You'd think we can just interchange the . and : so you should wake up 12:83…, or 2:83… the next day.
But no, it's apparently 2:75 metric time. Why?
Also nice how they IMMEDIATELY ran into a continuing fraction 1/3, which is exactly the point of the 60-minute hour, 24-hour day, or 360-degree circle: lots of factors.
> Also nice how they IMMEDIATELY ran into a continuing fraction 1/3, which is exactly the point of the 60-minute hour, 24-hour day, or 360-degree circle: lots of factors.
While I agree in principle, this example is not evidence of it. It's not like we measured the amount of sleep needed and it's precisely 8 hours. A more reasonable thing to say would be: You need to sleep for 3.5 metric hours.
It’s almost certainly a rational number as activity in the body is finite. There will be an exact point at which a certain molecule has interacted with another molecule marking the exact point of perfect sleep. Averaging many rational numbers can’t get you an irrational number.
Also it is invalid to compare infinite series like you do in your paracentesis argument, there are infinitely many irrational numbers and infinitely many rational ones. If you do it, you run in to contradictions. (Something which is related to Cantor's paradox)
The statement OP put in parentheses is a well-defined statement in mathematics. It means that the Lebesgue measure of the set of the rational numbers is zero in the space of real numbers.
"If the physical property that time meassuring devices meassure is continuous, it must also contain irrational numbers?" Is that what you are refering to?
Mabye... To me it seams like nothing is truly continuous i nature. But mabye there is such a thing somewere out there somewere.
But irrational numbers require definitions that contain or require recursion. Mabye physical time is built with such a recursive definition?
"There will be an exact point at which a certain molecule has interacted with another molecule" statement is contradictory with the uncertainty associated with time as described by quantum mechanics and molecular interactions due to Brownian motion. Material that is returned when searching for those topics will better answer your further questions about them and those above than I here.
The uncertainty principle is about unavoidable measurement "errors" (measurements that can’t be done). If you are going to measure the time (or anything) you will always get finite values. Finite values are not irrational.
Either one talks about the underlying physics or about the measurements of it. If one talks about the underlying physics all bets are of, it is unmeasurable by definition. Anything is possible bellow the measurement threshold(including irrational numbers).
If one talks about the measurements you will always get finite rational values.
Are you sure? Isn't the average calculated by dividing by n, an integer?
Edit: I mean, you're right that nearly every number is irrational. But I think averages are going to be some of the tiny fraction of numbers that aren't.
Base 12 being 3-smooth, any number with 2 or 3 as a factor has a reciprocal with a terminating expansion.
base 60 is 5-smooth, so any number with 2,3,5 as a factor has a terminating expansion.
Just as SI has to add in deg min sec for fields like astronomy.
As the practical numbers are quite dense up to 60, they could divide by multiplication of the reciprocal for many more numbers.
Floating point is more complicated than just the base as the radix also matters. C(++) finally got decimal radix support this year in the standards and IBM has had decimal floats for a long time.
FFT and encryption often use mixed radix despite being a binary base too.
It is a far more complicated subject than it appears on the surface.
But there are problems that aren't easily solvable in base 10. The degrees of a circle are an example and why navigation uses the nautical mile, where 1 nautical mile= 1 minute of latitude is an example.
12,60,360 are Superior highly composite numbers and 12 is the smallest 3-smooth and 60 is the smallest 5-smooth.
This also means that with using 360 degrees one can divide a circle or semicircle in 12 sections with just a square, 345 triangle and equalatrral triangle. Where decimal or even radians requires the square root of 2, pi, etc...
I am a fan of universal units of measurement, but had they been base 12 it would have been better IMHO. SI could be more broadly adopted if it has been base 12.
Seximal is actually great for hand counting. You can use fingers the classical way for one digit per hand or base 6² compression for two digits per hand, allowing you to count up to 1296 with two hands.
The whole argument started when somebody above claimed that we probably do not sleep exactly 8 hours, but some weird irrational number. That problem persists even if you measure in radians...
