They exist! The crystal structures of molecules are rather Platonic, for example. Nature is elegant when you get very small, requiring fewer and fewer core concepts as you work your way down to more fundamental levels of understanding. (At lower levels those fundamentals might be irrational ones, but still, those few primitives [like spirals in complex space] become the only tools you need.)
The inelegance, then, comes from modelling the interactions of mind-bogglingly huge collections of these fundamental things, at high levels of abstractions, and then expecting your abstraction (which is just that: a formula that allows you to make some useful prediction of these super-high-level interactions) to be as elegant as the fundamental forces operating at the lowest levels.
Say you want to measure the circumference of the visible universe. This is all the space we have, the rest is outside our light cone and we can't interact with it to the point that we can only infer it might exist.
You need about 55 decimals of pi to get this circumference to the accuracy of a Planck length. Or about that many, give or take.
So that's measuring the very largest real thing that could possibly matter, down to the accuracy of the very smallest thing we can conceive of.
So that's it for pi. Any further decimals are strictly theoretical. We can prove they must be these decimals and not others, using math, but it's only theoretical knowledge, these additional decimals serve absolutely zero purpose in nature or reality. You cannot get to them by measuring reality, you can only theorize about what these numbers would be if you could measure to infinite precision, which we can't, because there are limits.
The "Real" numbers is really a misnomer. Even if you don't buy the above accuracy argument, and want to describe nature as something infinite (even though we're strictly limited to interacting with a finite subset of it), then at least agree that it's countable. The real numbers are way too stretchy and insane (see the Banach-Tarski paradox). In nature you can't stretch things infinitely far, nor can you cut up things to arbitrary precision.
And yes occasionally we discover new smaller particles, or sub-particles, but what we don't discover is a continuum. And it would be really weird if we did, because you can do crazy tricks to the Real numbers.
My understanding is that almost all numbers are real, but not rational. They can be expressed as the limit of a sequence that can be made arbitrarily close to the limit. But the number is not the sequence, the number is the limit of the sequence.
[1] = I had to look up what a Planck is. Should have taken more physics along with the math.
Try to cut that square in half diagonally and things get irrational really fast!
FWIW I thought calculus made a lot of sense and helped make the world make more sense. Algebra is a fancy set of rules to manipulate rather abstract symbols, calculus actually explains how real things work!
Take, for instance, the Fundamental Theorem of Algebra, where the proof in terms of elementary complex analysis and/or topology closely related to complex analysis make the truth of the theorem geometrically obvious, the (mostly) algebraic proofs I've seen leave me with little more than the desire to re-check the proof, because I'm not at all certain that something equivalent to the F.T.A. hasn't been implicitly assumed at some point in the proof.
Now it may be "just me" — I've always had an easier time following analytical proofs than abstract algebraic ones — but just thinking of the necessary prerequisites — homotopy between maps defined by complex polynomials vs. what? Galois theory? — I don't think it's just me.
Incidentally, complex numbers are another case where, as with the irrationals, a poorly-chosen name has made simple and quite generally useful ideas seem esoteric to those not already familiar with the subject.
In theory. In reality you have an integer number of atoms in one half and another integer number in the other.
There are no irrational numbers to measure, and squares made from "continuum material" do not exist in reality.
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.
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)
Area of a platonic circle is a real number, area of an actual circle is discrete, but varies depending on how you draw the border.
You can count atoms and get your volume of the tree in integers that way. Bonus points for being temperature and pressure independent (more so than a volume of the spherical tree in vacuum anyway).
I guess they call those numbers irrational because they are never represented physically and thus are a pure figment of imagination. :)
I remember reading (might have been here) that math and art both get difficult at the same moment for the same reason. When you're a kid, you live in the rational world. By this, I mean numbers that can be expressed as the ratio of two integers. Human creations are rational. The volume of a square is a neat, tidy equation. So is the area of a square. You can draw them easily, too, using a ruler and clean nifty lines, and they look great. Squares are all over human creation. You can draw a car with straight lines and squares. Wheels and other things bring in this inconvenient number, pi, that is "irrational", but let's just go with three point blah blah and it'll be fine. At least the curvature is constant.
So, where are the squares in nature? Hell, where are the circles. Where is the constant curvature. How do you draw a leaf, a tree, a face? How do you calculate the surface volume of a leaf, or the volume of a tree?
All of a sudden, you can't measure it with the numbers you know. There is no neat ratio of integers that will calculate the volume of that tree trunk. Or even the volume under an easily expressed mathematical equation on a graph. In fact once you start measuring nature, rather than the things people make, rational numbers aren't anywhere. All of a sudden, you have to deal with limits, sequences, strange numbers that can be made arbitrarily close to zero as other numbers approach infinity. It turns out every number is "irrational", pretty much nothing is rational. So, instead of irrational, let's call it Real.
Where do math and art get hard? When you start to describe things as they are, rather than as we imagine the to be. You know, Real.