There is obviously a set of situations where one lack an algorithm to compute the exact solution of an equation via symbolic manipulation only, for example `x = sin(x)`

.

One has to resort to numerical analysis and iteration methods (for example) to estimate a value resolving the above.

My question is: is there an accepted mathematical proof or set of evidence demonstrating that it is impossible to resolve this equation via symbolic manipulation or is there a possibility that one comes with a solution through some clever trick in the future?

If such a proof exist, which mathematical concepts do I need to Google for further investigation?

`ax = sin(x)`

where`a`

is any positive real. I was only thinking about real solutions when I wrote my question. I could have chosen another example. Iterations are performed until a threshold of quality is reached around the intersection. I was wondering about mechanical, step-by-step, resolutions, like that of resolving a set of linear equations. $\endgroup$ – Jérôme Verstrynge Dec 10 '17 at 14:244more comments