I'm reading some notes about topology and homeomorphisms and there is an example of a homeomorphism from the unit ball in $\mathbb R^n$ to $\mathbb R^n$. The map $x \mapsto {x \over 1 - |x|^2}$. I assume absolute value means Euclidean norm here. What I don't understand is the square in the definition. Wouldn't $x \mapsto {x \over 1 - |x|}$ also be an homeomorphism and a simpler example? Is there a reason why one would want to square the absolute value to get some sort of better homeomorphism?