4 ms·

Please give me an example of how base 12 arithmetic is "much friendlier" than base 10. I've just skimmed the PDF... it looks interesting but a bit cranky. I'll

by chm 11y ago

Please give me an example of how base 12 arithmetic is "much friendlier" than base 10. I've just skimmed the PDF... it looks interesting but a bit cranky. I'll edit when I have a better idea. Table 9 is just... ridiculous to me.

ubernostrum 11y ago
Outside of scientific and engineering fields, most everyday interactions people have with their units of measure do not involve translating between orders of magnitude, and yet the metric system optimizes for that to the exclusion of all other conveniences, through its focus on base 10 and "it's so easy, just move the decimal point!" Meanwhile, in most other areas of life, dividing into halves, thirds or quarters is overwhelmingly more common than dividing into tenths, hundredths or other powers of ten. If you wonder why the "unintuitive" imperial units have such staying power, consider that 12 and 16 are more typical bases for them, and 12 is essentially ideal (evenly divisible by 2, 3, 4 and 6) while 16 is better than 10 (evenly divisible by 2, 4 and 8).
- dozzie 11y ago
  Order of magnitude is defined as power of 10 precisely because we work with base 10 system. The same would stand for base 12, only the order of magnitude would be defined as a power of 12.
- Oletros 11y ago
  > If you wonder why the "unintuitive" imperial units have such staying power, Or perhaps because the SI are not thought since the kindergarten
  - avmich 11y ago
    Yes, to me, as a SI person, argument for 12-based system looks ridiculous :) . Starting with "we don't have 12 digits" and then just saying there's no problem with division by 3 - you sure can handle the .3333... infinite tail in your head; by 4 - those .25 with properly adjusted comma - by 5, 6, 8, 9... It's all just a matter of habit.
- Symbiote 11y ago
  Nothing stops you taking the advantage of both. Kitchen cupboards, appliances etc are designed in multiples of 300mm, which has factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300. The smallest part is still in mm, not awkward fractions of an inch, and the whole wall can be put in mm: 3880mm. That's easy to understand, but 152 inches is less so (who uses inches for something that long?). 12ft 9in is presumably intuitive for those familiar with imperial, but annoying to convert as soon as it needs dividing or multiplying. Also, imperial units didn't have much staying power in the ex-British Empire, apart from in Britain.
  - ubernostrum 11y ago
    Nothing stops you taking the advantage of both. I think the main question is why we feel a need to impose a single one-size-fits-all universal unit set onto disparate domains with different ideas of what properties make a unit desirable. There are domains where base-10 units are a good fit, and other domains where base-12 units are a good fit, and I don't see why we need to try to force one set of units to serve both.
    - Symbiote 11y ago
