xC) kiwis the last month." The question how to denote the number as hexadecimal remains. Since people are used to the fact that decimal numbers are the standard, it would make sense to just denote the hexadecimal numbers as nonstandard. On the other hand, as soon as a person is more used to hexadecimal numbers than to decimal numbers, it might make sense to specifically mark the hexadecimal numbers instead of the hexadecimal numbers. More correctly and completely would be to always write which numbering system is being used since the hexadecimal system might also one day be replaced by an even more useful numbering system (maybe 4 because of the 4 DNA proteins?).

There seems to be a certain degree of competition between the hexadecimal and dozenal numeral system since both articles on Wikipedia are almost equally long. It is clear that the dozenal (and to some degree the 6-based system) have advantages that the hexadecimal does not have and that the decimal system only has to a lesser degree.

On Wikipedia's hexadecimal numeral system, there is also a counting scheme and a good overview which letters have been proposed for A-F and which markers are being used for the hexadecimal numeral system in every computer science and electrical engineering context. I will probably use the Unix/C notation 0x because it is easy to find on every keyboard.

One should also not forget that for example fractions require quite a (re)adjustment from the decimal to the hexadecimal system. Purely mathematically, that does not matter at all, but in reality, most people will have to think at least a fraction of a second longer if they are not used to a particular numeral system or several numeral systems.

Another idea is not to write the several different numeral systems if all commonly used numeral systems have a base that is above these numbers and therefore, it would be clear anyway. For example, if one is used to used the decimal, dozenal and hexadecimal numeral system, one would not distinguish numbers between 0 and 9. I will also handle it like that. (If the 6-based numeral system would be commonly used, only the numbers 0-5 could be written without information which numeral system is being used). Another question concerns when just one number is high enough, for example 10, if one has to rewrite all numbers, also the lower numbers. Maybe a pragmatic approach to just write the one number that is too high in several ways and not all other numbers (there is such a case further up on this page here).

## Numeral systems for whom?

I am not sure to which degree/extent the following statement is true: Always (keep in) mind in which communication context one is: Don't mathematically untrained linguists or other non-engineers and non-mathematicians think differently about numeral systems (for many of them, even a low hexadecimal number might already be a challenge to understand quickly)? And what about engineers or "mediocre" mathematicians? Isn't the issue of numeral systems and their priorities especially interesting for them? And what about the few extremely intelligent mathematicians? How do might they think about numeral systems? Do they care at all about them. Probably not, at least not as a fully developed mathematician in adulthood. Maybe as a child for some time when numbers were still beneficial for their concrete learning experiences? But I am not even about that sure. But in terms of the majority of children and people, numeral systems certainly matter and they will continue to matter at least in the short- and maybe medium-term.

## Advantages and disadvantages of a particular numeral system

Maybe a table with advantages and disadvantages instead of a list might also be an idea, but at the moment, I do not know how tables are made with/in (default installations of) MediaWiki.

### Abbreviations

- el.en.now = electric engineering now/nowadays
- DNA correlates with el.en.now since DNA is based on 4 and el.en.now is based on 2 and 16 (0x10) (4 somehow being the "link" in between)
- ndpd = numeric density per digit:
- negatively correlates with number length (independent of the numeral system "range")
- ndpd and memorability:
- low ndpd = low memorability for lower numbers: the binary system (low number) is hardly memorable (because its too limited set of characters) (for entities of now)
- medium ndpd = high memorability for medium numbers: the decimal, duodecimal and hexadecimal system have a medium ndpd and a high memorability (for entities of now)
- high ndpd = low memorability again for high numbers (like the low numbers): the sexagesimal system has a high ndpd and a low memorability (for entities of now)

- mem = memorability / easy of learning
- whmim = what might it mean?
- prime = prime number potential (might correlate with electric engineering potential in the future and even more in the future even with a future DNA), note that from a viewpoint from pure mathematics and mathematization, this factor is the only important factor and therefore, this factor will become increasingly important compared to the other factors and ultimately be the only important factor (therefore, the duodecimal numeral system might ultimately be the numeral system of choice but only if numbers matter which is not the case since numbers do not matter in pure mathematics, therefore, it might a paradoxical development: the more the duodecimal numeral system would/will matter, the less numbers or numeral systems matter...)
- ++++ = perfectly adapted to the task or obviously the best/perfect solution/choice
- +++ = very suitable
- ++ = suitable
- + = might be suitable somehow "too"
- -- = unsuitable
- --- = very unsuitable
- ---- = absolutely/most unsuitable

