Sign of Quotient of Factors of Difference of Squares/Corollary

Corollary to Sign of Quotient of Factors of Difference of Squares

Let $a, b \in \R$ such that $a \ne b$.

Then

$-\map \sgn {\dfrac {b - a} {b + a} } = \map \sgn {a^2 - b^2} = -\map \sgn {\dfrac {b + a} {b - a} }$

where $\sgn$ denotes the signum of a real number.

$\ds \map \sgn {\frac {b - a} {b + a} }$

$=$

$\ds \map \sgn {\paren {-1} \frac {a - b} {a + b} }$

$\ds $

$=$

$\ds \map \sgn {-1} \map \sgn {\frac {a - b} {a + b} }$

$\ds $

$=$

$\ds \paren {-1} \map \sgn {\frac {a - b} {a + b} }$

$\ds $

$=$

$\ds -\map \sgn {a^2 - b^2}$

$\ds $

$=$

$\ds \paren {-1} \map \sgn {\frac {a + b} {a - b} }$

$\ds $

$=$

$\ds \map \sgn {-1} \map \sgn {\frac {a + b} {a - b} }$

$\ds $

$=$

$\ds \map \sgn {\paren {-1} \frac {a + b} {a - b} }$

$\ds $

$=$

$\ds \map \sgn {\frac {b + a} {b - a} }$

$\blacksquare$