Division on Numbers is Not Associative

Theorem

The operation of division on the numbers is not associative.

That is, in general:

$a \div \paren {b \div c} \ne \paren {a \div b} \div c$

By definition of division:

$\ds a \div \paren {b \div c}$

$=$

$\ds a \times \paren {\dfrac 1 {b \times \dfrac 1 c} }$

$\ds $

$=$

$\ds a \times \paren {\dfrac c b}$

$\ds $

$=$

$\ds \dfrac {a c} b$

$\ds \paren {a \div b} \div c$

$=$

$\ds \paren {a \times \dfrac 1 b} \times \dfrac 1 c$

$\ds $

$=$

$\ds \dfrac a {b c}$

So we see that:

$a \div \paren {b \div c} = \paren {a \div b} \div c \iff c = 1$

and so in general:

$a \div \paren {b \div c} \ne \paren {a \div b} \div c$

$\blacksquare$