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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.)

Lawton
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    Can you explain why you're rejecting the ubiquitous base ten? – Matthew Leingang Dec 31 '19 at 22:33
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    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. – Jair Taylor Dec 31 '19 at 22:34
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    See here, here, and here. – Simply Beautiful Art Dec 31 '19 at 22:36
  • If you want the parents to be of no use in helping their children learn arithmetic, go with anything other than base ten. – kimchi lover Dec 31 '19 at 22:36
  • @MatthewLeingang Decimal's lack of divisors has always bugged me a bit, especially the way one-third doesn't have a terminating representation. I'm not trying to say that the world should switch to a new system, because as you point out the decimal system has centuries of inertia and we're pretty much stuck with it at this point. – Lawton Dec 31 '19 at 22:37
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    I'd vote for base 6, since it's so nice for counting on fingers. – littleO Dec 31 '19 at 22:47
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    @littleO We use base ten because we have ten fingers. – herb steinberg Dec 31 '19 at 22:49
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    @herbsteinberg But you see what I mean about base 6 right? In base 6 you can easily count to 35 on your fingers. Actually I don't see that having ten fingers makes base 10 a natural choice. – littleO Dec 31 '19 at 22:55
  • @littleO I don't get it. What does base 6 have to do with counting to 35? For base 10, count on your fingers. – herb steinberg Dec 31 '19 at 23:18
  • @herbsteinberg There are finger-counting tricks for many bases. The simplest is the unary counting method that is common today, where the number of fingers you extend is equal to the number you want to represent; this lets you count from zero to ten.

    In base-six, each "digit" has six representations, which can all be represented on one hand (zero fingers, one, two, three, four, five fingers). By using one hand as the ones-place, and the other hand as the sixes-place, you can count up to the senary number 55, which is represented in decimal as 35.

     – Lawton Dec 31 '19 at 23:21
  • @herbsteinberg To count to twelve on one hand, you can use your thumb to point at the bones or joints in the other fingers in that hand; four fingers of three bones/joints means you can count up to twelve on each hand. You could use that for a base-thirteen system with the same kind of two-digit system described above for base-six. You can also use each finger as a binary digit, letting you count up to one-thousand-and-twenty-three (1111111111 in binary). – Lawton Dec 31 '19 at 23:30
  • In base six we use the digits 0 through 5, which are naturally represented on one hand by holding up the corresponding number of fingers. – littleO Dec 31 '19 at 23:30
  • @Lawton FYI, in case you weren't aware, for future questions like this, you should consider if posting them on Mathematics Educators Stack Exchange might be more fitting & possibly get you more and/or generally better responses. Although cross-posting is generally frowned on, if you don't get what you consider to be an adequate answer to this question after waiting for at least a few days, you may wish to consider posting this there. If you do, though, please make sure to have each question link to the other to minimize any duplication of effort. – John Omielan Jan 01 '20 at 01:22
  • @JohnOmielan Thanks! I'll keep that in mind! – Lawton Jan 01 '20 at 17:32
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    Cross posting is NOT a good idea. This question now has answers in both places. – Ethan Bolker Jan 09 '20 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!.

wendy.krieger
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  • 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! – Lawton Jan 01 '20 at 17:44
  • 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.

     – wendy.krieger Jan 02 '20 at 09:21
  • 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. – wendy.krieger Jan 02 '20 at 09:24
  • 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. – wendy.krieger Jan 02 '20 at 09:28
  • What makes the dicker-dozen system better than the decimal system? – Lawton Jan 03 '20 at 13:58
  • It is considerably better at dealing with fractions than the decimal, and powers of 2 as well. The numbers on either side have a lot of divisors as well, and 12-10 is smaller than 10-6, which means that the arithmetic is easier. – wendy.krieger Jan 04 '20 at 08:27
  • I'm afraid I'm still not sure I understand. I think I might need to see a direct comparison of this system with more well-known ones, in long-form writing (and probably with diagrams) instead of in character-limited comments, to see what you're talking about. – Lawton Jan 05 '20 at 23:14