In a question on geometrical constructions of numbers, two constructions appear to be related by analogy.
In the first construction a specific straight line $L$ (blue) is constructed (going through $1$ and $ix $). The line $L'$ which is parallel to $L$ and goes through the point $i$ (red) yields the number $1/x$:
In the second construction a specific circle $C$ (blue) is constructed (going through $-1$ and $x$). The circle $C'$ with center $0$ and going through the intersection $i\sqrt{x}$ of $C$ with the straight line $\overline{0i}$ yields the number $\sqrt{x}$:
I wonder which arithmetic meaning might be given to this analogy of constructions, i.e. what might an analogy between $1/x$ and $\sqrt{x}$ be in arithmetic terms?
What do the arithmetic operations $1/x$ and $\sqrt{x}$ have in common – in arithmetic terms? Which single concept are they examples of?
(In arithmetic terms $\sqrt{x}$ is the inverse of $x^2$, i.e. $\sqrt{x^2} = \sqrt{x}^2 = x$. For $1/x$ we find $x^2/x = x$, but this doesn't make $1/x$ an inverse of $x^2$, because $(1/x)^2 \neq x$.)