This is good, but I think it's cyclical (heh). We want to compare our signal with a "pure tone". What's a pure tone? A sine wave. Why is a sine wave a pure tone? Because when we compare it with a pure tone, it's identical.
That's a question of why Fourier transforms are important though, not just how they're defined and computed. The next-level answer is presumably that sinusoids (or complex exponentials in general) are the eigenfunctions of general linear time-invariant systems, i.e. that if the input to an LTI system is exp(j*w*t), then its output will be A*exp(j*w*t) for some complex constant A. Some other comments here already alluded to that, noting that sinusoids are good for solving linear differential equations (which are LTI systems), or that the sum of two sinusoids of the same frequency shifted in time (which is an LTI operation, since addition and time shift are both LTI) is another sinusoid of that same frequency.
LTI systems closely model many practical systems, including the tuning forks and flutes that give our intuition of what a "pure tone" means. I guess there's a level after that noting that conservation laws lead to LTI systems. I guess there's further levels too, but I'm not a physicist.
That eigenfunction property means we can describe the response of any LTI system to a sinusoid (again, complex exponential in general) at a given frequency by a single complex scalar, whose magnitude represents a gain and whose phase represents a phase shift. No other set of basis functions has this property, thus the special importance of a Fourier transform.
We could write a perfectly meaningful transform using any set of basis functions, not just sinusoids (and e.g. the graphics people often do, and call them wavelets). But if we place one of those non-sinusoidal basis functions at the input of an LTI system, then the output will in general be the sum of infinitely many different basis functions, not describable by any finite number of scalars. This makes those non-sinusoidal basis functions much less useful in modeling LTI systems.
I appreciate the feedback. I’m person I talk about pure tones more, mostly because I love how pretty complex spirals are.
Here, I just tried to define them “Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency” and then use sine as an example.
Truly, complex rotations are a much better definition, but it’s somewhat reasonable to expect people to be familiar with sine waves.
A pure sine wave has only one frequency, that's why it is analogous to a "pure tone". It's not a circular definition; they key part is the "pure", which means no dilution by other frequencies, which admittedly is not obvious.
