I'm a freshman undergraduate so I have no idea if this question is pointless or infantile but, I've had this thought on my mind, what do infinitely repeating decimals or irrational numbers really represent in physical reality and how can they be conceptualized?; one thought I've had concerning this issue or rather another question that follows on this topic is then: what is the smallest amount of 'space' that can exist ignoring the smallest objects that are possible(assuming space exists independently of objects) theoretically let us conceptualize a device that we can measure objects through such a unit(such that the unit is equivalent to space at the smallest that it can be, as close to 0 as possible(0=black hole???)) would such a measurement always lead to infinitely extending decimals or would there be some other result. Perhaps I am missing something and this is very messy, but I would in essence like clarification on the idea if any can be provided, thanks.
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Take the smallest possible unit of space (planck length) and set that equal to 1, whatever you want to call your units. – Señor O Oct 06 '21 at 22:09
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Irrational numbers don't have to represent any real physical objects, and nearly all of them don't. – notovny Oct 06 '21 at 22:54
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The planck length perfectly describes what I was looking for, I was not asking in terms of whether or not there was a direct correlation between irrational numbers and real physical objects but rather trying to make a direct sense of what infinitely extending decimals really mean or rather what they are capable of describing – Lasaro Morell Oct 06 '21 at 22:56
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You might enjoy reading about surreal numbers – Oct 07 '21 at 00:54
2 Answers
what do infinitely repeating decimals or irrational numbers really represent in physical reality and how can they be conceptualized?
A repeating decimal is not particularly meaningful. There is nothing significant about base 10 or the representation of numbers in base 10. There are important distinctions between natural, integer, rational, real, hyperreal, and complex numbers, but rational numbers that are represented by finite or infinitely repeating strings of digits are equivalent.
Now, physically we use numbers to represent measurable quantities. Some quantities can be measured with natural numbers and integers, but other quantities require more refined numbers. However, all such quantities have a finite uncertainty. That means that any such measurement is consistent with an infinite number of rationals, reals, and hyperreals. Thus we are free to use all such number systems essentially interchangeably for representing physical measurements.
Now, physical theories are primarily based on calculus so we tend to use number systems that are naturally adapted to calculus. That is primarily the reals and hyperreals. So we tend to use those number systems most commonly. However, it is good to remember that the concept of a measurement with its associated uncertainty is not directly represented by a single number in any of the number systems.
The reals and hyperreals are convenient for calculus, and calculus seems to be good at representing the laws of nature, but the numbers themselves are not particularly physical since they represent a degree of certainty that we never achieve in physical measurements.
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Repeating decimals are just a consequence of an arbitrarily selected number base and have no significance. A decimal always terminates if its equivalent fraction's denominator is a product of one or more the prime factors of the number base. For instance,
$(20/6)_{\text{ten}} = 3.\bar3_{\text{ten}}= (202/20)_{\text{three}} = 10.1_{\text{three}}$
(Read those base three numbers as: "twice three-squared and two, divided by two threes, equals three and one third")
Irrational numbers have slightly more significance: they're irrational in every number base. However, irrational numbers are ideas, not measurements. For example, $\pi$ is the ratio of circumference to diameter of a circle. The diameter is measurable and the circumference is measurable, but the ratio is just an idea.
So the answer to "what do numbers with never-ending trains of decimals mean physically?" is a succinct "Nothing."
The other half of your question has a more interesting answer: "What is the smallest distance one can measure?"
To measure a distance, we need to interact with each end of the distance separately and compare what happened.
Conventionally, you can't measure a distance using a particle (photon or otherwise) whose compton wavelength is larger than the length you want to measure, because it'll interact with both ends at once. There are some ways around this, but they require starting with assumptions that you didn't get from direct measurement, like "the length is constant with time".
The smaller the Compton wavelength, the higher the energy density. The higher the energy density, the higher the spacetime curvature. Eventually, if you increase particle energy (hence: proportionately decrease its Compton wavelength), you end up with an interaction with so much energy that instead of a reflection telling you where the interaction happened, you just get a black hole where the interaction would have been. This sets an absolute lower limit to distance that can be measured directly: the size an which a particle's Compton wavelength is the same size as the event horizon of a black hole with that particle's energy.
That value is the Planck Length.
