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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?

Angelo
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aguiadouro
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 09 '23 at 17:14
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    @aguiadouro, if $,x_2\neq0,,,$ the square root of $,(x_1,x_2),$ is $,\left(!\sqrt{\frac{x_1}{x_2}},1!\right).$ Moreover every ordered pair $,\left(k\sqrt{\frac{x_1}{x_2}},k\right),,,$ where $,k\in\Bbb R\setminus{0},,,$ is a representation of the square root of $,(x_1,x_2),$ in projective line.

    The square root is a real number in projective line if and only if $,x_1x_2\geqslant0,,,$ otherwise it is a complex number.

     – Angelo Feb 09 '23 at 17:57

1 Answers1

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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?

David K
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  • Thankyou!!! this make all sense! the sqare root of [x² : y²] is equal to [x : y]. In other words! I only need to take the square root of each term and i get my result. SQRT([a : b]) = [sqrt(a) : sqrt(b)] thank you! – aguiadouro Feb 09 '23 at 17:58