Yeah, their assumption that there's a ternary relation between those immediately breaks down when you consider sets other than the real numbers. And you don't even need to go that far, roots as used on the real numbers don't make sense in C.
The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...)
I guess a different notation could have some meaning pedagogically, math notation is incredibly inconsistent at times, but there really is no "deeper truth" here.
Exponentiation does have some scenarios where it can be defined without an (obvious) exponential function, and roots may not be uniquely defined (or rather they are almost never unique), which means you need to be a bit careful, and which means that in theory the 3rd root could differ from the definition of x^(1/3).
However in the cases where you need to be careful most of stuff you'd use the more general notation for wouldn't be applicable anyway, you'd have a high chance of writing down an expression that has no unique value, or can't even be evaluated.
> The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...)
Why do you think this? It doesn't seem to fit historically or formally. 3^2 is a more elementary object than anything built with exp, and there's often no natural notion of exp(A) of a function A even when finite powers like A^3 are defined. exp(A) is defined with a power series that may not converge.
Premise: Forgive the sloppiness, I have some math background but I'm not a mathematician, any correction is more than welcome.
I don't really conceptualize them as the same kind of object, to be honest. I'm aware exponentiation is more fundamental, but the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers.
When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge, for example, (-2)^2 is well defined and equal to (-2)(-2), but (-2)^(1/2) isn't, unless you relate it to the exponential function.
Since the OP was about a notation proposal for working with real numbers, in that specific context I believe the more natural interpretation is to relate everything to the complex exponential and work your way up from there.
> the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers.
But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed.
> When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge
Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function.
> Since the OP was about a notation propos...
My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".
The thing is that ultimately notation isn't that important, what matters are the formal concepts at play (and to develop an intuition on how to manipulate them, if you're learning).
If this can help students "see the light", then sure, but I'm not entirely sold on the idea that the notation is actually the hardest part.
This is similar (with a different twist) to the various ideas going around that we should stop using base 10 and think in hexadecimal. It might be better in some absolute sense, but it's not something it's worth spending any energy into.
I'm just not sure I see much from this, but obviously I'm not the "intended target", so please take this with a pinch of salt.
The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...)
I guess a different notation could have some meaning pedagogically, math notation is incredibly inconsistent at times, but there really is no "deeper truth" here.