### The list with the numeral systems

- two (0x2; dd_2): http://en.wikipedia.org/wiki/Binary: el.en.now++++, DNA+++, ndpd-4, mem-4, prime?? (lowest prime number, whmim?)
- three (0x3; dd_3): http://en.wikipedia.org/wiki/Ternary: el.en.now--, DNA--, ndpd-3or4, mem-3(or -4?), prime?? (second lowest prime number, whmim?)
- four (0x4; dd_4): http://en.wikipedia.org/wiki/Quaternary: el.en.now+++, DNA++++, ndpd---, mem-1or2, prime- (only 2*2 => actually only 2)
- five (0x5; dd_5): http://en.wikipedia.org/wiki/Quinary: el.en.now--, DNA--, prime?? (third lowest prime number, whmim?)
- six (0x6; dd_6): http://en.wikipedia.org/wiki/Senary: el.en.now--, DNA--, prime+++ (2*3 => good, especially for such a low number)
- seven (0x7; dd_7): http://en.wikipedia.org/wiki/Septenary: el.en.now--, DNA--, prime?? (fourth lowest prime number, whmim?)
- eight (0x8; dd_8): http://en.wikipedia.org/wiki/Octal: el.en.now++, DNA++, prime-- (only 2*2*2)
- nine (0x9; dd_9): http://en.wikipedia.org/wiki/Nonary: el.en.now--, DNA--, prime?? (3*3, see 3, whmim?)
- ten (0xA; dd_A): http://en.wikipedia.org/wiki/Decimal: el.en.now- (a little bit better than the other prime numbers since at least the number 2 is implied), DNA--, prime+++ (2 prime numbers like 6, but 10 is higher than 6; 12 also has 2 prime numbers although 12 is a little bit higher, but the prime number 2 at 12 is double but to which extent does that matter?)
- twelve (0xB; dd_10): http://en.wikipedia.org/wiki/Duodecimal: el.en.now?? (clearly less apt than the hexadecimal system and probably also less apt than the quaternary system, but at least 2*2=4 is implied, therefore, clearly a better choice at least than the decimal system where 2*2=4 is not implied, but only (but at least) 2), DNA++ (in a similar way to 8), prime++++ (the prime number richness/density despite its low value is the main advantage and the main reason why to choose the duodecimal numeral system already now as a numeral system and to prefer it over the decimal system and at least in terms of prime number density also very clearly over the hexadecimal system), mem+4
- fifteen (0xF; dd_13): el.en.now--; DNA--; prime++
- sixteen (0x10; dd_14): http://en.wikipedia.org/wiki/Hexadecimal: el.en.now++++, DNA+++ (or only ++?), ndpd+1, mem+3, prime----
- twenty (0x14; dd_18): http://en.wikipedia.org/wiki/Vigesimal: el.en.now+??, DNA+??, prime++, mem+1
- twenty-four (0x18; dd_20): http://en.wikipedia.org/wiki/Base_24: mem-0
- thirty (0x1E; dd_26): http://en.wikipedia.org/wiki/Trigesimal: mem: -1
- thirty-six (0x24; dd_30): http://en.wikipedia.org/wiki/Base_36 (several designations/names)
- sixty (0x3C; dd_50): http://en.wikipedia.org/wiki/Sexagesimal: el.en.now+??, DNA+??, ndpd+4, mem-3, prime+++++ (the most inclusive prime number situation in the list, but 5 times higher than 12 and 60 different characters would be needed; a good idea for the future when learners can master 60 different numeral characters during the learning process?)
- two hundred fifty-six (0x100; dd_194): el.en.now++++???, DNA+??, ndpd+10, mem-10, prime-----
- what about (the unary, the bi-quinary, the pentimal,) the negative base, the square root of negative square numbers (complex numbers) (what did I mean with that?) and the quater-imaginary base (http://en.wikipedia.org/wiki/Quater-imaginary_base ) numeral systems?

The above list and the assigning process of the --s and ++s is not yet finished, for more information, see my personal diary.

Other factors to consider is primorial number (base 30 is a good choice), vulgar fractions (base 30 is a good choice), radix economy (base 30 would be a bad/suboptimal choice), highly composite number (dozenal, base 24 and base 36 and sexagesimal would be good choices), another important factor is the cognitive overload when having to learn and internalize a lot of symbols (see below). A technical issue is the number of alphanumerical characters available, base 36 is the highest when the uppercase and lowercase letters are not differentiated (see below).

