Exactly. The exact amount of sleep needed on average is almost certainly an irrational number. (Because almost every number is 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.
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.