In order to clarify my book selling example, we need to think carefully
about what we mean by negative numbers. This is exactly the kind of
thought-provoking exercise that we want our students to become engaged in.
Mr. Leibman defines positive numbers as "Outgoing = expense = negative.
Incoming = revenue = positive." Let's think about what is incoming and
outgoing without being restricted to just the money. If I buy a book, I have
spent money. We all seem to agree that cost is negative (-$10 in the
example). But don't forget that I have also received a book (+1 book, a
positive quantity). Now, if I sell the book instead, the book is an outgoing
quantity, and therefore negative (-1 book). Since the book has the same
price tag whether I buy or sell, I argue that the price is still -$10; only
the direction of flow of the book has changed.
There is nothing incorrect about Mr. Leibman's interpretation that the price
changes from negative to positive--his emphasis is on where the money is
going. My interpretation emphasizes where the object is going. Both make the
change from buying to selling by introducing one new negative sign. In the
end, we have shown (through our different interpretations) two things: 1)
Commutativity of the negative sign in a product of numbers, and 2) the
arithmetic fact that (-a)(-b) is equivalent to (a)(b).
I strongly feel that an example is only as good as the thinking it
generates. This one has served me pretty well.
George Alexander
UW Colleges Developmental Math Coordinator
UW Rock County
2909 Kellogg Ave.
Janesville, WI 53546-5699
(608) 758-6627
-----Original Message-----
From: Bob Leibman [mailto:bleibman@io.com]
Sent: Monday, November 08, 1999 9:52 PM
To: George Alexander; mathedcc@archives.math.utk.edu
Subject: RE: [MATHEDCC] Marilyn Vos Savant
I began this reply earlier, but I think it is not entirely redundant with
some of the most recent replies, so I will go ahead and add my thoughts.
At 10:14 AM -0600 11/8/99, George Alexander wrote:
>I like to address this question about multiplying two negative numbers by
>examining what we mean by the following questions:
>
>"You buy 5 books priced at $10 each. How much did you spend?" Most students
>will readily agree that this asks for the calculation (5 books) x (-$10)
>= -$50. The book price and the total cost are negative numbers because they
>represent expenses. The number of books is positive because the buyer is
>receiving the books while spending the money.
>
>Now what about the opposite statement:
>"You sell 5 books priced at $10 each. How much did you earn?" Note that the
>book price remains the same (negative $10!), so changing "buy" to its
>opposite "sell" introduces a second negative sign. The number of books is
>now negative since the seller is giving up the books. Thus we have the
>calculation
>(-5 books) x (-$10) = +$50. We know the result must be positive because it
>represents income. Most people will automatically change the double
>negatives and only write 5 x 10 = 50. This verifies that the product of two
>negative numbers must be positive in a common sense argument that students
>find convincing.
>
>
>George Alexander
>University of Wisconsin Colleges
>Developmental Math Coordinator
>UW Rock County
>2909 Kellogg Ave.
>Janesville, WI 53546-5699
>(608) 758-6627
First, with respect to the example given above, I am unconvinced. There are
5 books (+5) whether they are coming or going. Moreover, you are not really
dealing with the books, but with the money which was handed over in the
other direction. Outgoing = expense = negative. Incoming = revenue =
positive.
Buying 5 books @$10 means there is an outward flow of $10 for each of 5
books, hence 5(-$10) = -$50.
Selling 5 books @$10 means there is an inward flow of $10 for each of 5
books, hence 5(+$10) = +$50
I see no double negative here.
To me, a more reasonable example (although I'm sure there might be better
ones) is to consider the activies involved in keeping a ledger of some sort
in which debits are negative and credits are positive. The activity of
entering might be considered positive while that of removing is negative.
This produces a sort of 2x2 matrix of possible activities in which the
entries are the net effect on the account of multiplying the sign of
(enter/remove) with the sign of (credit/debit):
credit debit
enter (+,+) = + (+,-) = -
remove (-,+) = - (-,-) = +
Thus entering a credit of $10 has the same net effect on the value of the
account as removing a debit of $10.
If we can equate taking the "opposite of" with multiplying by -1, then
there are many real life examples of two-state systems in which taking the
opposite twice returns you to the original state, -e.g., a light switch
(particularly in those situations where you can control the light from
either switch).
I think it is harder to find simple real life examples for products like
-3(-4).
Clearly they are not immediately obvious or the question wouldn't arise.
I would like to address the question somewhat differently. I am somewhat
bothered by the need to justify everything in terms of its immediate
application to "real life," whatever that is.
First of all, would a business calculus problem involving limited growth
modeled by an equation of the form P = L - ae^(-kt) qualify as "real world"
whether or not everyone is going to want to be able solve this? Looking at
an example in some book of the form
S = 5500 - 5000e^(-.04t), letting S = 2750, the solving leads to t = (ln
0.55)/(-0.04) = approx 15 days. Of course, real life is probably much more
complicated.
It seems to me that in teaching math we are also teaching something about
the different number systems and how and why they arise. It seems to me
that it is worth pointing out to students that math often takes basic
simple ideas which are clearly real life and available to us at an
intuitive level, abstracts them, and then "pushes" that abstraction as far
as it can - way beyond the bounds of intuition.
Thus, if multiplication arises as repeated multiplication, n*m meaning the
sum of n numbers all of which are m (or more concretely, the total number
of objects when you combine n piles of m objects each.) It is still
reasonably intuitive to talk about 3(-400) once we have given some concrete
meaning to -400. (The total debit due to 3 identical debits of 400, or the
total descent in feet achieved by descending in three stages of 400 feet
each.
It is less obvious what -3(400) or even -400(3) means. We tend to justify
this on the basis of extending the commutative property, or by showing the
pattern in a sequence of products. Finally, to justify (-3)(-2) most books
now look at sequences of products. Of course, if we accept that -3 = -1(3),
etc.
Likewise, when we move into negative exponents in Elem. Algebra (or even 0
as an exponent) we are moving from a basically intuitive notion (shorthand
for repeated multiplication) in which the exponent represents the number of
identical factors to a purely formal definition which is intended to expand
the notion (and notation) in a way which is consistent with the
already-established rules of exponents and also the results of working with
ratios of exponentials with the same base but higher exponent in the
denominator.
Just some thoughts.
Bob Leibman
Austin Community College
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