y^2=x vs y=sqrt(x). Is they is or is they ain't the same thing.

[Jragon]

@leahcim

The important thing to be clear on is that the arbitrariness is necessary because it would be impossible to have a function from R to R return both. It is the nature of functions that for each input in their domain they return exactly one output. Even though for any x, there are two possible values of y such that y^2 = x, the sqrt function must pick and return only one of them in order to be a valid function from R to R.

The human conceit that positive numbers are preferable to negative ones makes the choice easy, but basic math demands that a choice must be made.

Well, to be fair, the fact that we assume it should be a function from R to R is kind of arbitrary and biased too. Admittedly for good reason, since R is kind of nice to work in, but still arbitrary. There’s no reason it can’t be a non-function mapping to some finite deterministic set of values in R; or even a function from R to some space I’ll randomly call P (for pair!), which I’ll define as { (x,y) | x,y in R; y = -x}. So sqrt(0) uniquely maps to (0,0) and sqrt(1) uniquely maps to (1,-1) and so on.

[Indistinguishable]

Or a function from positive reals to positive reals…

Speaking of which, 1 doesn’t have just 2 square roots. It has 4… in the split-complex numbers. Or in the integers modulo 8. Of course, it has just 1 in the positive reals. (While a value like 3 also has 2 square roots in the reals, 4 in the split-complex numbers, and 1 in the positive reals, but 0 in the integers modulo 8 or in the rationals).

Everything is always context-dependent.

[Indistinguishable]

(And for me… Dinah Washington.)

[ultrafilter]

@leahcim

The human conceit that positive numbers are preferable to negative ones makes the choice easy, but basic math demands that a choice must be made.

The choice looks a lot less arbitrary when you look at the general power equation x[sup]n[/sup] = k. In that case, the roots are laid out in the complex plane on the vertices of a regular polygon, and the only place where you’ll always find one is on the positive real line. That’s why we take the positive solution to be the principal root.

[Chronos]

That is, if k is itself a positive real. It’s not well-defined to say which of the square roots of -1 is the principle one. Or rather, we claim without any particular support that one of them is the principle one, and use that in all of our other definitions.