      Because mistakes happen when people use systems they aren't familiar with, and the mistakes can be expensive, dangerous, and sometimes even fatal. Here's one example: https://psnet.ahrq.gov/webmm/case/293 https://psnet.ahrq.gov/webmm/case/293 If the nurse had been used to working in kilograms, she would have realised 25kg was a huge sack of coal / much bigger child / whatever. (She would have known that her own weight as a child was 40kg or whatever.) Or, if the country had been properly metric, the scale wouldn't have had the option of pounds. (Hopefully, you see no reason to divide a child into 3, 4 or 6 equally massed parts.)
      - ubernostrum 11y ago
        So... the metric system magically prevents data-entry errors? Or somehow you can't believe that there actually are common everyday tasks involving splitting something into halves, thirds, quarters, etc. and so feel a need to engage with a hyperbolic child-dissecting strawman instead? You're really not doing a great job of presenting your case here.
ciconia 11y ago
"Every third number is a multiple of three; yet no power of ten, no matter how high one multiplies, is ever also a multiple of three. Similarly, every fourth number is a multiple of four; but one must get to 102 before the fourth becomes an even divisor. Every power of ten also misses six as an even factor, and one must wait until 103 (1000) before eight divides in evenly. All told, ten simply skips many of the most important fractions, and those it does catch it generally catches imperfectly. Ten is certainly a poor base from this perspective." and "This shows that all bases have irregular wholenumber fractions; for example, the reciprocal of the base less one is always 0.1. However, it also shows that one base stands out in the regularity of the most common and important fractions (1/2, 1/3, 1/4, 1/6, 1/8): that is the base of twelve."
jacobolus 11y ago
This PDF is just an example, showing some of the simplification you could get with a base 12 metric system. I’m not advocating this particular design. In general I think the “dozenal society” people need better numeral glyphs and better names for things. When reading their materials, try to look past the choices of names. As for the advantage of base 12, try writing out multiplication and addition tables for both base 12 and base 10. Here are those multiplication tables using modular arithmetic: multiplication table mod 12 multiplication table mod 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 2 4 6 -4 -2 0 2 4 6 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 3 6 -3 0 3 6 -3 0 3 6 -3 0 3 -4 -1 2 5 -2 1 4 -3 0 4 -4 0 4 -4 0 4 -4 0 4 -4 0 4 -2 2 -4 0 4 -2 2 -4 0 5 -2 3 -4 1 6 -1 4 -3 2 -5 0 5 0 5 0 5 0 5 0 5 0 6 0 6 0 6 0 6 0 6 0 6 0 -4 2 -2 4 0 -4 2 -2 4 0 -5 2 -3 4 -1 6 1 -4 3 -2 5 0 -3 4 1 -2 5 2 -1 -4 3 0 -4 4 0 -4 4 0 -4 4 0 -4 4 0 -2 -4 4 2 0 -2 -4 4 2 0 -3 6 3 0 -3 6 3 0 -3 6 3 0 -1 -2 -3 -4 5 4 3 2 1 0 -2 -4 6 4 2 0 -2 -4 6 4 2 0 -1 -2 -3 -4 -5 6 5 4 3 2 1 Or try converting a list of common fractions to decimal / duodecimal notation.
- chm 11y ago
  Here's a table showing the fraction i/n in decimal (white columns) and in dozenal (blue columns) for i=1,12 and n=1,12.[1] There's absolutely no way I can get most people I know to mix the alphabet and numerals. I'll definitely look more into it, but for the moment I doubt it's immense usefulness. And honestly, I think the naming of units is very important. Numeral prefixes are important and I think the metric prefixes are quite easy to work with (maybe that's because I grew up with them). Gravyard is a bad choice I think! [1]: http://i.imgur.com/LNhsEOe.png http://i.imgur.com/LNhsEOe.png

