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 Educators Stack Exchange as suggested by @JohnOmielan.

  • $\begingroup$ 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

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
  • $\begingroup$ 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
  • $\begingroup$ @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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