-2 x -3 = +6
A.S. Chamove
multiply two negatives you get a positive result?
--
-----------------------------------------------------------------
Arnold Chamove
Massey University Psychology
Palmerston North, New Zealand
A.S. Chamove
Craig Powderkeg DeForest
Can anyone give a simple, common-sense explanation as to why when you
multiply two negatives you get a positive result?
For the same reason that you get a positive when you multiply two positives!
Multiplication does an exclusive-OR operation on the sign of the numbers.
That's just how it is.
What else would you want to get? Another negative? Then multiplication
by a constant wouldn't be reversible: you'd have
-2 x -3 = -6
-2 x +3 = -6
so you couldn't switch the terms around to find that
-2 = -6 / -3
for example, because
+2 = -6 / -3
would be equally valid.
--
DON'T DRINK SOAP! DILUTE DILUTE! OK!
John C. Fowler
>Can anyone give a simple, common-sense explanation as to why when you
>multiply two negatives you get a positive result?
This is difficult only because you don't often run into negative numbers
in real life. But I'll try.
Say you have a car that can travel a straight highway east or west, at
an average speed of 60 miles per hour. Let's orient our coordinate
system so that 0 is at the car's starting point, East is positive x,
and West is negative x.
So say you travel East at 60 miles/hour for one hour. Your end position
will be 60. If you had traveled West, you'd be at -60. So in a sense,
your position is 60 x 1 if your car is facing East, and 60 x -1 if your
car is facing West.
{ 1 if car is facing East
60 x <
{ -1 if car is facing West
Now say some vandal redid your engine so that the car only runs in
reverse. Now, your average velocity will be -60 instead of 60, because
you're going backwards. If your car is pointing East, you will end
up at -60, and if your car is pointing West, you will end up at 60.
{ 1 if car is facing East
-60 x <
{ -1 if car is facing West
The 1 and -1 haven't changed because we didn't do anything to the
coordinate system: only our car. Hence, -60 x -1 = 60.
--
John C. Fowler, fow...@boulder.colorado.edu
CSM...@mvs.oac.ucla.edu
So 3 * 6 is really 0 + 6 + 6 + 6
3 * -6 is really 0 + (-6) + (-6) + (-6)
-3 * -6 is really 0 - (-6) - (-6) - (-6)
Pete Moore
>Can anyone give a simple, common-sense explanation as to why when you
>multiply two negatives you get a positive result?
It's just like double negatives in English, something with which I'm sure
you're not unfamiliar? 8-)
2 x 3 = 6 Familiar?
(-2) x 3 = -6 `unfamiliar' means the opposite of `familiar'
2 x (-3) = -6 `not familiar' means the opposite of `familiar'
(-2) x (-3) = 6 `not unfamiliar' means the same as `familiar'
--
+------------------- pe...@bignode.equinox.gen.nz -------------------+
| The effort to understand the universe is one of the very few things |
| that lifts human life above the level of farce, and gives it some |
| of the grace of tragedy - Steven Weinberg |
+---------------------------------------------------------------------+
Roland Hutchinson
Or, making the above explanation a bit less abstract (and thereby even
more commonsensical, I hope): think of +6 as a credit of six dollars
on an account, -6 as a debit on the account.
3 * 6 ==> add a credit of 6 dollars three times ==> you're up by $ 18
3 * -6 ==> add a debit of 6 dollars three times ==> you're down by $18
-3 * 6 ==> remove a credit of 6 dollars three times ==> down by $18
-3 * -6 ==> remove a debit of 6 dollars three times ==> up by $18
--
Roland Hutchinson Visiting Specialist/Early Music
Internet: rhut...@pilot.njin.net Department of Music
Bitnet: rhutchin@NJIN Montclair State College
Upper Montclair, NJ 07043
Carl J Lydick
>Can anyone give a simple, common-sense explanation as to why when you
>multiply two negatives you get a positive result?
Start with a pile of tokens. Take away 2 of them. Repeat the operation three
times (That's multiplying a positive number by a negative number). OK, now,
invert the operation (that's multiplying a negative number by a negative
number). QED.
--------------------------------------------------------------------------------
Carl J Lydick | INTERnet: CA...@SOL1.GPS.CALTECH.EDU | NSI/HEPnet: SOL1::CARL
Disclaimer: Hey, I understand VAXen and VMS. That's what I get paid for. My
understanding of astronomy is purely at the amateur level (or below). So
unless what I'm saying is directly related to VAX/VMS, don't hold me or my
organization responsible for it. If it IS related to VAX/VMS, you can try to
hold me responsible for it, but my organization had nothing to do with it.
Rob Chansky
>....
>--
>DON'T DRINK SOAP! DILUTE DILUTE! OK!