For a converter between the different numeral systems, see http://www.translatorscafe.com/cafe/units-converter/numbers/calculator/ or http://www.efunda.com/units/base_n.cfm, this are the two most comprehensive converters that I have found so far.

## Limits of cognition of the children / New characters/symbols? (Character repertoire/set)

Alternative title: Limits of human cognition (now) / Limits of present human cognition

It is unclear how big the number/character set can be that the children can still learn it. In respect to this question, one simply has to gain experience by letting children learn numeral systems other than the decimal system.

It is also unclear to which extent the children can parallely learn several numeral systems or how much they should focus on only one or two numeral systems and if these numeral systems should be in the lower range (probably the vigesimal system or lower) or the upper range.

This is a general question. Should, in order to clearly distinguish the numeral systems from each other, a new set of characters be introduced or should the present characters be used? A disadvantage would also be if the children learn totally new number-symbols and afterwards, it turns out these new symbols are not used widespread enough and they have to return to conventional (Latin) alphanumeric characters. If several numeral systems are learned, it would probably also be additionally complicated if for every numeral system, new symbols would have to be introduced and not just one single set of characters for all new numeral systems. For the sexagesimal numeral system, new characters would have to be introduced but the sexagesimal system with its multitude of symbols is almost certainly already out of range for most children of nowadays (but maybe not in the future (thanks to better/improved didactics and genetic engineering even more in the future), so there might be a gradual process towards numeral systems with a larger base).

A general question concerns the extent to which a community would be able to parallely provide the different numeral systems, so that the children would be exposed to and how much the children could invidualize their exposure to the different numeral systems so that they are neither bored nor overstrained. Thanks to the fact that most reading would be on computer or robotic screens, this individualization of the learning/exposure process could be implemented with sophisticated software.

### Conclusion

The most promising candidates are the hexadecimal numeral system in terms of engineering (at least for now) and (especially in the longer term) the duodecimal system in terms of mathematics (with the limitation that numeral systems actually do not matter at all in pure mathematics). The 3 numeral systems of the decimal numeral system, the duodecimal and hexadecimal numeral system also share a high numerical density per digit and a good readability and learnability at the same time with slight advantage for the hexadecimal system over the other two numeral systems in terms of numerical information density per digit (and the same is true to a lesser degree for the duodecimal system over the decimal system). (However, the question remains to which extent increasingly intelligent people or entities might be able to master more and more dense digits which might mean that numeral systems with higher bases might become more/increasingly feasible to learn, for example 24, 60, 120 or 256.) Therefore, it is recommendable to especially use and get accustomed to the hexadecimal and the duodecimal numeral system (in addition to the decimal numeral system) at least for now, but to prepare to shift to higher and ever increasing numeral systems 12->24->120 and 16->256->65536... in the future. Actually, every combination numbers (and especially prime numbers) is possible. Therefore, the numeral systems might also develop in a way like 6->30->210->2310. In the most extreme case, it might also be possible that the highest possible prime number or the product of all known (adjacent or non-adjacent) prime numbers might be used as a numeral system more in the future with an ever increasing base (as long as the symbolic/character set/repertoire suffices). It is also possible that several numeral system paradigms (technological, prime-number/mathematical) might coexist with each other for a long time. Flexibility in terms of general numeral system learning and use is advisable and should be learned from birth in a similar way like multiple parallel natural language learning. A list of priorities might be constantly assessed, generated and adapted to the ever increasing learning abilities and the increasingly complex technologies. As for now, I would recommend the following numeral system priority (learning in terms of simply time or numeral-systems-specific-exercises) for a young learner (baby, toddler, child) with the potential for having an IQ of maybe 120-140 as an adult: 25 percent hexadecimal system, 25 percent duodecimal system, 15 percent decimal system, 10 percent quaternary system, 10 percent binary system (time-consuming) and the remaining 15 percents divided to other systems like 6, 3, 8, 15, 5, 20, 7, 30, 9, 60, 11, ... based numerical systems (maybe in that order). For a mathematically more intelligent child (IQ maybe between 140-170), I would suggest reducing the percentages in the beginning to maybe 5-10 percent each and to reserve more learning with the remaining numeral systems and add more numeral systems. In case of mathematically extremely intelligent children (IQ maybe >170), no numeral systems might be used at all since they might no be involved in engineering at all or if numeral systems are used when they are still a baby, toddler or a very young child, they would learn in as many numeral systems as possible and probably soon after stop using any numbers at all. Regardless of the mathematical intelligence of a particular learner, every learner would individually select the numeral system being used anyway autonomously. The numeral systems would just have to be different so that no learner is underchallenged and as long as a learner is intrinsically motivated to switch between the different numeral systems, it should be encouraged as much as possible by providing the necessary options in the UMIS and in engineering as much as possible. And in order to prevent that the mathematical and engineering mind of a child is increasingly formed in less numeral systems than it could have mastered and/or be motivated for, it is necessary to give every learner the chance to realize the full extent of diversity of numeral systems so that it can select best which and how many numeral systems the learner want to learn with. It might be possible that every learner might develop an individual/personal preference based on the own mathematical and engineering intuition in a similar (but more abstract) way that one has a preference for a particular color (now, the colors from the E. A. Girtanner page are here...). The particular/individual preferences and priorities that a numeral systems learner might develop might also be reflected in other learning processes of other (including more purely) mathematical topics where where individual learning preferences and priorities might equally be developed by a learner and lead the intrinsically motivated learner to one or the other topic of pure mathematics more than to other topics of pure mathematics, although selecting or preferring a particular number or numeral system is only comparable to a limited degree to selecting or preferring the learning of a particular topic from pure mathematics, isn't it? (see also speculative details of mathematization?)

