I have a new theory. It may be true or false, I'm not sure how to proceed in proving it.
My theory is that all numbers in the physical world are actually transcendental, and that we humans mostly use integers and rational numbers because we're rounding it to terms we understand.
A quick primer, which you can skip if you're a mathematician. Numbers come in various kinds. Natural numbers are the first we humans learned to deal with like 1, 2, 3, and so on as if you were counting apples. Integers are those and also 0 and negative numbers like -3. Rational numbers are all those that can be represented as a fraction of integers, including integers themselves. 6/1, for instance, for 6. This was all discovered by the times of the ancient Greeks. Their next discovery was irrational numbers, which cannot be expressed as a fraction of integers. The square root of 2 being the first example. If represented with our decimal system, the pattern after the decimal point would both go on forever and never repeat at any point.
Transcendental numbers are irrational, and also not any sort of square root. The most commonly referred to ones are pi and Euler's number, which are useful for circular constructs and natural growth modeling respectively. There are uncountably many transcendental numbers, but most have no easy way to reference. Mathematicians can now begin reading again.
In Engineering, there is a concept of precision. All measurements are slightly wrong. This error can be reduced by measuring more carefully, but all measurements are to within some plus-or-minus of the true value. Most serious projects reduce this error to ludicrously small values. Not zero, however, as that would take infinitely long.
Evidence against this theory includes Max Planck's discovery of graininess in the universe, where measurements below a certain threshold are no longer meaningful. This implies rational numbers.