And while we're at it, how about this guy, Dr. Bronner? (For those who don't
know, he makes a brand of soap you're likely to see in small health-food
stores around the country. The attraction here is the label: pages of tiny-
print exhorting Bronner's version of God's own truth. Maybe I'll post some
of it, but to get the full effect you have to get a bottle and read it in
the shower one bleary-eyed morning.)
Anyone have insights into Dr. Bronner's character? Later on I'll tell you
what little I know..
Rob
bay...@nebula.ssd.lmsc.lockheed.com
>Can anyone give a simple, common-sense explanation as to why when you
>multiply two negatives you get a positive result?
>
that way so that mathematical operations work consistantly regardless of
whether you are working with positive or negative numbers.
For example, a*(b+c)=a*b+a*c regardless of whether a,b, or c.
is positive or negative.
Or better yet, (x+1)^2 = x^2 + 2x + 1).
This is the same type of reasoning by which X^0 (X raised to the zero
power) is always equal to 1.
Once you define the concepts of multiplication and of negative
numbers, the rest follows.
Craig Powderkeg DeForest
zo...@daedalus.stanford.edu (Craig "Powderkeg" DeForest) writes:
>....
>--
>DON'T DRINK SOAP! DILUTE DILUTE! OK!
And while we're at it, how about this guy, Dr. Bronner?
Dr. Einstein Heilbronner, Soapmaker, is dead. His son runs their business
out of (I believe) Escondido, CA. They make a number of products, including
corn chips (HEALTHY HUNZA FOOD), corn puffs (HEALTH IS OUR ONLY REAL WEALTH),
and some other packaged foods.
I forget the son's name. I think He's a Heilbronner too. They had a
special issue of the bottle, in around 1982 or 1983, that explained it
all. Of course, that was back before they came out with these newfangles
eucalyptus and jasmine and etc. flavors of soap: peppermint used to be
the only one you could get.
Geoff Lane
: Can anyone give a simple, common-sense explanation as to why when you
: multiply two negatives you get a positive result?
Think of a point in the middle of a field. Call north the +ve direction,
south the -ve; east the +ve west the -ve. Walk 2 steps south and 3 steps
west.
What is the area of the rectangle on the ground you have just walked around
two sides of? 6 square `steps'.
The assignment of +ve and -ve are arbitary, it works so long as you are
consistent.
--
Geoff. Lane. | Internet: zza...@uts.mcc.ac.uk | Janet: zza...@uk.ac.mcc.uts
UTS Sys Admin, Manchester Computing Centre, Oxford Rd, Manchester, M13 9PL
"If the code and the comments disagree, then both are probably wrong."
-- Norm Schryer
wede...@vxdesy.desy.de
> In article <1993Mar16.2...@massey.ac.nz>, ACha...@massey.ac.nz (A.S. Chamove) writes:
>>Can anyone give a simple, common-sense explanation as to why when you
>>multiply two negatives you get a positive result?
>>
> The one line answer is that it was designed by mathemeticians to work
> that way so that mathematical operations work consistantly regardless of
> whether you are working with positive or negative numbers.
> For example, a*(b+c)=a*b+a*c regardless of whether a,b, or c.
> is positive or negative.
> Or better yet, (x+1)^2 = x^2 + 2x + 1).
This is of course a correct answer. But there is also a simple "common sense"
explanation of this.
For example: If you add -2 bank accounts to your posessions (i.e. you give
away two bank accounts) which contain -100$ each, then you earned
(-2)*(-100$)=+200 $. Is'nt this logical?
(Well, you could also say, you _lost_ +2 bankaccounts and thus you _lost_
(+2)*(-100$)=-200$. Which is the same as to say you won +200$! - As said above,
it is all a question of consitancy!)
Greets from hamburg, germany! Volker
lrud...@vax.clarku.edu
>A.S. Chamove (ACha...@massey.ac.nz) wrote:
>: Can anyone give a simple, common-sense explanation as to why when you
>: multiply two negatives you get a positive result?
>Think of a point in the middle of a field. Call north the +ve direction,
I was told we wouldn't have to do field theory in this newsgroup.
Lee "in fact it was supposed to be a GUT" Rudolph
Andrew C. Plotkin
Rob Cha...@adobe.com (690)
> And while we're at it, how about this guy, Dr. Bronner? (For those who don't
> know, he makes a brand of soap you're likely to see in small health-food
> stores around the country. The attraction here is the label: pages of tiny-
> print exhorting Bronner's version of God's own truth. Maybe I'll post some
> of it, but to get the full effect you have to get a bottle and read it in
> the shower one bleary-eyed morning.)
He's a loon. And the world absolutely needs more loons like him.
(Especially if he's dead now, as someone posted. :-( )
--Z
Darryll Hobson
>> Can anyone give a simple, common-sense explanation as to why when you
>> multiply two negatives you get a positive result?