Do the physicist thing and just define all your constants to be one.
"How long is a day? 1 day."
"I slept for .37 days."
This nicely unifies things with the tao manifesto[1] people since 1 day = 1 turn = 1 tau = 2 * pi radians and reinforces the periodic nature of these things.
I don't really care how much sleep I get as long as it's enough. I care greatly when the bus leaves and when my meetings are. Anything that hopes to replace a clock had better be adding precision rather than removing it.
Decimal seconds are slightly shorter than normal seconds, so times that use seconds are more precise. The other fun thing is that decimal time doesn't really need names for the levels, it's just numbers of significant figures so if you only want to give the first digit of the seconds that works too.
It's less precision for a given number of symbols. Fewer usable factors mean you can only reasonably divide up segments of time into larger, less precise chunks.
This is far from the truth. The closest time before midnight you can conveniently represent in the current time system is 23:59:59 which is six symbols for a standard second before midnight.
In decimal time the closest time before midnight you can conveniently represent would be .9:99:99 which is noticeably closer (0.163 of a second closer) to midnight and uses only 5 symbols.
This should not be surprising - our current system is very inefficient in its use of digits.
45 minutes and 00 seconds is very close (less than a quarter of a second difference) to :23:33. But there's no magical or particular reason why 45 minutes is an important time period to make a nice round number. :24 is a very convenient number (so many factors!) that's very close to the same time duration. That's 46.2963 minutes.
Or maybe you are referring to three quarters of the way through the major division. That's actually pretty natural too :75 like a percentage. Now dividing the major division into thirds is a little less convenient but I do that far less than I need to add and subtract times which this system makes much more convenient.
How close do you want to get? For what purpose are you representing 12:15?
.5:10 is 12:14 and 24 seconds. That's pretty close and uses only 2 sig figs - even fewer than the four you needed to represent 12:15.
But also, why would you obsess about these specific times? If we were using the proposed system and someone made the argument to switch, you'd be saying 'How well does your system represent the time .5:1 ? It's only 2 sig figs, but you need to go down to seconds to accurately represent it - 12:14:24.
It can get worse too - .5:01 is 3 significant figures, but to represent that time in the current system requires that you go down to tenths of a second with 7 significant figures - 12:01:26.4. Or if I go down to second equivalents, you sometimes need to go down to milliseconds - .5:09:01 is 12:12:58.464
Now apply that same logic to how we write our numbers. Duodecimal notation makes very clean multiplication tables and makes arithmetic easier to learn.
I'm a twelve maximalist. Let's convert SI to duodecimal prefixes. One kilometer should be 1728 meters (of course that's 1000 in duodecimal). One centimeter should be 1/144 of a meter.
I keep hoping for a chance to welcome some highly composite overlords.
Is 1/3 (1/6) such an important concept? 1/2 (1/4) is natural in metric.
I would think a bigger benefit is that you can drop up and down without any math depending on the precision you need. Or use decimals, no math involved either, since it's the same thing.
Yes, 1/3 and 1/6 are quite important. Do enough making and you will run into those fractions plenty. It just turns out that in practice, as you say, easy unit conversions outweigh the benefit of clean division. Metric's important benefit is that it's a consistent base, not that it's specifically base-10.
And now that we use computers for a lot of measurement, those infinite decimals aren't as big of a deal. (Well, most of the time, anyway. Representation of infinite fractions famously creates some programming problems. But I mean for the user making the measurement.)
Base-10 is easy to learn since it's the same we use for counting, but I think there's an interesting argument to be made that purely from a science/engineering/making perspective it would be better to use a consistent base-12 measurement system.
Now apply the same logic to currency. What is the combined value of thirteen shillings and ten farthings? Simple, it's 317 halfpennies, or £317/480. It's much easier to deal with than a more difficult calculation like $0.25 + $0.37 = $0.62.
