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I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or grade-school level.

I know that different numerical bases (i.e. decimal/base-ten, senary/base-six, ternary/base-three, dozenal/base-twelve) have different patterns and quirks and tricks. Many historic cultures used bases other than decimal (some have even hung around to modern times, like how we divide days into 24 hours and hours into 60 minutes), and most of them did quite well for their time.

There is a similar question on this site, What could be better than base 10?, but the question and its answers do not address my main question: ease of use for humans just starting to learn basic mathematics, while still remaining reasonably efficient for advanced mathematics.

Note: I'm not trying to suggest the world change to something other than the decimal system, or start teaching different bases to elementary schoolers. I'm just curious as to how other systems compare if we imagine parallel universes where each base has the same global presence, inertia, and educational/social infrastructure that is currently enjoyed by base-ten in our own universe.

Primary Considerations

  • Ease of mental arithmetic (addition, subtraction, multiplication, division)
    • In particular, prevalence of shortcuts/patterns that can be used to simplify mental calculation
    • Multiplication tables are easy to learn, either because they're small or because they have intuitive patterns
  • Compactness, in two contradicting categories that need a compromise:
    • Numbers don't get long too quickly, to save time and space when writing
    • Doesn't use too many symbols, to simplify learning
    • Examples of poor compromising: Numbers stay really short in base-one-hundred-and-twenty, but it uses a ton of symbols. Base-two only uses two symbols, but numbers get really long really fast.

Bonus Points

  • The most common/basic fractions terminate (1/2, 1/3, 1/4)
  • Interesting mathematical properties beyond simple arithmetic
  • Many factors, like how dozenal divides evenly into halves, thirds, quarters, and sixths
  • Simple conversion to/from binary, for binary computers
  • Simple conversion to/from balanced ternary, for balance-scale math (or balanced ternary computers)

Note: Cross-posted to Mathematics Stack Exchange as suggested by @JohnOmielan.

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  • $\begingroup$ Wellcome to Matheducators.SE! I think that having ten fingers, five in each hand, makes it almost natural for youngsters to use decimal or pentadic systems for their calculations. $\endgroup$ – Βασίλης Μάρκος Jan 9 at 15:22
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    $\begingroup$ It seems to me that your question is a duplicate of this one at Mathematics Stack Exchange: What could be better than base 10? $\endgroup$ – Joel Reyes Noche Jan 9 at 15:22
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    $\begingroup$ This is a really interesting questions, and one that I have a whole soapbox on (bases should be prime or maximally compost!), but I wonder if this is right forum for the question. At the end of the day, this isn't really an education question. It does make me wonder if there should be a place for sociology of math style questions that don't really fit on the current three stack exchanges. $\endgroup$ – Nate Bade Jan 9 at 16:45
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    $\begingroup$ @JoelReyesNoche I did see that question, and a few others that are also similar, but none that I've seen have looked at it from the perspective of "easy to teach people and convenient to use". $\endgroup$ – Lawton Jan 10 at 0:26
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    $\begingroup$ After the French Revolution, a committee was charged with making recommendations about measurement standards. Some supported base-twelve, others base-ten. Lagrange, who was on the committee, suggested in jest that base-eleven was a good compromise. $\endgroup$ – user52817 Jan 12 at 1:16
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Clearly there is no historical data that addresses this question

I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or grade-school level

since we have ten fingers and humans have learned only decimal arithmetic for everyday use.

I just finished four weekly sessions with fifth graders, learning arithmetic on Siff (the planet of the Six-fingered-folk) where, of course, numbers are written in (our) base 12. They invented new symbols and names for 10 and 11 and new names for 12, 144 and 1728 (10, 100 and 1000 on Siff). The game we played was that they were to learn the arithmetic operations from scratch, as if they were Sifflings, not convert back and forth to decimal.

The material progressed from counting through addition and subtraction, multiplication and fractions, decimals and percentages, all in a new language, roughly covering the work of grades 1-5.

We rediscovered is that arithmetic is hard. It takes a lot of practice to develop what the elementary school curriculum calls "number sense".

Finally, in answer to (part of) your question. I think that everyday arithmetic would be a little bit easier in base 12 than in our base 10.

You can play here: https://www.cs.umb.edu/~eb/heath.pdf , http://www.dozenal.org/

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    $\begingroup$ "Clearly there is no historical data that addresses this question" why is that so clear? $\endgroup$ – Michael Bächtold Jan 9 at 18:57
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    $\begingroup$ @MichaelBächtold I would be surprised and delighted to hear of a place or culture with a recorded history where grade school arithmetic was taught in other than base 10, so we could compare with current practice. I don't think we know enough about commerce in Babylon for that comparison. $\endgroup$ – Ethan Bolker Jan 9 at 21:35
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    $\begingroup$ I've definitely thought a lot about base-twelve, but I didn't want to just jump on it as the better option without knowing more about other bases. There's lots of information on base-twelve thanks to the various Dozenal Societies, but not nearly as much about the others. $\endgroup$ – Lawton Jan 10 at 0:34
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    $\begingroup$ @EthanBolker: Base 60 was used by the Babylonians. This use persists in the use of 360 degrees, 60 seconds, 60 minutes. See books of Neugebauer for details. $\endgroup$ – Dan Fox Jan 10 at 8:42
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I am thinking that, if you had someone who was an expert in this field, they would observe that the immediate concept of counting does not logically imply running out of numerals and having to invent the idea of “ten”. 

They would then observe that different values for “ten” result in different artefacts under processes such as addition and multiplication… and choosing one would be about choosing which type of artefact one wanted. For instance, a prime number such as 7 or 29 would have features that are over the author’s head, such as might interest a cryptographer. Conversely, a number with several prime factors, such as [our] 30 (=2*3*5) or 12 (=2*2*3) or 6 (=2*3) yields interesting patterns when multiplying and dividing (for instance). (There is some discussion of these features in the page linked in the question.)  They would also observe that base 2 is interesting on account of having only one numeral other than 0.

I think this person would suggest that what the OP is really thinking is that it would be interesting to teach young students about the artefacts that arise differentially across different bases.

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  • $\begingroup$ Wouldn't a highly composite base make large prime numbers more visible than if a prime base were used? Since a prime base would make multiples of only that prime obvious, making it harder to find the primes between those multiples, whereas a highly composite base would make multiples of all the factors of that base obvious. $\endgroup$ – awe lotta May 5 at 22:57
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The concept of zero and ten was a huge advancement! Roman numerals did not run out of numerals but it was nearly impossible to compute something like MXVII divided by IXXI. They had to use precomputed books of multiplication and division tables to look up what today is trivial in an exponential based number system.

Teaching another number base is an opportunity to teach the eponential principles involved instead of the mechanics that, let's be honest, most people never move beyond.

At the elementary level, base 3 & 6 make for convenient comparisons of repeating base 10 decimals like 1/3. The classic 1 = 0.999... is instantly resolved in base 3 or 6.

Quickly in your head, compute:

Base 7 (4356.5512) divided by Base 10 (49) and provide the answer in Base 7.

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