Pete and Repeat Are in a Boat
This one's going to be fun:
I'm not sure how to make this clearer. Dare I open it for comment?
OK - in what *mathematical* manner of speaking is multiplication not a function? I've probably always taken it for granted that multiplication is a function. This could be my big awakening moment now - the big Aha! That I can't just assume that multiplication is a function for just everyone --- I'm holding my breath!
I'm not sure how to make this clearer. Dare I open it for comment?
Labels: education, general, mathematics
Comments:
"operations" are indeed functions.
in particular, multiplication
*is* a function ...
a function of two variables.
i'm not at all clear on why
anybody would think otherwise.
product(2, 7) = 14, for example
(in the usual "f(x,y)" notation
for functions of two variables).
maybe the "infix" notation
could cause some confusion
but there's nothing in the world
going on here that can't be
described (to our profit!)
in terms of sets-of-ordered-pairs ...
vlorbik
The problem, it seems to me, with considering operations in and of themselves as functions is that when you have two different operations, A and B, the notion of function equivalence can be extended inappropriately to the operations in and of themselves.
Yes, if A(x, y) = 10 and B(x, y) = 10 (with A and B, again, being operations), then of course A(x, y) = B(x, y), but A and B are not necessarily, as a result, also equivalent (or the same, or whatever).
If that were the case, then the equivalence A(4, 2) = B(4, 2) would allow us to say that division (A) and subtraction (B) are the same operation.
Thoughts?
Yeah, that's what I thought. : )
sorry for the silent treatment...
the fact that "A(x,y) = B(x,y)
for some particular values
of x & y" does not imply "A = B"
(i.e., does not imply that
A(x,y) = B(x,y) for *all*
choices of x & y) may be confusing.
if so, it would appear to be
an opportunity to clarify
what is meant by saying that
one function is equal to another
(rather than a reason to just
give up altogether on using,
say, the "function concept"
to understand [what are called]
operators).
obviously examples from calculus
are wildly inappropriate in actual
elementary-ed settings; i hope
the following won't appear to be
useless here (following).
lots of texts, and almost all
lecturers, will at least occasionally
confuse "f" (the name of a function)
with "f(x)" (the name of a
function *value* [usually a number]).
most of the time, no confusion ensues.
trouble is, a lot of (anyway)
students appear to be *unable*
to make the distinction when it
*does* matter.
the clearest example known to me
(as of right now ... i'm doing
a calc I class) occurs in defining
the "differential operator"
d/dx. as of last week, several
of my students routinely confused
this symbol with dy/dx.
"d/dx is the name of the *procedure*
that you 'do to y' to get dy/dx"
seems to've been a pretty useful
little intervention (the next
homework will tell).
y & dy/dx are the names of
(real-valued) *functions* here;
d/dx is of course an *operator*
(which, the heart of the matter,
is itself, yes, a certain type
of function--one whose domain
and range are *sets of functions*
[rather than the more familiar
sets of *numbers*].
helpful?
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