<<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.>>
I would argue that the $10 is only -$10 to the bookseller who is now selling
the book back to you. Relative to your point of view, the result is +$10
(income to you).
Whether something is positive or negative is a result of the frame of
reference in which you are viewing the problem. The sign change error can
be thought of as a switch in the point of view occurring during the analysis
of the problem.
> ----------
> From: George Alexander[SMTP:galexand@UWC.EDU]
> Reply To: George Alexander
> Sent: Wednesday, November 10, 1999 9:29 AM
> To: 'Bob Leibman'; mathedcc@archives.math.utk.edu
> Subject: RE: [MATHEDCC] Marilyn Vos Savant
>
> I appreciated Bob Leibman's comments on multiplying negative numbers. He
> has
> added some worthwhile thoughts.
>
> 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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