College Math Teaching

February 16, 2012

The “equals” sign: identities, equations to be solved and all that…

Here is the sort of thing that got me thinking about this topic: a colleague had a student complain about how one of her quiz problems was scored. The problem stated: “show that \sqrt{2} + \sqrt{3} \neq \sqrt{5} “. She was offended that her saying “\sqrt{x} + \sqrt{y} \neq \sqrt{x+y} ” wasn’t enough to receive credit and would NOT take his word for it. In fact, she took this to the student ombudsman!!!

But that raised the question: “what do we mean when we tell our students “\sqrt{x} + \sqrt{y} \neq \sqrt{x+y} “?

Of course, there are some central issues here. The first issues is that our “sure of herself” student thought that “\sqrt{x} + \sqrt{y} \neq \sqrt{x+y} ” meant that this relation is NEVER true for any choice of x, y , which of course, is false (e. g. let y = 0 and x \ge 0 .) In fact, \sqrt{x} + \sqrt{y} \neq \sqrt{x+y} is the logical negation of the statement \sqrt{x} + \sqrt{y} = \sqrt{x+y} ; the latter means that “this statement is true for ALL x, y and its negation means “there is at least one choice of x, y for which the statement is not true. “Equal” and “not equal” are not symmetric states when it comes identities, which can be thought of as elements in the vector space of functions.

So, \sqrt{x} + \sqrt{y} \neq \sqrt{x+y} means that \sqrt{x} + \sqrt{y} and \sqrt{x+y} are not equal in function space, though they might evaluate to the same number for certain choices in the domain.

So, what is the big deal?

Well, what about equations such as x^2 + 3x + 2 = 0 or y^{\prime \prime} + y = 0 ?
These are NOT equalities in the space of functions; the first means “what values in the domain does f^{-1}(0) take given f(x)=x^2 + 3x + 2 and the second asks one to find the inverse image of 0 for the operator D^2+1 where the domain is the set of all, say, twice differentiable functions.

But, but…would the average undergraduate student understand ANY of this? My experience tells me “no”; hence I intentionally allow for this vagueness and only address it as I need to.

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