Road across some mathematics

Mathematics isn't just about answering questions; even more so, it is about asking the right questions, and that skill is a difficult one to reach. However I am working on a specific strategy: starting with small toy problems and investigate the nature of examples. As much as I can.

So, I will include such example here for sure. There will be a problem with the mathematical notation, but I'll see how I can work it out.

Here is an appetizer. Imagine a pendulum clock. The overall accuracy of such clock can be no better than the adjustment of the pendulum, which requires continual intervention. But for subsidiary timekeeping functions there is another kind of error one need to deal with. Even if the mean time is exact will the solar indicator keep pace correctly? The answer depends on how well celestial motions can be approximated by the arithmetic of the rational numbers. I mean, gear ratios. For example the day is 23 hours, 56minutes, 4.09053 second whereas the mean solar day is exactly 24 hours (by definition!). Therefore, question here is (not the only one of course) can we work this problem out with gear ratios really? well, It seems to me there are plenty of examples of this achievement (Strasburg great clock for instance). Moreover, if the rational numbers do the job most of the time, then where and how do the irrational number show up? I was wondering if this kind of numbers (they do exist right!?) was definitely one of the first steps of mathematical abstraction. Let me think about it.....Csar