by chm 11y ago

ubernostrum 11y ago

Outside of scientific and engineering fields, most everyday interactions people have with their units of measure do not involve translating between orders of magnitude, and yet the metric system optimizes for that to the exclusion of all other conveniences, through its focus on base 10 and "it's so easy, just move the decimal point!" Meanwhile, in most other areas of life, dividing into halves, thirds or quarters is overwhelmingly more common than dividing into tenths, hundredths or other powers of ten. If you wonder why the "unintuitive" imperial units have such staying power, consider that 12 and 16 are more typical bases for them, and 12 is essentially ideal (evenly divisible by 2, 3, 4 and 6) while 16 is better than 10 (evenly divisible by 2, 4 and 8).

dozzie 11y ago
Order of magnitude is defined as power of 10 precisely because we work with base 10 system. The same would stand for base 12, only the order of magnitude would be defined as a power of 12.
Oletros 11y ago
> If you wonder why the "unintuitive" imperial units have such staying power, Or perhaps because the SI are not thought since the kindergarten
- avmich 11y ago
  Yes, to me, as a SI person, argument for 12-based system looks ridiculous :) . Starting with "we don't have 12 digits" and then just saying there's no problem with division by 3 - you sure can handle the .3333... infinite tail in your head; by 4 - those .25 with properly adjusted comma - by 5, 6, 8, 9... It's all just a matter of habit.
Symbiote 11y ago
Nothing stops you taking the advantage of both. Kitchen cupboards, appliances etc are designed in multiples of 300mm, which has factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300. The smallest part is still in mm, not awkward fractions of an inch, and the whole wall can be put in mm: 3880mm. That's easy to understand, but 152 inches is less so (who uses inches for something that long?). 12ft 9in is presumably intuitive for those familiar with imperial, but annoying to convert as soon as it needs dividing or multiplying. Also, imperial units didn't have much staying power in the ex-British Empire, apart from in Britain.
- ubernostrum 11y ago
  Nothing stops you taking the advantage of both. I think the main question is why we feel a need to impose a single one-size-fits-all universal unit set onto disparate domains with different ideas of what properties make a unit desirable. There are domains where base-10 units are a good fit, and other domains where base-12 units are a good fit, and I don't see why we need to try to force one set of units to serve both.
  - Symbiote 11y ago
    Because mistakes happen when people use systems they aren't familiar with, and the mistakes can be expensive, dangerous, and sometimes even fatal. Here's one example: https://psnet.ahrq.gov/webmm/case/293 https://psnet.ahrq.gov/webmm/case/293 If the nurse had been used to working in kilograms, she would have realised 25kg was a huge sack of coal / much bigger child / whatever. (She would have known that her own weight as a child was 40kg or whatever.) Or, if the country had been properly metric, the scale wouldn't have had the option of pounds. (Hopefully, you see no reason to divide a child into 3, 4 or 6 equally massed parts.)
    - ubernostrum 11y ago
      So... the metric system magically prevents data-entry errors? Or somehow you can't believe that there actually are common everyday tasks involving splitting something into halves, thirds, quarters, etc. and so feel a need to engage with a hyperbolic child-dissecting strawman instead? You're really not doing a great job of presenting your case here.

ciconia 11y ago

"Every third number is a multiple of three; yet no power of ten, no matter how high one multiplies, is ever also a multiple of three. Similarly, every fourth number is a multiple of four; but one must get to 102 before the fourth becomes an even divisor. Every power of ten also misses six as an even factor, and one must wait until 103 (1000) before eight divides in evenly. All told, ten simply skips many of the most important fractions, and those it does catch it generally catches imperfectly. Ten is certainly a poor base from this perspective." and "This shows that all bases have irregular wholenumber fractions; for example, the reciprocal of the base less one is always 0.1. However, it also shows that one base stands out in the regularity of the most common and important fractions (1/2, 1/3, 1/4, 1/6, 1/8): that is the base of twelve."

jacobolus 11y ago

This PDF is just an example, showing some of the simplification you could get with a base 12 metric system. I’m not advocating this particular design. In general I think the “dozenal society” people need better numeral glyphs and better names for things. When reading their materials, try to look past the choices of names. As for the advantage of base 12, try writing out multiplication and addition tables for both base 12 and base 10. Here are those multiplication tables using modular arithmetic: multiplication table mod 12 multiplication table mod 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 2 4 6 -4 -2 0 2 4 6 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 3 6 -3 0 3 6 -3 0 3 6 -3 0 3 -4 -1 2 5 -2 1 4 -3 0 4 -4 0 4 -4 0 4 -4 0 4 -4 0 4 -2 2 -4 0 4 -2 2 -4 0 5 -2 3 -4 1 6 -1 4 -3 2 -5 0 5 0 5 0 5 0 5 0 5 0 6 0 6 0 6 0 6 0 6 0 6 0 -4 2 -2 4 0 -4 2 -2 4 0 -5 2 -3 4 -1 6 1 -4 3 -2 5 0 -3 4 1 -2 5 2 -1 -4 3 0 -4 4 0 -4 4 0 -4 4 0 -4 4 0 -2 -4 4 2 0 -2 -4 4 2 0 -3 6 3 0 -3 6 3 0 -3 6 3 0 -1 -2 -3 -4 5 4 3 2 1 0 -2 -4 6 4 2 0 -2 -4 6 4 2 0 -1 -2 -3 -4 -5 6 5 4 3 2 1 Or try converting a list of common fractions to decimal / duodecimal notation.

chm 11y ago
Here's a table showing the fraction i/n in decimal (white columns) and in dozenal (blue columns) for i=1,12 and n=1,12.[1] There's absolutely no way I can get most people I know to mix the alphabet and numerals. I'll definitely look more into it, but for the moment I doubt it's immense usefulness. And honestly, I think the naming of units is very important. Numeral prefixes are important and I think the metric prefixes are quite easy to work with (maybe that's because I grew up with them). Gravyard is a bad choice I think! [1]: http://i.imgur.com/LNhsEOe.png http://i.imgur.com/LNhsEOe.png