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Topic: Re: Multiple Multiplication Mea
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Ladnor Geissinger

Posts: 55
Registered: 12/4/04
Re: Multiple Multiplication Mea
Posted: Jul 30, 1997 6:01 PM
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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
repeated addition idea for all products (we can for some of course).
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.

Ladnor Geissinger
Math Prof at UNC Chapel Hill & Math Chair at IAT
email: ladnor.iat@mhs.unc.edu
or ladnor_geissinger@unc.edu
phone: 919-405-1925
address: Institute for Academic Technology
2525 Meridian Parkway, Suite 400
Durham NC 27713 USA
IAT phone: 919-560-5031
IAT fax: 919-560-5047
IAT web home page: www.iat.unc.edu
LEARN NC home page: www.learnnc.org
Mathwright Library: ike.engr.washington.edu/mathwright/





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