[reikonomusha]

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:

    Do integrals of elementary functions give us elementary functions?

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.

[eru]

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.

[cyberax]

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.

[reikonomusha]

The Bring radical has a great geometric interpretation: BR(a) is where the curve x^5 + x + a crosses the x axis.

Like sine or exp, it also has a nice series representation:

    sum(k = 0 to inf) binom(5k,k) (-1)^(k+1) a^(4k+1) / (4k+1)

We can compute its digits with the very rapidly convergent Newton iteration

    x <- x - (x^5 + x + a)/(5x^4 + 1)

and so on.

Why not invite it to the table of functions?

Ellipses are simple and beautiful figures known to every child, but why do we rarely invite the elliptic integrals to the table too?

I guess my point is that "nice geometric interpretation" is a little subjective and hasn't led to much consistency in our choice of which functions are popular or obscure.

[cyberax]

> The Bring radical has a great geometric interpretation

Erm... No. It's not great.

> Why not invite it to the table of functions?

Because it's too arbitrary.

[thaumasiotes]

> 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.

[reikonomusha]

You can calculate, measure, draw, construct, write a power series for, express as hypergeometric function, etc. the Bring radical too.

All of these concepts, from sine to real numbers, Bring radicals to complex exponentials, can all be defined in different, equivalent ways. What is interesting are the properties invariant to these definitions.

It still doesn't seem to me that a square root should be any more or less contrived than a Bring radical. Maybe we should call it a ultraradical instead?

[xigoi]

For me, what makes the square root more “natural” is that, although it’s usually introduced as an “answer by definition”, it can also be arrived at by wondering what happens if you take something to the halfth power.