Question:
A question in a sixth grade text as: An algebraic expression ______
suggests an operation. The book says the answer for the blank is "always."
A parent challenged this answer saying that a single variable for instance
can be an algebraic expression which does not suggest an operation. Our
question: Can a single variable be defined as an algebraic expression, or
does that variable have to be tied to a sign such as n + 5? or 2n? Perhaps
what we need is a specific definition of expression. Does it mean the same
thing as equation?

Replies:
I vote with the parent. I think that you are correct that a more careful
definition of "algebraic expression" is required in order to decide this
issue. As one of Lewis Carroll's characters in Alice in Wonderland said,
"Words mean exactly what I want them to mean, neither more nor less."
Definitions in mathematics are up to definer. Least you think this leads to
anarchy, be aware that there are standardly accepted definitions which
mathematicians agree to but some words have not reached this status. The
questions that you should ask are about what is true in light of the
definitions you accept. It is widely accepted that algebraic expressions
are made up of variables and operations combined according to certain rules
of formation (e.g. x+ would not be a valid expression). Single variables
are accepted as valid expressions. Note that operations do not have to be
binary. There are unary operations (e.g. square root).

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