In this Wikipedia article, I found the Projective Line: https://en.wikipedia.org/wiki/Projective_line.
I want to know how to take a square root of a number in this representation.
Let's take one example: the number $9$ is represented on projective on Projective line by the tuple $(9,1)$. Then the $\sqrt{(9,1)}$ is equal to $(3,1)$.
But there is so many other representations to $9$ in projective line... for example: $(18,2)$ and the root of this is $(3,1)$.
How can I get one equation to solve a square root of any number in projective line?
As you know, $(9,1)$ and $(18,2)$ both represent the same number on the projective line. But they are not identical pairs of real numbers. However, in the notation used on the Wikipedia page, $[9:1]$ and $[18:2]$ are the same identical number on the projective line. So let's use that notation.
Suppose you have the number $[a:b]$ and you want to find its square root.
You know that for any numbers $[x_1:y_1]$ and $[x_2:y_2],$ $$ [x_1:y_1] \cdot [x_2:y_2] = [x_1x_2:y_1y_2]. $$
You want to find $[x : y]$ such that the multiplication of $[x : y]$ by itself has the result $[a:b].$ That is,
$$ [x : y] \cdot [x : y] = [a:b]. $$
Using the definition of multiplication, can you solve for $x$ and $y$ in terms of $a$ and $b$?
It is possible that in your solution, two different ways of naming the same number $[a:b]$ will result in different names of the square root. Does that make the solution invalid? Why?