Still think life would be simpler if we all just adopted .beats (Swatch Internet Time): https://en.m.wikipedia.org/wiki/Swatch_Internet_Time
Then it just kinda fizzled out. I'm still upset it didn't take off.
We would all develop an intuition for what different times in UTC 'mean' (first thing in morning, after lunch, mid afternoon, etc) relative to the one we're in so we're not looking it up all the time. Maybe keep a UTC clock showing on relevant devices.
I don't think there's a need for a completely new system.
Same as the world adopting the metric system, and in science specifically SI-units. Surely you agree even though that was a bit of a hassle when countries changed over from their old unique units at the time, in the long run it was the right decision. Imagine what it would be like working in an international team always having to convert units of measurement. It's still an issue in engineering sometimes, because outside of science the US didn't make the switch and it's extremely annoying.
Unless you're a savant who can remember all regions with corresponding time zones and all the exceptions, you'd be looking this up, which is what people normally do.
It was unlikely that a rebrand by a greedy Swiss corporation would make it more popular.
Plus, current time keeping already use superior base systems. 12 and 60 are both highly composite numbers. Using a decimal system would be a step back. I wish we had symbols for 10 and 11 however.
The fact they did not is proof enough for me that using the same base than our number system isn't relevant. It makes sense intuitively because it's extremely rare that we have to go above 24 for hours and sixty for minutes while we divide these time periods fairly often.
I would gladly switch to base 12 just to have simpler multiplication tables (0, 1, 2, 3, 4, 6, 8 and 10 are really simple, 9 and A are fine while 5, 7 and B are annoying) but I am no fool and realise it's never going to happen.
My personal observation is that nobody divides hours into more than halves or quarters, and .5, .25, .75 are pretty simple.
Consider this: 1 USD is 100 cents, and people are just fine with its common fractions.
People do divide hours in twelfth, sixth and third all the time. Thinking in blocks of 5 and 10 minutes is very common.
Using base 12 or 60 you are in the same situation.
If you want elegance, you should cut down on the number of special cases.
Hence I would suggest to use a base like 11 or 59. That way more fractions will be covered by the general case.
Base 11 also works well for counting on your fingers. (Though base 6 works for that one as well.)
That's what highly composite numbers are: the set of natural numbers having more divisors than all smaller ones. 12 is the smallest number having 6 divisors and 60 the smallest one having 12.
Both 11 and 59 would be inconvenient bases. They are primes.
Prime numbers have no non-trivial divisors. So more fractions get handled by the general case, the case of infinitely repeated digits.
> 12 and 60 are both optimal cases for what you are describing.
Optimal in what sense? They are certainly more convenient. I was describing mathematical elegance.
I don’t understand what you mean by special cases and general cases.
R\Q is dense in R so most numbers can’t be written as repeating digits at all whatever the base you are using and in every bases all numbers which have a finite representation also have a repeating one by subtracting 1 to the last digit and adding an infinity of the biggest available digit after (decimal representations are not unique).
Generally you want more divisors for your base. The more divisors you have the nicer the fractional part is likely to look like for common operations.
(I'm not sure why anyone would want to bring up R? Why R and not eg p-adic numbers? Or algebraic numbers?)
General case: your fraction p/q will be expressed as some repeating decimal fraction, like 22/7 = 3.|142857 (where | marks the start of the repeat.)
Special case: your fraction p/q will be expressed by a decimal fraction that stops. Like 2/5 = 0.4 or so.
You can't get rid of the general case. No matter what base you pick, the representation of the vast majority of your fractions will have infinite repeats.
You can however minimize the occurrences of the special case by picking a base with fewer divisors.
In contrast, picking a base with more divisors, like 12 or 60 gives you more special cases to worry about.
(And yes, you can apply the trick you suggested, to sort of remove the special case by writing 0.4 as 0.3|9. Or for that matter as 0.4|0; but I'm not sure that last one buys you anything.)
Just to be clear: this is all a bit tongue in cheek and just talking about what system is mathematically more elegant, by having fewer special cases. In practice, having more divisors is convenient; though in practice just rounding after some number of digits is also fine.
Last I checked, most of us have only 10 digits
Base six is probably most practical for this: You can count from zero to five on your left hand. When you go to six, you close your left hand (go to zero) and increment the count on your right hand by one. Two hands let you count from zero to thirty-five (inclusive).
That can be interpreted mathematically or practically.
ALL decimal, base 12 and base 60 expansions of rational numbers end in a repeating sequence. Just like base 11 and 59.
The only difference between 1/7 and 1/5 in decimal is that for the 1/7, the repeating sequence is 7 digits long (142857) and for 1/5 the repeating sequence is 1 digit long (0). In base 12, 1/5 has a 4 digit repeating sequence (2497).
Here's my small contribution to furthering the effort: https://beats.wiki
(1 s = 0.011574 .beats)
Plus acceleration is metres per second per second and that isn't unitless.
One solution though would be to make use of one of the freed up words and use minute instead of beat. So we could say minutes per beat. Does that deal with your concern?