Zach I am with you on disagreeing on the interpretation of the error and more specifically the example used to illustrate it. Mathematical abstraction and operations are free of contextual interpretations, at least in its basic inceptions. While it can be practical to use such examples they can somehow be misleading.
The point on properly use equivalency and equality is valid but I do not agree on the reasoning exposed in the article. Perhaps the teacher did want to illustrate such matter but I am not sure on the context or method of the test.
Multiplication by definition is commutative (property) in which a x b = b x a. Also, structurally one can say the (a x b) will render the same result as (a + a + a +a +a +…b times) and (b + b + b + … a times). In this context one can say both operations are equivalent as rendering the same result.
But let’s go slightly deeper. If my memory does not betray me equality can be seen as an operation (“comparison” in a very mundane view if allowed) or “=” be seen as an operand, also by definition has the symmetry property. As such if a=b then b=a. Then let’s add the transitive property in which if a=b and b=c then a=c. That being said:
If a x b = b x a =R (R as in result)
(a + a +a +a +…b times) = R so does
(b + b + b + … a times) = R
then we can say
a x b = (a + a +a +a +…b times) or
b x a = (a + a +a +a +…b times) or
a x b = (b + b + b + … a times) or
b x a = (b + b + b + … a times)
In a broader sense equality complies with the properties of equivalency and as such it is one.
- All this within a well defined set of study, if you want, real numbers, etc.
- Nice reference articles (within a more abstract set theory frame of reference)