alt Hacker NewsThis 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:
Do integrals of elementary functions give us elementary functions?
Risch gave us an algorithm to compute the answer, when it exists in elementary form.
Replies
Sure. But the square root and the sine function also have nice geometric interpretations.
Bring radicals don't. They're just defined as a solution to this particular quintic.
Kinda the similar story with the Lambert function.
> This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known.
> Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved.
But that's not how sine was introduced. It's been around since classical geometry. It was always easy to solve the differential equation y'' = -y, because the sine had that property, and we knew that.
Heck, you can tell this just by looking at the names of the functions you mentioned. "Sine" is called "sine", which appears to have originated as an attempted calque of a Sanskrit term (referring to the same function) meaning "bowstring".
"Square root" is named after the squaring function that was used to define it.
Introducing an answer-by-definition gives us negative numbers, rational numbers, imaginary numbers, and nth roots... but not sines, come on. You can just measure sines.
That's one way to get at complex numbers and the sine function. But it's not the only one.
Eg you can get complex numbers from matrices.
But if you want to go in your direction: you can say we get fractions and negative numbers this way.