Spoiler alert: When people talk about "group theory," this is the sort of thing that it does. The operations of addition, multiplication, and functional composition share certain symmetries that group theory makes formally sound. Once you grok the relationships, and how multiplication is "the same type of thing as" rotations, then you can see how square roots are similar to fractional rotations.
Less abstractly... the Exponent operation lets you turn addition into multiplication. It takes a little while to wrap your head around that, but once you do, it's straightforward to see how square roots correspond to fractions.
This is also concretely visible in Matrix algebra. Some matrices literally are rotations, and the square roots of such matrices are rotations by half as much. Matrices are nice to study in group theory, because they bridge the gap between numeric operations and functional composition.
Reminds me of category theory, which I tried to understand a little of yesterday, based on a discussion of Haskell. I don't know enough of either to know what the difference is between the two.
Nice, I didn't realise the relationship between rotations and the matrix square root: multiplying a vector by the sqrt of the pi/2 rotation matrix rotates it by pi/4!
Less abstractly... the Exponent operation lets you turn addition into multiplication. It takes a little while to wrap your head around that, but once you do, it's straightforward to see how square roots correspond to fractions.
This is also concretely visible in Matrix algebra. Some matrices literally are rotations, and the square roots of such matrices are rotations by half as much. Matrices are nice to study in group theory, because they bridge the gap between numeric operations and functional composition.