Irrational

My only guess as to what this could mean is that since quantum mechanics is quantum, i.e. discrete, the universe therefore cannot be continuous as the reals are. But this is a category error. Just because you could never find an object that is, say, exactly pi meters long, does not mean that the definition of pi is threatened. There's nothing infinite that we can observe, but infinity is still a useful concept. And it works both ways; just because quantum mechanics is our best model of the universe doesn't mean the universe is therefore quantum. 150 years ago everyone believed the universe was like a big clockwork mechanism, perfectly deterministic, because Newtonian physics are deterministic. And who knows, maybe they were right, and we just don't have the framework to understand it so we have a nondeterministic approximation!
- We could make an object that is exactly pi meters long. Make a circle of 1 meter in diameter, and then straighten it out. We would not be able to measure the length more accurately than we can calculate it (that might be the largest understatement ever) but to the tolerance with which we could make a 1 meter diameter circle, you should have the same tolerance to the circumference being pi.
  
  I mean, you only need 39 digits of pi to calculate the circumference of a circle with a diameter the size of the universe to the width of a hydrogen atom. So no matter how detailed you get it's impossible to determine if a circles circumference is anywhere close to exactly pi.
  To ops point, you could set up your thing theoretically and we can math out that it should be pi. But we could not make that object.
  
  No, by our current understanding there is no length smaller than a Planck length, and any distance must therefore be divisible by an integer. That is, the length is made up of discrete quanta. Pi, or any other irrational number, is by definition not divisible by an integer, or it would be a ratio, making it rational. This has nothing to do with the accuracy or precision of our measures.
  
  Two issues:
  Planck length The resolution of the circle would be too low to be precisely pi. (I'm not sure whether everything moves only in Planck lengths or whether we live in a voxel world but either way it's not as precise as however many digits of pi we know.
  Matter waves Assuming that the circle has energy (which it must due to Heisenberg's uncertainty principle(I haven't actually read this but I assume that like a vacuum according to the unruh effect matter can't be devoid of energy)) the particles making up that circle would have a wavelength in which they can interfere with other particles and also have an area in which they might be.

There is no fact about reality that can ever threaten facts about mathematics. Mathematic definitions exist independent of reality.
- In fact, you can build a system of definitions that very clearly doesn't exist in real world, like hyperbolic geometry.
  
  Sometimes we find that obscure pure mathematics does describe reality when no one expected it to. Riemannian geometry is one such example.
- Kurt Gödel:

But irrational numbers aren't the same as imaginary numbers. Also, there are irrational imaginary numbers. And quantum physics loves using imaginary numbers. So that sentence in the image is nonsense, right?
- “Imaginary” was merely poor word choice from long ago.
- The definition of irrational numbers is that they are the real numbers that are not rationel. So we need to look at the definition of real numbers. A real number is a number that can be used to measure a continuous one dimensional quantity.
  Quantum physics says that reality is not continuous. Particles make "discrete" jumps instead of moving continuously. So irrational numbers can't exist.
  
  That is not a definition of the real numbers, quantum physics says no such thing, and even if it did the conclusion is wrong
  
  They don’t make “discrete jumps” as in teleportation. They exist stable in discrete energy levels, but that doesn’t imply things don’t move continuously.

What? You use these words, but I do not think they mean what you think they mean.
Quantization is probably the result of vibrational modes, that doesn't mean irrational numbers don't exist, just that we can't measure an infinitely precise value. Tau and root-two exist, they arise naturally in the most basic geometric shapes.
- This sounds really interesting but I'm afraid it's a bit high level for me. Can you explain how vibrations would cause quantisation? I'd also be happy with a link to the correct Wikipedia article or a paper which explains it. :)
  
  This text book seems to cover the idea. https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/30%3A_Atomic_Physics/30.06%3A_The_Wave_Nature_of_Matter_Causes_Quantization I guess I'm drawing my ideas mainly from the Bohr model.

The premise here is completely wrong.

I know those words, Someone please explain .
- https://www.sciencenews.org/article/quantum-physics-imaginary-numbers-math-reality
  
  Imaginary numbers are a rotational operator.
  You don't need quantum mechanics to observe rotation in the real world.
- Yes please!

https://www.sciencenews.org/article/quantum-physics-imaginary-numbers-math-reality
Nontechnical article explaining the concept.
- But that's imaginary numbers, not irrational numbers.
  The issue between quantum physics and irrational numbers is different than the use of imaginary numbers: Irrational numbers have infinite decimals, while quantum physics is quantized.
- So because quantum mechanics is well modeled by imaginary numbers, the existence of quantum particles threatens the definition of irrational numbers? That doesn't make any sense.
  
  Yes, it does not make any sense. If the link above is what it appears from the summary, some students unknowingly attempted to square the circle.
  https://en.m.wikipedia.org/wiki/Squaring_the_circle