In Chemistry, you learn about electron configurations, which involves learning the rules for 4 quantum numbers. Three of these numbers are integers. But students tend to be mystified by where these suddenly and almost magically appear. Why integers? Why the values that you are forced to memorize?
It all starts with the heart and soul of quantum mechanics, which is the Schrodinger equation. This is the F = ma of quantum land. For this case, our system is an electron oscillating back and forth between two walls. It is a nice, neat 1-D system. When we see what the Schrodinger equation looks like in this case, it will be identical to what we get for a mass on a spring oscillating back and forth. Since we know the solution of simple harmonic motion is a sine or cosine, then the solution of what turns out to be the wave function for our electron is also a sine or cosine. We will see that the electron will be restricted in its energy, that it will have restricted energies determined by an integer that we get as part of our solution! It is a quantum number!
These quantum numbers are simply part of solutions to complicated equations you get from the Schrodinger equation. An exact solution exists for a hydrogen atom, for example. You get three integers for an electron in an atomic orbital because it is a 3-D system rather than the 1-D system we deal with here. But the idea is the same. Integers are natural parts of these solutions, and they then are part of energy solutions for atoms and particles, which tells us that the energies of the electron have specific, allowed values, and not a continuum of energy that we see for a superball bouncing between two walls in a big, macro-world. The micro-world follows a separate set of rules in quantum mechanics. I hope this helps.
By the way, for an example of how the quantum numbers play out for the periodic table, check out some rules of the game.