Subtraction

It matters very much which of the two parts of a difference are named first. Taking $500 from an account with $300 in it is very different from taking $300 from $500. For this reason, subtraction is not commutative: a - b does not equal b - a.

Subtraction is not associative either: (a - b) - c does not equal a - (b - c).

An example will demonstrate this: (20 - 10) - 3 is 7, but 20 - (10 - 3) is 13.

Often one encounters expressions such as 5x² - 2 - 3x², with no indication of which subtraction is to be done first. Since subtraction is non-associative, it matters. To avoid this ambiguity one can agree that subtractions, unless otherwise indicated, are to be done left-to-right. This is a rather limiting agreement, therefore, it may be more convenient to use some other order. Another agreement, which is the common agreement of algebra, is to treat the minus sign as a plus-the-opposite-of sign. Thus one would interpret the example above as 5x² + (-2) + (-3x²). In this interpretation it becomes a sum, whose terms can be combined in any order one pleases.

In certain sets subtraction is not a closed operation. The set of natural numbers, for instance, is not closed with respect to subtraction. If a merchant will not extend credit, one cannot buy an article whose price is greater than the amount of money one has.