Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Relationships Among Operations

As with addition and subtraction, multiplication and division of whole numbers and integers have characteristic properties.  For instance, given any two whole numbers a and b, their product is a whole number.  Furthermore, a× b = b× a for all whole numbers a and b: If a = 4 and b = 5,  4× 5= 5× 4. Consequently, multiplication is said to be closed and commutative on the set of whole numbers.  Multiplication is also closed and commutative on the set of integers.

These properties do not hold, however, with respect to the operation of division.  Given any whole number a and non-zero whole number b, it is not always the case that a/b is a whole number For instance, if a = 2 and b = 3, the quotient a/b = 2/3 does not belong to the set of whole numbers.  Furthermore, a/b ≠ b/a for all non-zero whole numbers a and b: If a = 5 and b = 2,  5/2 ≠  2/5.   Consequently, division is neither closed nor commutative on the set of whole numbers.  The same finding applies to the set of integers.  These results are summarized in Table 3.1.

	Whole Numbers	Integers
Multiplication	closed commutative	closed commutative
Division	not closed not commutative	not closed not commutative

Table 3.1: Properties of Multiplication and Division

In order to consider the topic of fact families for multiplication and division, it is useful to consider these operations on the set of rational numbers, a topic taken up formally in the next chapter.  For now, though, we may think of the rational numbers as the set of all fractions.  Because every whole number or integer n may be written as n/1, the set of rational numbers includes the set of whole numbers and integers.  In particular, the set of rational numbers includes the number zero.  Because division by zero is undefined, the operation of division is not closed on the set of rational numbers. 

Given any two non-zero rational numbers a and b, we may write the following fact family associated with their product: a× b = c; a = c/b; b× a = c; and b = c/a.  This same fact family is associated with the indicated quotients c/b and c/a.  Consequently, we see that

· Multiplication and division are inverses of one another; and

· Related multiplication and division operations share the same fact family.

 These observations are summarized in Figure 3.10.