# Subtraction on Numbers is Not Associative

## Theorem

The operation of subtraction on the numbers is not associative.

That is, in general:

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

## Proof

By definition of subtraction:

 $\displaystyle a - \paren {b - c}$ $=$ $\displaystyle a + \paren {-\paren {b + \paren {-c} } }$ $\displaystyle$ $=$ $\displaystyle a + \paren {-b} + c$

 $\displaystyle \paren {a - b} - c$ $=$ $\displaystyle \paren {a + \paren {-b} } + \paren {-c}$ $\displaystyle$ $=$ $\displaystyle a + \paren {-b} + \paren {-c}$

So we see that:

$a - \paren {b - c} = \paren {a - b} - c \iff c = 0$

and so in general:

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

$\blacksquare$