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I found following books useful: A Guide to Distribution Theory and Fourier Transforms By Robert S. Strichartz. Not very rigorous and not much content either. But good book to start from. Generalized Functions: Theory and Applications By Ram P. Kanwal. Not very rigorous. This book starts with chapter on Dirac delta function and then slowly builds the ...

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In the underlying physics, c = 1 (Planck units). 1 is rational. But your unit system might not have a rational length. Speed of sound is rational in nature if macroscopic quantum mechanics holds (this is still open to debate that I will not enter). We should be able prove given macroscopic quantum that speed of sound is an integer multiple of Planck length ...

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Well it's a tricky question in some way. You can for example consider the second as a rational number because its definition (a number of times the time needed for some atom to change state) is rational in nature (you can see it like this at least): you're technically just counting a number of occurrences of an event. Then if you consider the speed of ...

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It depends on the unit you want to express it. If you choose c/100 as the speed unit, c will be expressed with a rational number. If you choose c/π, you'll have an irrational one. That depends on measure, not on nature.

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Something I posted on reddit answers this question quite well, I think: "Rational" and "irrational" are properties of numbers. Quantities with units aren't numbers, so they're neither rational nor irrational. A quantity with units is the product of a number and something else (the unit) that isn't a number. By choosing the unit you use to express a ...

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$$D(\hat{X}_i):= \left\{ \psi \in L^2(K, d^nx)\:\left|\: \int_K x_i^2|\psi(x)|^2 d^nx< \right.+\infty \right\} = L^2(K, d^nx)$$ where the last identity holds true if $K$ is bounded (in particular compact) because $x_i^2$ is bounded as well thereon (in this case the operator is bounded, too). Moreover D(\hat{J}^2):= \left\{\psi \in L^2(\mathbb S^2, ...

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