alt Hacker NewsBring radicals are just cheating.
You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.
Bring radicals are just cheating.
You can't solve an equation? Why not just introduce a function that is equal to the solution of the equation! Problem solved.
This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known.
Can't solve the differential equation x^2 - a = 0? Why not just introduce a function sqrt(a) as its solution! Problem solved.
Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.
A lot of 19th century mathematics was essentially this: discover which equations had solutions in terms of things we already knew about, and if they didn't and it seemed important or interesting enough, make a new name. This is the whole field of so-called "special functions". It's where we also get the elliptic functions, Bessel functions, etc.
The definition of "elementary function" comes exactly from this line in inquiry: define a set of functions we think are nice and algebraically tractable, and answer what we can express with them. The biggest classical question was:
The answer is "no" and Liouville gave us a result which tells us what the answer does look like when the result is elementary.Risch gave us an algorithm to compute the answer, when it exists in elementary form.