Question: what are some of the things that seem simple but actually have a lot of maths to it?

Giuseppe Cotugno answered on 27 Jan 2020:

A radio is a good example.

A radio needs to convert signals of different types (AM – amplitude modulation, FM – frequency modulation, or digital) very quickly in order to enjoy some good music. A radio also has to be able to adjust the strength of basses and treble and to filter interference and noise. All those operations are quite complex mathematically, so much that engineers and mathematicians have to use “tricks” to make their life easier.

The transmission of radio signals is even more complex, as radio signals must have a decent quality and need to be properly spaced on the spectrum to be audible.

All this stuff is so complex that the mathematics calculations are not performed in the conventional cartesian space, but all the calculations are converted to a different space, typically using a thing called Fourier transform. This is one of the tricks I mentioned earlier. Using a Fourier transform it is possible to observe radio signals in terms of frequency and amplitude (instead of evolution over time). It is much more simple to create a filter which modulates some frequencies rather than modulating some part of the wave over time because the mathematical model became simpler (e.g. a filter in the frequency domain can be drawn as a trapezium).

All those operations, like decoding and filtering a signal, are very complex and yet the radio can do everything extremely quickly thanks to the underlying electronics which is also another very complicated part of the radio as it has to perform mathematical calculations very quickly.