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The most striking example in my opinion of complex mathematics producing realistic results is quantum mechanics. The Schrödinger equation seems very intuitive and reasonable when you look at it, since it says something as simple as "the total energy of a particle-wave is the sum of its curvature component and its interaction energy with the local force fields". This is very reasonable - after all, if you squeeze something into a small space - high curvature - you need energy, and by definition particles have different energies in different parts of a force field. But if you follow through what this equation implies, you'll predict a result called quantum tunnelling, where a particle goes through a high barrier it lacks energy to jump over. And this produces an immediate effect, the high acidity of the hydrogen ion, vs. other acidic things such as boron. These are larger and since quantum tunnelling reduces immensely with increasing mass, only the proton experiences it to significant effect. To caricature, without this rust could be corrosive as acid, which is clearly not the case.
For an example of systems rather easy to think about but difficult to accurately predict, consider systems that contain oscillators that respond to external vibrations. In MRI, the body is subjected to an extremely high magnetic field. This causes a small proportion of hydrogen nuclei to change in spin to line up with the field. Reversing this orientation can be done in the range of FM radio or television broadcast band radio frequencies. The exact frequency depends on the field strength. This is the trick in MRI: extra coils change the magnetic field, so you can tell which proton you're talking to based on its frequency. Now the problem is that that if you individually try each frequency, the data is too noisy for any real use. This problem needs some math. Rather than trying each individually, "all keys in the piano" are slammed, and the resulting noise is split into individual frequencies. The integral transform called Fourier transform can do this. It involves multiplying the noise with a sinusoidal function, taking the area under this curve and measuring that per each frequency gives the distribution of frequencies. The amplitude at each frequency and the known coil magnetic field allow to work back to the amplitude at each location inside the body. Collecting these for all points gives a MRI image
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