It depends entirely on the context -- mathematical notation, as always, depends upon the level of instruction, textbook, context, mathematician etc. In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. When you're teaching it for the first time (as is the context we are discussing right now) it's quite important to reinforce how many square roots there are. Context generally makes it clear (among people who already understand what is happening) which version you mean as is usually the case. Similarly with logs, though we have the nice log/Log distinction which I suppose has become standard-ish. There is literally no reason we select the positive branch other than notational convenience and one of the biggest mistakes I've seen with freshmen students trying to learn calculus is algebra mistakes like this -- taking a square root and only considering the positive branch. Because they were taught square root of 4 is 2. And that's true sometimes (when you are computing sqrt(x)), and not others (when you take the square root of something for the purpose of solving an equation). I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician.
Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.
> In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values.
But that's just overloaded syntax, isn't it? We're concerned with real numbers here.
> There is literally no reason we select the positive branch other than notational convenience
Agreed, it's just a convention, though a very helpful one.
> I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician.
I had to look up that one, since my maths education wasn't in English. I agree that it's nicer than what appears e.g. in my textbook: "the square root is always a positive number or zero", which conflicts wiwitthe definition that's on Wikipedia. It's probably the conflicting uses of "square root" to refer to two different things (all solutions, which is more relevant analytically, or the unique positive solution, which is more helpful for notation) that causes the kinds of freshman errors you mention.
> You understand the concern here: simplifying notation masks the actual mathematics.
My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.
>But that's just overloaded syntax, isn't it? We're concerned with real numbers here.
Yes, you're 100% correct, I was just giving an example for why it's natural for some (me included) to think of sqrt(x) as a solution set, I suppose.
>My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.
Well, the normal notation is a little confusing yes, but the new notation makes it worse -- they propose marrying the principal sqrt function, which makes an arbitrary choice and drops information, to the log and exponential function, which both do not over R, and for which the exponential function does not over C.
At least we teach three separate concepts and then unify them later as best as we can, as opposed to trying to pretend they are all the same.
Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.