|
Number & Operations for Teachers Copyright David & Cynthia Thomas, 2009 |
|||||||||
|
Properties of Whole
Number & Integer Operations It is both normal and desirable that, in the process of doing addition and subtraction problems, students realize that these operations have properties. For instance, addition and subtraction are inverse operations. Another important property is that of closure. If an operation on a set is closed on that set, operations on numbers in the set always produce answers that are in the set. For instance, if you add any two whole numbers, you obtain a whole number as their sum. One way to state this property is to say that The set of whole numbers is closed with respect to addition. The same cannot be said about subtraction of whole numbers. For instance, 2 5 = -3, but -3 does not belong to the set of whole numbers. So, the operation of addition is closed on the set of whole numbers, but the operation of subtraction is not. Another important property of operations on sets is commutivity. An operation is commutative on a set if the same result is obtained regardless of the order in which the set elements are used. For instance, a + b = b + a for all whole numbers a and b. Consequently, addition is said to be commutative on the set of whole numbers. Subtraction is not commutative on the set of whole numbers, that is, a b ≠ b a for all whole numbers a and b. For example, 2 3 ≠ 3 2. These properties are summarized in Table 2.2. Why is subtraction closed on the set of integers but not closed on the set of whole numbers?
Table 2.2: Closure and Commutative Properties |
|||||||||
|
|
