Systems of coordinates don't have to linear, they don't even have to be monotonic. The underlying object is unchanged by a human making a choice of a set of coordinates in which to describe it, nor by switching to a different system of coordinates.
Your keyboard's "Q" key does not know when you think of it in terms of you-centric Cartesian-ish or spherical-ish coordinates. It doesn't know when someone nearby thinks of your "Q" key in a different set of coordinates, or with a different coordinate-origin, and it doesn't matter to the "Q" key or its behaviour even if those coordinates aren't relatable by e.g. a Lorentz transformation. A physicist might care about that. The engineers who blueprinted the keyboard might care about that too, for electronics-timing reasons, for example. The person nearby you might not.
It is that you are using a real valued coordinate system -- chosen by you, rather than by some law of nature, and certainly not by the "Q" key or any of its molecules or subatomic components -- that lets you encode the coordinates of its depressed position as containing some decimal representation of your login password. You are free to choose coordinates in which that information vanishes, just like you can choose a system of coordinates with the origin on the depressed Q key in which the entire works of Shakespeare can be found in the base-36 expansion of real-valued coordinates at some small spot in your left fovea.
> an irrational number cannot describe a physical object
Use a system of coordinates in which the centre of mass of your pineal gland is at the coordinate origin and the tip of your nose is at (x, \pi, z) at all times.
Simply bisecting a square creates an irrational number. If you refuse that then you must refuse that the sides of a square are integers.
By your logic, a fraction 0.1 1/10 is infinite because it can be expressed as an infinite series of nontrivial powers of pi or even in base 3.
2. This is not my logic nor what I wrote about someone else's argument. I didn't say that all infinite decimals store infinite information. Obviously you can talk about how much information is stored in a given infinite string but I didn't go there and it's not relevant. It doesn't matter whether you think all real numbers store the same amount of information or not.
I described the claim that if you could physically realize an arbitrary real number you could store unlimited information in the tiniest piece of matter. Which goes against the intuitive idea based on experience that more matter is required to store more information.
Position isn't quantized; energy is.
An object having a position described by an irrational number does not store infinite information. The object isn't storing anything at all; it's just sitting at that location.
The person doing the decimal expansion has to store an infinite amount of information. What if the person just uses an infinite-series expression?
Another question - what if I just shift my reference frame to be one irrational unit offset from yours? Does the information encoded by object-positions in my reference frame now change?
https://physics.stackexchange.com/questions/32665/why-positi...
Does a slide rule or an abacus store information, or does it just "sit there"?
Yes, an abacus stores information, but not an infinite quantity; the information is only a few bits, and it's interpreted that way because we take a very low-precision view of the state of the abacus by dividing the rungs in half, mapping each side to 0 or 1. This is the key; the units of position used for measurement.
The object itself can be at any 'true' position (including an irrational position) but no measurement device has enough precision to say for sure; so I think the answer is that objects can take any position, but the fact that measurement-devices are limited makes it a moot point.
And even if we could, is there a law of physics that would contradict? (I'm actually asking, I have no idea)
I thought that because nature is made of quantized things, that the opposite is true - all numbers are really rational, and it's irrational numbers that don't exist except in human imagination.
One way to look at why an irrational number cannot describe a physical object-
Suppose you had an object with a variable position in one dimension that could be described with an irrational number. Then that single object can, in principle, store an infinite amount of information in the decimal expansion of that number simply by positioning it and measuring its position.