I didn't check the details of that metric time system but there would be no problem living a 10 hours day with 100 minutes hours and 100 seconds minutes. The second would be defined with a different number of "cycles of the radiation produced by the transition between two levels of the cesium-133 atom" and that's all. Instead of rounding our lives at "our" quarters of hours or ten minutes, five minutes marks we would round to the metric quarters of hours, maybe eights of hours (we could have found a name for that, like for coins) etc, and nobody would notice because that would be what we are born with.
Similarly, people buy 50 cm x 70 cm frames in metric countries and 20" x 30" frames in the USA. Nobody thinks about that except frame factories in China, that have to cut them in two sizes.
Nobody questions the logical consistency of the system, I think.
Presently, we use 24-hour time (AM/PM or otherwise), so the loss in precision is enormous going from 24 to 10.
Think about it.
A tenth of a standard hour is 6 standard minutes.
A tenth of a metric hour is 144 standard minutes, more than two standard hours.
A hundredth of a standard hour is 36 standard seconds, or half a minute. It is in the context of hours a negligible amount of time.
A hundredth of a metric hour is 14.4 standard minutes! Imagine being 0.01 metric hours late for a meeting.
So everywhere you now have to specify time to two decimal places /at least/, three to have sub-standard-minute precision.
Layer on top of this the fact that the new system has significantly fewer distinct prime factors, so you quickly run into continuing fractions. Disaster.
What? A metric hour is 144 standard minutes - or do you imagine that a day of metric hours is 1441010 = 144,000 standard minutes long?
A tenth of a metric hour is 14.4 standard minutes. A hundredth of a standard hour is 1.44 minutes. Being 0.01 metric hours late, or 1 metric minute late, is not too much more than being 1 standard limit late.
And current time is specified to 6 digits for second precision - hours, minutes, and seconds.
Don't we have the same basic criticism going from Fahrenheit to Celsius? You go from a 10 degree swing being big to ridiculously huge.
That is, arbitrary number is arbitrary, at large. Most of the "benefits" of any system won't actually be realized by most people that are using it. Consistency, on the other hand, is hugely important.
> Don't we have the same basic criticism going from Fahrenheit to Celsius?
Sure, and one shouldn’t do that either. Fahrenheit’s range (0 being the freezing point of brine, almost but not quite intolerable to a human, and 100 being roughly body temperate and also almost but not quite intolerable to a human) is far more human than Celsius’s freezing-to-boiling range.
Having never used Fahrenheit in any capacity, the math for it feels inhuman.
Having grown up in metric, the values dont feel 'inhuman' just a scale that i'm familiar with. I know that my body is usually at 37, I know that water boils at 100, i know that I don't like anything below 8c or above 40c.
>Presently, we use 24-hour time (AM/PM or otherwise), so the loss in precision is enormous going from 24 to 10.
>Think about it.
No, YOU think about it. When do you use hour-only precision today ?
When you say "I have meeting at 8" that never means "any time period between 8 and 9", that means "it starts at eight". That doesn't change with change of the length of the hour.
You'd still use hours on their own only if you mean "an hour and close to zero minutes around that hour". You'd still add minutes for anything else.
> A hundredth of a metric hour is 14.4 standard minutes! Imagine being 0.01 metric hours late for a meeting.
...no ? it's 100 seconds so ~1.66 of standard minute
Imagine being 1.66 minutes late instead of one!!! Such horror!
Instead of changing the definition of second, it might make sense to separate day-time from scientific time. Decimal hours and minutes would be normal time keeping. If needed more accuracy, then would switch use centi-minute for casual use or second for scientific use.
One nice feature is that the day-time would be different on other planets. There would Mars-day and Mars-hour. But the second would be the same.
An hour is just an another name for a deciturn, or a deciday.
Tau is not a good unit for angles in general, unless you have circles, or you need lengths of circumferences somewhere. (You want to paint your clock and calculate the amount of paint, or something.) Turn is the natural angle unit. And since the earth rotates 1 turn/day, the amount of turns is the same as the amount of days.
Metric time is about factors too, but it prioritizes the factors that simplify the comparison across (literally) many orders of magnitude. This is way less helpful for everyday life because we usually only reason within 2-3 orders of magnitude, but we deal with tons of harmonic subcycles that the 24h clock makes easy.
