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Root notation isn't really _concealing_ anything. The fact that it's mostly equivalent to exponentiation by half is a theorem. Do we need to admit that 2 is con

Root notation isn't really _concealing_ anything. The fact that it's mostly equivalent to exponentiation by half is a theorem. Do we need to admit that 2 is concealing 1+1, thus making the game impossible? Given the prevalence of quadratic polynomials over higher-order ones, sqrt does feel somewhat more fundamental than arbitrary exponentiation. Lots of people have pointed out the farcity of the game after allowing fancy functions, but IMHO, there is a lot of fun in just finding satisfying solutions, without the need for specific rule limitations.

kazinator 2y ago
The root notation is a syntax with a place for a number, to specify the degree. For instance ∛ -> cube root(s). When the degree is 2, it can be omitted. So yes, the notation conceals something: its default value.
- xelxebar 2y ago
  I have a function f: R -> R such that f(0)=1 and f'(x)=f(x). Is this concealing a parameter? It turns out we can deduce that f(x)=e^x, so f(x)=pow(e,x), so maybe? But wait, is pow concealing another parameter? Indeed, exponentiation is just (a completion over) repeated multiplication; multiplication is repeated addition, and addition repeated succesorship. It's straightforward to show that pow(x,y)=f(2,x,y) where f(n,x,y) encodes said hierarchy. But wait! f is a function, which is a set of 2-tuples, so we have another "hidden" parameter. It's also essentially part of Fast growing hierarchy[0] functions. Or what if we extend to complex numbers? Is the imaginary axis degree of freedom another hidden parameter? Et cetera ad infinitum... So, if you want to say that sqrt(x) has a hidden parameter, you're just making an arbitrary claim unless you accept it is concealing an unbounded number of hidden parameters. Or we can just do the obvious thing and say that sqrt(x) has one parameter. [0]:https://googology.fandom.com/wiki/Fast-growing_hierarchy https://googology.fandom.com/wiki/Fast-growing_hierarchy

kazinator 2y ago

The root notation is a syntax with a place for a number, to specify the degree. For instance ∛ -> cube root(s). When the degree is 2, it can be omitted. So yes, the notation conceals something: its default value.

xelxebar 2y ago
I have a function f: R -> R such that f(0)=1 and f'(x)=f(x). Is this concealing a parameter? It turns out we can deduce that f(x)=e^x, so f(x)=pow(e,x), so maybe? But wait, is pow concealing another parameter? Indeed, exponentiation is just (a completion over) repeated multiplication; multiplication is repeated addition, and addition repeated succesorship. It's straightforward to show that pow(x,y)=f(2,x,y) where f(n,x,y) encodes said hierarchy. But wait! f is a function, which is a set of 2-tuples, so we have another "hidden" parameter. It's also essentially part of Fast growing hierarchy[0] functions. Or what if we extend to complex numbers? Is the imaginary axis degree of freedom another hidden parameter? Et cetera ad infinitum... So, if you want to say that sqrt(x) has a hidden parameter, you're just making an arbitrary claim unless you accept it is concealing an unbounded number of hidden parameters. Or we can just do the obvious thing and say that sqrt(x) has one parameter. [0]:https://googology.fandom.com/wiki/Fast-growing_hierarchy https://googology.fandom.com/wiki/Fast-growing_hierarchy