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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 Educators Stack Exchange as suggested by @JohnOmielan.

There are now answers on both sites. (So cross posting wasn't such a good idea after all.) (However, no answers on either site have fully answered the question as of yet.)

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    $\begingroup$ Can you explain why you're rejecting the ubiquitous base ten? $\endgroup$ – Matthew Leingang Dec 31 '19 at 22:33
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    $\begingroup$ If you're just looking for ease-of-use, surely base 10 suffices, since it is the one they are most used to, and whatever speed advantages there could be from using other bases would be discounted by the time it takes to learn. Of course, there are other reasons to learn other bases - to increase understanding, etc. $\endgroup$ – Jair Taylor Dec 31 '19 at 22:34
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    $\begingroup$ See here, here, and here. $\endgroup$ – Simply Beautiful Art Dec 31 '19 at 22:36
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    $\begingroup$ I'd vote for base 6, since it's so nice for counting on fingers. $\endgroup$ – littleO Dec 31 '19 at 22:47
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    $\begingroup$ Cross posting is NOT a good idea. This question now has answers in both places. $\endgroup$ – Ethan Bolker Jan 9 at 18:15
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I've had considerable success with base 120, with nothing more than the 12-times tables. The trick here is to use 'alternating arithmetic', that is, realise that 73 is 7T + 3U, and then provide multipliers for T and U.

The other table one would master is the 'dicker-dozen' table: being to convert instantly, 73 to 6.1 (ie six-dozen-one).

The common algorithms one does on paper, such as long arithmetic, (multiplication, division, square roots, criss-cross multiplication), all translate easily into alternating arithmetic.

The periods of 1/7 and 1/11 are short (ie 0:17.17... and 0:10.V9.10.V9...). The factorials are shorter and have simpler reciprocals, so eg

10! = 2.12.00.00 and 1/10! = 0:00.00.00.57.17.17.17...

I have used this base for nearly forty years with proficiency as good as the decimal algorithms. Sometimes i do the decimal on alternating digits!.

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  • $\begingroup$ Could you clarify this a little? I can't find information on "alternating arithmetic" or the "dicker-dozen table" outside of another thread you posted that talks about using it. That thread doesn't seem to explain the theory behind the system, or what makes it better (subjectively or objectively) than the decimal system, or how you found/developed the system. I'd love to see other resources on the topic! $\endgroup$ – Lawton Jan 1 at 17:44
  • $\begingroup$ It is my invention from 1988. It's all done without words, just process. When I explained it at DozensOnline, there were no words, so i had to invent them. One might note there is very little on calculating in any base, let alone one that changes the nature of the arithmetic. Because a digit no longer corresponds to numbers up to the base, this caused a lot of confusion, and hindered the discovery process. The article at DozensOnline is a direct translation from a Shilling Arithmetic (ie a 'middle-school' maths text), and the algorithm is no worse than what's in there. $\endgroup$ – wendy.krieger Jan 2 at 9:21
  • $\begingroup$ There is very little theory in it. You have two multipliers or two divisors, one 10 times the other. The dicker is a group of ten, the dozen is a group of twelve. The dicker-dozen is converting tens to twelves, eg 73 (7 tens and 3) is 6.1 (six tulfs and 1), so 10*73 = 6.10. So when you see '73' in the units, you write 6.1 in the tens. Apart from keeping parity of the places, it's pretty much long division and long multiplication. $\endgroup$ – wendy.krieger Jan 2 at 9:24
  • $\begingroup$ I serched for 10 or so years for algorithms that one could do with pen and paper. None of the maths books had one. None of the history books had one. The sumerians used multiplication of inverse to do division. The 'times tables' is not well understood why so, until you figure out it's for division. If 7 is the divisor, you run down the seven-times tables to you find one bigger, and step back one. But once you see what;s inside, it's hard to unsee it. $\endgroup$ – wendy.krieger Jan 2 at 9:28
  • $\begingroup$ What makes the dicker-dozen system better than the decimal system? $\endgroup$ – Lawton Jan 3 at 13:58

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