For a fun, a more justified usage of metric time, check out Vernor Vinge's "A Deepness in the Sky" (also IMO one of the best sci-fi novels of all time). A spacefaring humanity that stretches their life across journeys and projects that span centuries, and which has to artificially produce their own daily cycles, gets a lot more value out of metric time.
Came here to mention/contrast Vinge's metric time with with my (quick, approximate) understanding of this approach...
The problem with this approach is it attempts to "redefine" hours and minutes... part of what I liked in Vinge's/Deepness' approach was, it ignored all that and just talked about seconds in standard exponential scientific notation... it makes a lot more sense in space though, where there's no need/logic to connecting or synchronize to a specific solar cycle, so they just think/talk in kilo-seconds (about 40 mins) and mega-seconds (about 11.6 days)... or at least those are the two I recall them using enough that I got semi-used to thinking in them while reading the book... I had to convert, but I didn't have to redefine any existing unit or remember/remind myself which meaning of hour/minute they're using, because nothing changed.
I'm a huge fan of both, but you're not the first person I've heard who had problems with the writing style / dialogue / characterization of Fire. In that respect, I'm not sure Deepness will feel different. Vinge has a pretty consistent style and cultural stance, which IMO is an interesting balance of Heinlein-style techno-libertarianism and genuine "humanism" (which extends to dogs, plants, and other thinking creatures).
Structurally, though, it's pretty different. The characters have much more opportunity to develop, and it's purely hard sci-fi where Fire used its setting to veer into fantasy ideas. So my biased advice remains: give it a shot :)
> You'd think we can just interchange the . and : so you should wake up 12:83…, or 2:83… the next day.
> But no, it's apparently 2:75 metric time. Why?
You can, they just messed up the example by unnecessarily rounding off the times. The night from 9:50 till 2:75 metric time only lasts 3.25 metric hours, or 7h48m in standard time; less than the 8 standard hours they started with.
You know what other measuring system is based around highly composable numbers?
It's not metric. It's the imperial system. Halves, thirds, quarters are all neat numbers. It does get a little weird because we only use feet, yards, and miles today, but there is a progression from inch to mile that all make sense to some degree.
Similarly with volumes. You can get from teaspoon to gallon without having to worry about any weird decimals.
The only thing metric really has going for it is uniformity of conversion.
People shit on it, but the imperial system is not actually that horrible.
I disagree, it’s not equal but different. The French famously tried to switch to a decimal system for angles before, but failed in no small part because of the relatively few unique prime factors. Being able to divide evenly by three turns out to be more important than five.
To quote yet another time format: NTP 64-bit timestamp format (rfc8877), which is 32 bits seconds since epoch + 32 bits fixed point second fractions. (Outside of Network Time Protocol, you'll find this baby for instance in ISOBMFF ProducerReferenceTimeBox(prft)).
Here seconds are just 1/(24*60*60) of a day as expected, but the base 2 fixed point part, where a tick "is roughly equal to 233 picoseconds" makes you want to pull your hairs out if you just want to accurately express milliseconds. (Similarly for other timescales frequently used in media processing, like 90kHz, 25, 60 or 29.97)
The answer to all this is of course: hand waving — "you don't need that". Your time can be perfectly accurate in itself (ie. an accurate discrete sample of continuous time), even if no accurate conversion exists to some other time system.
It's tongue in cheek or wink wink. No one uses it seriously but there is sometimes an undercurrent of resistance to what's seen as an external cultural standard. But you'll never see it used as a kind of rallying cry.
You sleep 3.33… metric hours.
You fall asleep 9:50 metric time.
You'd think we can just interchange the . and : so you should wake up 12:83…, or 2:83… the next day.
But no, it's apparently 2:75 metric time. Why?
Also nice how they IMMEDIATELY ran into a continuing fraction 1/3, which is exactly the point of the 60-minute hour, 24-hour day, or 360-degree circle: lots of factors.