I decided to not write the corresponding hexadecimal numeral systems in this section here but instead, I switched back to exclusive decimal numeral system writing because this section here would look quite chaotic if I used both numeral systems next to each other at least in this section here. A solution might be to change the section from time to time forth and back between the hexadecimal and the decimal system and this might also be partially the case for learning environments in order to prevent a lack or overview.

This page here has really been work in progress and just in the course of writing and thinking about the numeral systems, I developed some new ideas in addition to my older ideas. It has now become clear that the discussion about the use of a numeral system in a learning environment generally and the task of selecting and setting priorities (for) specific numeral systems is a complex and interesting task, depending on learning abilities, technologies involved and mathematical complexity of the base number (and not on tradition) with a possible increase of the importance of the factor of the mathematical complexity of the base number in the longer term compared to the other two factors.

## To do

- square root of negative number / complex number numeral system is wrongly written, I have to correct it
- for analogous neural networks (non digital circuits), 2, 4 and sixteen (0x10) is relatively irrelevant
- big DNA=4 advantage of twelve (0xC) over ten (0xA), another reason why to use the duodecimal system instead of the decimal system (but what about the hexadecimal system with even better DNA=4 compliance?), furthermore, there is also a much better congruence with the binary system since at least 4 is included too and not only 2 like in the decimal system => strengthening both: duodecimal and hexadecimal at the expense of the decimal that has no such advantages since the 5 in 2*5 is not really important at least now (but maybe sometime in the future surprisingly again because of 5-(2-)based technologies)?

### Just some details

- the hexdec system somehow comprises the quaternary and the binary system and to a lesser degree the octal system since it is possible to relatively directly read the numbers out of the hexdec system when writing quaternary or binary numbers.

