On commutative and associative

While working on a review of a Math textbook from a popular series, I came across a problem. The book did not correctly explain or use the associative property. A quick search of the web also did not turn up an explanation that would be useful to most parents, so I thought I'd take a crack at it.

The official or technical definitions are usually given by equations such as these.

commutative a * b = b * a
associative (a * b) * c = a * (b * c)

The * represents the operator, such as + for addition. a, b, and c are operands (what the operator operates on).

What do the equations mean? To put it simply,

• The commutative property allows operands to be interchanged across a single operator but does NOT allow changing the order of computation.
• The associative property allows the operations to be computed in any order but does NOT allow rearranging of operands.

A string of numbers can be added or multiplied in any order because addition and multiplication are both commutative and associative. Neither property by itself will permit this. Here's why. Let's consider:
a + b + c =? c + b + a
where =? means "does (it) equal?"

To begin, we must realize that there are implied parentheses around the first 2 operands in both cases, because we are really asking: "If we add b to a and then c to the result, will the answer be the same as if we add b to c and then a to the result?" Thus, the question becomes:
(a + b) + c =? (c + b) + a

By the commutative property, we can move the operands across each plus sign
(a + b) + c = c + (a + b) = c + (b + a)

But we need the associative property to move the parentheses:
c + (b + a) = (c + b) + a

So
(a + b) + c = (c + b) + a
ONLY if addition is both associative and commutative.

We could do this the other way around, too, applying the associative property first, but we would then need to apply the commutative property afterwards in order to move the operands around. The associative property doesn't allow that!

As I mentioned, all this came up because the math book I was reviewing (Teaching Textbooks Math 7) said that the associative property (by itself) was the reason that strings of numbers can be added or multiplied in any order. This series compounds the error by saying in a more advanced text (Algebra 1) that the associative property is an extension of the commutative property.

Interestingly, another book I checked (Saxon's Math 5/4) said that we can add up strings of numbers in any order because of the commutative property of addition. No mention of the associative property. However, the associative property was correctly defined and used later.

For the benefit of those of our children who may wish to pursue math beyond high school, it is important that the two properties not be confused with one another. In reality the two are distinct and independent. A function can be one but not the other. For example, matrix multiplication is associative but not commutative. Unsigned difference is commutative but not associative.

Following TT's logic, the student would expect that three matrices to be multiplied can be rearranged in any order, but this is not true. Similarly, by Saxon's logic, the student might expect that the unsigned difference of three numbers could be computed in any order, but that's not true either.