>zza...@uts.mcc.ac.uk (Geoff Lane) writes:
>Think of a point in the middle of a field. Call north the +ve direction,
>south the -ve; east the +ve west the -ve. Walk 2 steps south and 3 steps
>west.
>
>What is the area of the rectangle on the ground you have just walked around
>two sides of? 6 square `steps'.
This explanation is a joke...
What if I walk 2 steps south and then 3 steps east?? The area of the
rectangle is still gonna be 6, but (-2)*(3) = -6.
Carl J Lydick
=>Think of a point in the middle of a field. Call north the +ve direction,
=>south the -ve; east the +ve west the -ve. Walk 2 steps south and 3 steps
=>west.
=>
=>What is the area of the rectangle on the ground you have just walked around
=>two sides of? 6 square `steps'.
=
=This explanation is a joke...
=
=What if I walk 2 steps south and then 3 steps east?? The area of the
=rectangle is still gonna be 6, but (-2)*(3) = -6.
That means the turf's upside-down (remember cross-products and how they work
:-).
Robert Roth
AC> Newsgroups: alt.folklore.science
AC> Subject: -2 x -3 = +6
AC> Message-ID: <1993Mar16.2...@massey.ac.nz>
AC> From: ACha...@massey.ac.nz (A.S. Chamove)
AC> Date: Tue, 16 Mar 93 23:46:35 GMT
AC> Organization: Massey University, Palmerston North, New Zealand
AC> Can anyone give a simple, common-sense explanation as to why when you
AC> multiply two negatives you get a positive result?
I tell my students that -3 could represent a debt of 3 dollars. The quantity
-2 x -3 will take away two debts of 3. What will take away two debts of 3?
Why +6 does this very nicely. Therefore -2 x -3 = +6
... "You keep saying that, I don't think it means what you think it means"
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Mike Ross
)In article <1993Mar16.2...@massey.ac.nz>, ACha...@massey.ac.nz (A.S. Chamove) writes:
)>Can anyone give a simple, common-sense explanation as to why when you
)>multiply two negatives you get a positive result?
)
)Start with a pile of tokens. Take away 2 of them. Repeat the operation three
)times (That's multiplying a positive number by a negative number). OK, now,
)invert the operation (that's multiplying a negative number by a negative
)number). QED.
)--------------------------------------------------------------------------------
)Carl J Lydick | INTERnet: CA...@SOL1.GPS.CALTECH.EDU | NSI/HEPnet: SOL1::CARL
There's also the direction approach. The first negative can be thought of
as facing backward, then multiplying by another negative is like being told
to face backward again, turning you around to positive, or 'forwards'. In
vectorial terms, those numbers are vectors in a 1-dimensional space, like
a line.
-mike
--
******************************** mi...@drseus.jsc.nasa.gov *****
* Michael L. Ross/C33 | Lockheed Engineering & Sciences Co. *
* Robotics Department | 2400 Nasa Rd. 1, Houston, TX 77062 *
*(713)333-7094 voice,(713)333-7201 fax**********boring, eh?****
Elizabeth Schwartz
LISTEN CHILDREN ETERNAL FATHER ETERNAL ONE WE'RE ONE ALL ONE
EXCEPTIONS ETERNALLY? NONE! ABSOLUTE NONE!
But here's something for alt.folklore.science, from the same bottle:
"Essene & Chinese birth controls must reduce birth or Easter Isle type
overpopulation destroys God's Spaceship Earth! God's Law prevents all
contraception below PH3. Therefore, Essene contracepted for 400 yrs.
with rosehips, pH2! So, absolute clean, apply vaseline, oil, butter,
or cream, insert teaspoonful juicy lemon pulp, pH2. OK! Next day,
douche with quart soapy water, pH8, restoring pH5 balance God made!
Eggwhite is pH9, Dr Bronner's Soap, pH8, guaranteed the mildest made;
below pH8 soaps biodegrade synthetic sulfides cannot. At conception,
10 grams contain 100 million humans...or10 humans in one invisible
microgram-- smaller than dust!" -- Dr Bronner's 8oz peppermint soap.
So does this really work? You sure don't see too many Essenes around
these days.
--
System Administrator Internet: bet...@cs.umb.edu
MACS Dept, UMass/Boston Phone : 617-287-6448
100 Morrissey Blvd Staccato signals
Boston, MA 02125-3393 of constant information....
Daniel F Boyd
> multiply two negatives you get a positive result?
Like everything else in mathematics, it's a convention. It follows
from the rules.
Feel free to define a system of mathematics (your own set of rules)
where it doesn't work that way; if you can find something to model
with your new system, great! If you can't, great!
But remember that conventions in mathematics tend to get adopted on
the basis of how useful they are.
What kind of result would you like it to be?
--
Daniel F. Boyd -- bo...@cs.buffalo.edu
"Here, a piece of the True Cross. And in this case, we have the skull
of John the Baptist at the age of twelve."