American weights are normally a 45lb barbell, 45lb plates, and then additional smaller plates. The large plates are just called "plates" and counted on one side. 1 plate -> 135 (45*2+45bar), 2 plates -> 225, 3 plates -> 315, 4 plates -> 405. These seemingly weird numbers become "round" when you've been lifting them a while.
What about jumps smaller than that? It gets a little weird. 45 doesn't exactly break up nicely. You might have 35s, definitely 25s, maybe 15s (especially olympic lifters) then 10s, 5s, and 2.5s (called "twos"). "A plate, a twenty-five and a two" is 190. Then you get into "washers". Maybe a 1.25. Or if you have a few pairs of 0.75s and 0.5s you can make any integer.
This is not ideal. You need to have more different types of plates than is most necessary. The math is summing up lots of odd numbers.
Enter dozenal plates. I'll put a d at the front for dozenal measurements. A dozenal plate would be d40, or 48lbs. Pretty darn close to the 45s. The bar is also d40. So a bar with a plate on each side is... d100. Well that's nice. 2 plates -> d180, 3 plates -> d240, 4 plates -> d300. And the smaller plates: d20, d10, d6, d3. If those jumps of 6 total pounds are too big, you could add d1.6, and d0.9 washers. All perfectly split.
I find it easier to lift weights in metric. It helps that I'm EU based, so my plates are in KG.
One plate is 20kg, and the bar is also 20kg
Jumps smaller than that,
10kg, 5kg, 2.5kg, 1.25 kg. I've never needed to get down to washers, 1.25 kg jump is small enough to make even with 'small muscle' exercises (wrist curls)
Powers of two all the way down, sums are easy.
The trick is to do sets of 12. Sets of 10 are too easy. So the dozenal still comes in :)
Also, with d(6/3/1.6/0.9), you're still summing up odd numbers.
You have one more plate in dozenal before you have to get fractional. And 6 and 3 aren’t weird numbers in dozenal. They’re like a half and quarter, respectively. 6+6=10.
11 is awkward, but using one hand for the ones place and one hand for the sixes place works quite well.
Start with the pinky, and 4 becomes an interesting number
But clearly you are right, the pinky is by nature the smallest digit. :)
<Hand raised with 9> Rock on!
I was told it was used to make 3/4/5 right triangles, which was kinda fascinating.
I'm having trouble explaining more succinctly why this seems like a totally pointless feature of a woodworking square - so it's possible I've misunderstood how it could be used to extend beyond the length of its arms.
>The clerks then described a typical problem, getting the cubic content of a package measuring 2'6" x 3'6" x 4'2"
Duodecimals (or any positional system) are a bad representation of these numbers. The author proceeds with a very tedious computation of 2,6 x 3,6 in base 12.
Instead, you can do it all in your head:
2'6" x 3'6" x 4'2"
= (5/2)" x (7/2) " x (4 1/6)"
= 35 x (1 1/24) ft^3
= 35 + 35/24 ft^3
= 36 11/24 ft^3
There is more to say for the "duodecimal" musical notation, where each note lives on its own line (abolishing the sharps and flats).
This notation is called the piano roll, as it was already in use in mechanical player pianos. It's not great for writing with a pencil, but excellent for screens and editing - which is why most producers would go straight to the piano roll.
It's not, however, connected to base-12 in any way.
You are increasing the length of one of the sides by 33%, but decreasing by 25% to get to roughly-a-cube. The product of ratios is 3/4 * 5/4 = 15/16, or about 6% underestimate.
Of course, this is already pretty close, and the third side being 107 and not 100 corrects for that.
In the end, that's because (1-x)(1+x) = 1-x^2, so if if the amount we're doing this with is under 20%, our error is under %4 (i.e. pretty good for rough estimates).
And we can count using fingers! Just discard thumbs! They are not standard anyway.
8's prime factors are 2×2×2, so the only nice patterns it can represent involve twos, fours and eights; but we can already do that using binary, so it's rather a waste of factors. Base 8 is useful as a 'compact notation' for binary (hence why it's easy to convert); along with base 16, base 24, base 32, base 64, etc. Even then, I prefer hexadecimal over octal.
If we change one of octal's factors to a 3 we get base 2×2×3 = 12, which can represent patterns involving twos, threes, fours, sixes, eights and twelves. That's a lot more 'bang for the buck' than octal.
Dozenal is still 'wasteful' by having two factors of 2. If we get rid of one we get seximal, which can represent patterns involving twos, threes, fours and sixes. In that sense, dozenal almost a 'compact notation' for seximal, but not as easy to convert as octal<->binary; we get easy conversion for base 36, base 216, etc. but they seem unwieldy.
The next meaningful jump up from seximal/dozenal is base 2×3×5 = 30, which also seems unwieldy. Dropping the factor of 3 gives us decimal that we all know and hate ;)
> And we can count using fingers! Just discard thumbs! They are not standard anyway.
We can count dozenal on one hand, using the finger joints. Our non-standard thumbs can be useful for pointing at the joints.
Not really. Multiplication by two is by far the most common thing that happens in life (when cells divide, they divide by 2, not by 3). So easy multiplication by 2 AND 4 is far more important than the occasional by 3, which is no better, really, than in the decimal system.