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.
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.
> I'd guess that you're using rational numbers in the actual calculation anyways.
Well, actually one uses floating point arithmetic, which isn't rational numbers either (as shown by the classic example a = 1/3, b = 3*a, then b != a)
The rotation of a hand of the clock, expressed as the arc length of the curve of its endpoint :)
A sleeping time of 0 is always a problem, though.
We sleep for 2/3 pi radians of the day in that case.
"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.
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.
Quartiles and hand counting would be my argument for 12 but am probably biased based on familiarity.
The set of rational number is a “null set”: https://en.wikipedia.org/wiki/Null_set
It only takes a single irrational input to make the average irrational.
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)
You're not being clever by being pedantic.
Try again, sweaty.
Just because the best current theory suggests a smallest observable time span does not mean as a consequence that time is discrete.
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?
> irrational numbers require definitions that contain or require recursion
The computable reals are also known as the recursive reals, but almost every real number is not computable.
Was it just a "out of context comment" on something that poped up in your mind as you read the text?
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.
I have some terrible news for you about statistical mechanics.
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.
Try converting something like 45 minutes into this convenient time system and let us know how efficient it is
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.
.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
"Eight hours for work, eight hours for rest and eight hours for what you will."