# Duodecimal by Stealth

It is widely recognised that the Duodecimal number system is superior to the decimal system. However, it is plainly obvious that trying to introduce such a system would be difficult, especially in a classroom.

However, could some advantages of such a system be gained by using 12 (and its multiples and divisors) more in the classroom, while still using the decimal system for writing down numbers.

Examples I could think of are:

• I High school: Graph paper with major lines at every 12th minor line, rather than the usual 10th line
• In Primary school: "Coins" of denominations 1,2,3,4,6,12

What are the pros and cons of such an initiative?

• Not to be snarky, but I feel like you would get more traction out of doing sexagesimal arithmetic with actual Babylonian examples. I can't think of any reason to try this in an actual classroom (as opposed to a college math class about some related topic, or a supplemental high school course). If you really had to do this, I would recommend using historical stuff like old-style British shillings or the notion of one gross. Dec 13, 2018 at 2:44
• Alternately, just introduce a number theory side course and say you're going to use clocks to indicate each "digit" of a number - that might be fun for some students, and perhaps prepare them for hex later on (though hex is not really any harder). But I don't think that is what you mean. Dec 13, 2018 at 2:46
• Historical note: the coins we actually had (up to a shilling) were ½, 1, 3, 6 and 12 (old) pence. After that it was 2 shillings then half-a-crown (2 shillings and sixpence: ⅛ of £1.) But I think learning to use coins of the wrong denominations compared to what they'll actually buy things with is a horrible idea, at least before they're totally fluent in real money.. Dec 29, 2018 at 1:49

A. Cons:

1. Distraction from normal topics (which many kids need to work on, are not meeting state standards). Just in that it is "extra material".

2. Potentially confusing for kids struggling to master intricacies of the normal decimal system. (after the decimal point, scientific notation, repeating decimals, etc.)

B. Pros:

1. Tees up the idea of alternate bases (this has been a GT "new math" topic for decades). However, I think it is better to use base 2 or 16.

2. I guess some forms of mental arithmetic might be easier (but this only if you do serious mastering of the new base so that it is automatic...which means a lot of time (Con A.1) as well as serious confusion unless they really master "two math languages" (and many kids struggle to master one), thus serious issues with Con A.2.

C. Analysis:

1. I think in effect it would just be a short special topic. So you would have a mild amount of Con A.1 but not Con A.2. Thus you don't get Pro B.2 either. You do cover Pro B.1 but at the expense of a more difficult to learn (versus 2) and less useful base than 2 or 16 (either).

2. (As with many questions on this site) the intro statement itself is in debate (that 12 is better). I have gotten by fine without 12. And I don't feel like debating/researching the topic (would not be appropriate for this board anyway since it is a math topic, not math ed). So bottom line, I don't "know" that the statement of it being better is correct (as I know 1 plus 1 equals 2 or even like smoking makes me at more risk for cancer).

3. The idea of trying to stealth change the world like this is sort of crankish. Worry about teaching the kids and getting them to learn fractions and stuff that they have a hard time on. Not stealth changing the world (since you never will anyway.)

• @DaveLRenfro: What does that have to do with the question? Dec 13, 2018 at 16:07
• Well, to be fair, sometimes there is a real reason to change such mathematical conventions (like using base 10 instead of base 60). Witness the eventual amazing success of so-called Hindu-Arabic notation in Europe over the course of several centuries. I don't know if Leonardo of Pisa could be called "stealth" but anyway it took a while and people had "sides". Woodcut that is kind of a troll of this Dec 13, 2018 at 16:08
• @Daniel Hast: This was intended to be a very tangential comment on the answer, and hence likely even more unrelated to the question. I thought it was a bit oxymoronic to see the terms "crank" and "stealth" associated like this, and to help make my point (in case anyone didn't get it), I gave a well known example of what is more typical of crankish behavior, namely behavior that would not be considered stealth by any stretch. Dec 13, 2018 at 21:03