## Other points

- paradoxical: the more intelligent and faster learning entities of the future might be able to both memorize low ndpd and high ndpd better so becoming better at numeral systems like the high hexagesimal and the low(est?) binary at the same time: an end at the lower end (2) but infinite openness upwards?
- a somewhat historical observation: I remember dimly that I might have advocated the decimal system on philosowiki.org in the year 2006 because of shortsighted convenience considerations (I proposed reforming time measures for convenience to the decimal system if I remember it correctly) - and now exactly or at least partially the opposite: the duodecimal instead of the decimal system? but wouldn't that also be convenience? a new kind of convenience? not necessarily, since the development upwards from twelve (0xC; dd_10) to 60 (0x3C; dd_50) and beyond (for example 420) is open too and even encouraged, therefore, a constant change - but for what? Constant "convenience"? But wouldn't everything be convenience then? Every measure of efficiency just serving "convenience"? (The "convenience considerations" might be a topic for a new page, actually, aren't they?)
- the term "percent" might also have to be changed in order to conform with the hexadecimal and the dozenal and hexadecimal system. 144 (dd_100) or 256 (0x100) as a new reference number? respectively per...144/256...s (word instead of "percent" missing)
- new wording would be necessary for numbers like 0x20, 0x30, 0x40, 0x50
- new words for hexadecimal system and duodecimal system (dozenal - there is already one word and I will use it in the future).
- to which extent might gradual changes be recommendable? would it be necessary to do a full numeral system change in one generation? for example time could still remain in the conventional system (that is by the way already more duodecimal than it is decimal) and just written in all three versions: hexadecimal, duodecimal and decimal. and if a new system for time is introduces, it would have to be truly revolutionary in order to be worth the change and this would probably only make sense if entities are already partially or mainly live in space. (so actually another topic: a new time and measurements. but not really important at the moment probably, isn't it?
- children prefer dd_10 over dd_A as it is written in Wikipedia
- is dd_ a good prefix for dozenal? I invented it spontaneously today. or another term? Maybe doz_ or do_ or dz_ would be a better term since dozenalists reject the term "duo-decimal"? and what about a better name for hexadecimal? (see next point)
- better name for hexadecimal? binbin(binbin)? bb? bbbb? 4b? techsys? ts? duoduo(duoduo)? dd=duoduo (but how to distinguish it from 4)? dddd? 4d? techduo? duotech? bintech? techbin? ee? eesys? esys? ees? hex? (but shouldn't the six be avoided because it is ten-based?) fourfour? ff? techtech? et=eltech? et=engineering technology? te=technology engineering? tt=techtech? it is probably too difficult to find a naturally well sounding name that is clearly better/superior than the other names since the hexadecimal numeral system a very unnatural and somehow artificial numeral system anyway with prime number poverty and maybe only limited importance in the longer term (despite DNA=4)? or just keep hexadecimal for the time being? and isn't such "definite name" important at all, now?
- won't the hexadecimal system become almost completely obsolete in the long-term than the dozenal system since technologies change while prime numbers don't change? and the only reason why the dozenal system might become obsolete too might be that it does not contain as many prime numbers as higher numeral systems?
- paradoxical: the more mathematized the future is and the more numeral systems become technologically and in terms learning intelligence feasible on a "low" mathematical level, the less important numeral systems become because the mathematical "level" increases? no, that is not true. it is not a matter of levels of the future because mathematics is already now far above numbers. so, it is a question of intelligence, of being able to do pure mathematics instead of numbers or numeral systems and it will be this development that will make numeral systems the more obsolete the better the mathematical ability becomes. this is the paradoxical effect from an engineering perspective. for pure mathematicians, this paradoxical effect does not exist probably because numbers do not matter already now.
- therefore, doesn't the increasing obsoleteness of numbers and numeral systems just reflect a property what mathematization is: (increasingly) pure mathematics instead of engineering and compromising with reality?
- maybe conclude this page here and focus on other topics?

## Conclusion(s) of the conclusion(s) ;-)

(This section "Conclusion(s) of the conclusion(s)" here was added on 2010-07-12.)

- adapt/fit the numeral systems to the context where they are used: for example in the context of time and date, the dozenal numeral system might often make more sense than the hexadecimal (and the decimal) numeral system. In a context of electrical engineering or computing with conventional microprocessors, the hexadecimal numeral system might make more sense than the dozenal and decimal numeral system. In terms of pure mathematics, no numeral system at all might be possible or if a numeral system is still used, a (an ever changing) numeral system with prime number richness like the dozenal or sexagesimal numeral system might make sense.
- the following/remaining three points matter especially for the universal mathematical information system (UMIS):
- prioritize the various numeral systems when necessary, but generally try to increasingly decrease prioritization (in the future): try to avoid favouring just a couple of "agreed upon" or "traditionally taught/learned/used" numeral systems like the decimal, dozenal, hexadecimal, octal, quaternary, binary and sexagesimal numeral system. Give all numeral systems an equal or at least sufficiently equal (in case prioritization is still necessary for practical reasons) chance to be learned (used and getting accustomed to).
- first commonness with multimodal parallel natural language learning: the more mathematically and especially "numerically" intelligent (not necessarily the same) a learner is, the more diverse the numeral systems that the learner encounters should be. even and especially when/if only one numeral system is used respectively at one particular place/moment, vary the numeral system as much as possible sequentially.
- second commonness (common/shared property?) with multimodal parallel natural language learning: maximize or rather optimize the number of simultaneously presented numer of numeral sysetms between the extreme goal of an infinite amound of numeral systems on the one hand and the practicability of tuition respectively learning environments design on the other hand.

- (be aware that) the importance of numeral systems decreases the more abstract and pure ("higher") the mathematical level becomes. (the issue of) numeral systems diversity primarily matters for engineering learning environments and not for mathematics learning environments.

## Copyright

Copyright by Lukas Girtanner, 2005-2012.