# Algebra/Division is not commutative

 Algebra ← Real Numbers Division is not commutative Exponents →

Division is not commutative. That means usually a ÷ b is not equal to b ÷ a, and can be demonstrated simply by example.

${\displaystyle {\frac {1}{2}}\neq {\frac {2}{1}}}$

While division itself is not commutative, there are two special cases where the answer is the same if you reverse the order of operation. These cases occur when the answer (quotient) is 1 or when the answer is -1:

${\displaystyle a\div b=b\div a\iff {\mbox{(rewrite as fractions)}}}$

${\displaystyle {\frac {a}{b}}={\frac {b}{a}}\iff {\mbox{(multiply both sides by}}\ ab)}$

${\displaystyle a^{2}=b^{2}\iff {\mbox{(take both square roots)}}}$

${\displaystyle a={\sqrt {b^{2}}}\quad {\mbox{ or }}\quad a=-{\sqrt {b^{2}}}}$

${\displaystyle a=b\quad {\mbox{ or }}\quad a=-b}$

${\displaystyle a\div b=1\quad {\mbox{ or }}\quad a\div b=-1}$