We all know from school that multiplying two negative numbers together gives a positive number, but can you think of a common-sense example of this rule in action that would convince someone who asked why?
I got asked why recently and I couldn’t.
Neither could anyone I asked. Plenty of examples involving mirrors and vectors etc. but nothing that didn’t sound rather like illusion and trickery. Nothing convincing.
So – after some thought, here’s an example that convinced the person who asked me. (Well, they say they’re convinced-ish, but I think that’s about as good as it’s going to get!)
This example is about getting two everyday, dependent variables that we can set a zero point on both and thus deal with the positive and negative values in both. Imagine I have a big bucket of sweets, and I have been giving you ten sweets a month for years.
How many more or less sweets do you have, six months from now?
Intuitively, you have 60 more sweets, and we can calculate that because you get +10 sweets/month and we want to know how many you have in 6 months;
10 x 6 = 60. (plus x plus = plus)
How about me? How many more or less sweets do I have, six months from now?
We know that I have 60 sweets less, because you have 60 sweets more. We can calculate this because I get -10 sweets/month;
-10 x 6 = -60. (minus x plus = minus)
That’s the easy ones done.
How many more or less sweets did you have, six months ago?
It should be easy to convince that if you have 60 sweets more in six months’ time, then you had 60 sweets less six months ago. We can calculate it using the same 10 sweets/month, but -6 months to go back in time.
10 x -6 = -60 (plus x minus = minus)
Finally, how many more or less sweets did I have, 6 months ago?
I still get -10 sweetsmonth. Just because we’re considering the past, you don’t start giving them to me or anything. In the case above, we used -6 to represent ‘six months ago’. So…
-10 x -6 = +60 (minus x minus = plus)
Which gives us the intuitively correct answer, that if I give you 10 sweets a month then I had 60 more sweets, six months ago.
It’s a tough one to argue with, because the answers are pretty obvious. Do you have a better way to explain why minus times minus equals a plus?
crossedstreams.com 
How’s this for a rationale for the influence of negative factors on the sign of a product (for the integers at least)?
If -a = 0 – a, and
if a * b = 0 + a + a + a + … + a wherein a appears b times as an addend,
then -a * b = (0 – a) * b = 0 + (0 – a) + (0 – a) + (0 – a) + … (0 – a) wherein 0 – a appears b times as a addend.
Hence -a * b = 0 – a – a – a – … – a wherein a appears b times as a subtrahend,
and thus -a * b = -(ab).
However if a * -b = 0 – a – a – a – … – a wherein a appears b times as a subtrahend,
then -a * -b = 0 – (0 – a) – (0 – a) – (0 – a) – … (0 – a) wherein 0 – a appears b times as a subtrahend,
then -a * -b = 0 + a + a + a + … + a wherein a appears b times as an addend,
and thus -a * -b = ab.
I’ve heard one which is sort of intuitive that might be interesting, perhaps you have too.
Suppose you have a transparent tank of water with an outlet with a tap at the bottom. You turn on the tap and the water begins to drain out of the tank.
Now you set up a video camera and film the tank draining out.
Take the tape and play it _backwards_.
Sort of like saying _backwards_ * _draining_ = filling where “backwards” and “draining” are “negatives” and “filling” is a “positive”. 🙂
Hey @Joe, @Ludwig, thanks for taking the time! Nice alternate views.
This is one of the approaches I take with my students, generally _after_ it’s been taught and I just want them to feel like it makes sense.
5 x 7 = 35
-5 x 7 must be -35
5 x -7 must be -35
The rationale: -5 x 7 can’t be the same as 5 x 7. You can’t just make one of the numbers negative and not have it affect the result.
Finally:
-5 x -7 can’t be -35 (that would be -5 x 7 or 5 x -7)
so it must be 35.
It’s kind of logical.
The best analogy I have heard is to imagine a money box stuffed with checks (payments to you, which increase your net worth and thus are positive) and bills (payments you owe, which decrease your net worth and thus are negative. Depending on how business goes each day, you add or subtract to your total value:
Received three checks for $5 each,
+3 x +5 = +15, net increase.
Gave back (refunded or canceled) three checks for $5 each:
-3 x +5 = -15, net decrease.
Received three bills for $5 each:
+3 x -5 = -15, net decrease.
Gave back (canceled) three bills for $5 each:
-3 x -5 = +15, net increase!!
Nice one Denise.
Maybe in a class you could have a number of fake notes – red being a $5 iou so the value is -$5, black being $5 cash. Students can work out their value based on how many of each they have. The teacher either takes some notes or gives some notes to students (of one colour). Taking is a negative operation so taking 3 means -3 * the value of the notes. So if the notes were black the student’s value would be reduced, if the notes were red the student’s value would be increased.
I think the idea is good – probably could have explained it better.
1 * a = a, – – ”here ‘a’ is any number”
hence, 1 * (-1) = (-1)
also -1 * a = -a, – -”here ‘a’ is any number”
hence, -1 * (-1) = – (-1)
Negative of +1 = -1
negative of (-1) = 1, because a number plus the negative of the
number adds up to zero. i.e. -1 + x = 0, that means x=1
From above, (-1) * (-1) = – (-1) = negative of (-1) = 1
That shows (minus * minus ) yields plus
Another Soln
This is not a proof, however a demonstration for a curious kid why negative times negative is positive. It may be helpful to think that when dealing with the numbers ‘-’ or ‘+’ signs are identical to ‘-1 times’ or ‘+1 times’. Here is an example:
-5 = ‘-1 times’ 5
A number plus a negative number can result zero. Lets take the number ‘1’ that is ‘+1’, then take ‘-1’. It could have been any other number. The MAIN assumption is:
1 -1 = 0
Without losing the effect we can times the equation by ‘-’ or ‘-1’ then:
-1 -(-1) = 0
Since both equations are the same, that means:
-1 -(-1) = 1 -1
that means:
-(-1) = 1
it shows:
‘-’ times ‘-’ is plus
—————————————————-
This approach is also helpful to show why ‘+’ times ‘-’ is ‘-’
Let us safely assume ‘1’, which is ‘+1’, times a number results in the number, i.e. ‘1 times’ 10 is 10. Then ‘1 times’ ‘-1’ is ‘-1’ . That is:
‘+1 times’ -1 = -1
That shows: ‘-’ times ‘+’ results ‘-’
————————————————-
I think the best answer is because ” It has been fixed to be so as a convention”. Negative numbers, like Complex numbers are after all artificial constructs used in the study of physical phenomena and maths by us, humans. Instead of looking too much to justify each and every one of our formulas and conventions, we should accept them to be so.
i got this explanation from 6th grade textbook
2 x 3 = 4
2 x 2 = 2
2 x 1 = 0
2 x -1 = -2
2 x -2 = -4
this tells you why + x – equals to negative
in a same way
-3 x 3 = -9
-3 x 2 = -6
-3 x 1 = -3
-3 x 0 = 0
-3 x -1 = 3
-3 x -2 = 6
the pattern explains – x – equals to positive
Thank you simon I have just been looking for this kind of example, an number line example sort of. All the other examples have been confusing or not common sense to me.
I attempted an answer to this question in my own way in my article :
https://ytelotus.wordpress.com/2013/06/24/arithmetic-operations/
Let me know if that makes sense.