Many of my students tends to believe that associativity of a given operation is always true.
We have seen in the previous post that addition is associative so that we always have . This actually hold true for all integers (positives and negatives).
But clearly you would agree that subtraction is not associative. .
This has obviously to do with the way subtraction is define. One way to think of it is to write as the additive inverse (or opposite) of , such that: . The subtraction is then defined as . The rule is that we can replace directly by .
It follows that the additive inverse of is as . Therefore we chose to write the additive inverse of as .
Let us now have a look at which we write so that clearly subtraction is not associative