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Topic: Re: Multiple Multiplication Mea
Replies: 0

 Ladnor Geissinger Posts: 55 Registered: 12/4/04
Re: Multiple Multiplication Mea
Posted: Jul 30, 1997 6:01 PM

A recent note to amte had a statement that caught my eye:
<<<<<<<<<<<<<<<<<<<
However, if you
take "plus" and "minus", there you start running into ambiguities.
"Plus"
can be a binary operator of addition, or it can be a unary operator of
direction in the case of defining the integers.
>>>>>>>>>>>>>>>>>>
Let's not go overboard. Plus (+) refers only to the operation of
addition, there is no unary operator of positive direction. Just look at
your calculator. There is no + put in front of positive numbers (it
would be redundant), the button that has + on it is for addition and that
alone. There is a button with plus/minus
which is used to change the sign of the number in the display (i.e. 3
goes to -3, or -7 goes to 7). It is enough of a problem that minus (-)
is used in two quite different senses -- don't compound this by
unnecessarily having two senses for +!

The questions about one/more meanings for subtraction and for
multiplication seem to be forgetting history -- that both of these
operations came into existence by generalizing/extending previous
operations and only part of the original meaning could be carried along
in the extended domain of the operations. This is typical. When you go
from the reals R to the complexes C in order to solve more equations, you
have to give up the possibility of having an order relation on C
compatible with the rational operations (you can still use it on R of
course). When you generalize multiplication from R or C to 2x2 square
matrices, you retain many of the properties but you give up commutativity
and you get zero-divisors! When you generalize multiplication to
real-valued functions on some domain you keep commutativity but you get
lots of zero-divisors, etc.

Originally subtracting 7 only applied to numbers bigger than 7 and meant
taking away 7. Then negative integers were included with the usual order
extension, and we extended the domain of subtraction to all integers so
that 3 - 7 meant decrease 3 by 7 to get -4, and similarly for -10 - 7.
But in fact we also extended addition (+) to apply to negative numbers
and so we had to worry about 3+(-7) and whether that was the same as 3 -
7 (subtraction), and then 10 - (-7), etc. Eventually people checked
this all out and by 1600 or before it was agreed that it seems to work
consistently and efficiently -- which is what we need for modeling and
applications.

Similarly for multiplication. Multiplication such as 5*39 may indeed
have meant repeated addition with 5 summands each being 39. [And from
counting rectangular arrays of objects we observe that we get the same
result if we calculate 39*5 that way.] When multiplication is extended
to cover negative integers as well, then we still can think of 5*(-23) as
repeated adding of -23, but as the recent discussions of neg*neg=pos
point out we can't reasonably do that for (-5)*(-23). And of course when
we extend multiplication to rational fractions we also can't retain this
That's the way it is. You give up simple meanings/explanations for more
powerful tools/techniques. When you first start working in the extended
domain you mostly proceed by analogy with the earlier situation
(mathematicians used to talk a lot about retaining the same "form").
Then you practice a lot until it becomes automatic and you can start
thinking about the important stuff -- how to use this newly acquired
skill to model new phenomena and solve new problems.

Math Prof at UNC Chapel Hill & Math Chair at IAT
phone: 919-405-1925
2525 Meridian Parkway, Suite 400
Durham NC 27713 USA
IAT phone: 919-560-5031
IAT fax: 919-560-5047