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Re: neg * neg = pos; why?
Posted:
Jul 22, 1997 6:39 PM
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Hello y'all,
I am sure there are many ways to explain why negative times negative equals
positive. I give three "reasons" to my students, each of which I hope will
provide them with an opportunity to reflect upon an aspect of mathematics.
I do not remember where I saw these in the past, so I will not attempt to
reference them. I suspect most of you know and use these, but it is the
middle of July and I am sitting in my office, and what the heck, I'll take
this as an opportunity for a diversion. By the way, I find it interesting
to note that many of my preservice teachers are left dissatisfied by the
first two "explanations." The third seems to satisfy more.
Be well,
Randy
1) It is defined this way because it works. Sometimes mathematical
definitions are designed to maintain the system. For example, it is
difficult to imagine why 0! should equal 1, but by so defining it this way
many of our formulas continue to work (e.g.,. consider the case where n=r
in the formula for the # of combinations of n objects taken r at a time: n!
/ [r!(n-r)!)]
2) Consider the pattern:
3 x -4 = -12
2 x -4 = -8
1 x -4 = -4
0 x -4 = 0
-1 x - 4 = ?
What would you expect this to equal? Why?
3) Consider the situation:
+ $50 = increase of 50 dollars per day; - $50 = decrease of 50 dollars per day
+ 3 = three days in the future -3 = three days in the past
CASE 1 (I have a job and am earning money.)
If I increase my amount by 50 dollars a day (+$50), what will my situation
be 3 days in the future (+3)? I will be $150 richer than I am today.
(+ $50) x (+ 3) = (+$150)
CASE 2: (I am on vacation and am spending money.)
If I decrease my amount by 50 dollars a day (-$50), what will my situation
be 3 days in the future (+3)? I will be $150 poorer than I am today.
(-$50) x (+3) = (-$150)
CASE 3: (I am back at my job and earning money.)
If I increase my amount by 50 dollars a day (+$50), what was my situation 3
days in the past (-3)? Three days ago I was $150 poorer than I am today.
(+ $50) x (- 3) = (-$150)
CASE 4: (I am back on vacation and am spending money.)
If I decrease my amount by 50 dollars a day (-$50), what was my situation 3
days in the past (-3)? Three days ago I was $150 richer than I am today.
(- $50) x (- 3) = (+$150)
Randolph Philipp, Associate Professor
School of Teacher Education, and
Center for Research in Mathematics and Science Education (CRMSE)
San Diego State University
Mailing address:
CRMSE
6475 Alvarado Road, Suite 206
San Diego, CA 92120
U.S.A.
Telephone: 619-594-2361
Fax: 619-594-1581
EMAIL: RPhilipp@mail.sdsu.edu
Web Site: http://public.sdsu.edu/CRMSE/rphilipp.html
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