When a minus times a minus equals a plus

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 sweets\month. 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?

The views expressed here are the personal views of the author.

Posted on September 19, 2009 at 11:22 am by Paul Brabban · Permalink
In: maths · Tagged with: 

18 Responses

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  1. Written by Walking Randomly » Carnival of Maths #58
    on September 27, 2009 at 9:36 pm
    Permalink

    [...] we have a submission from Paul Brabban who’s non-mathmo fiancee asked him ‘Why is a minus times a minus equal to a plus?’  How would you answer this question if [...]

  2. Written by Joe
    on September 28, 2009 at 1:17 am
    Permalink

    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.

  3. [...] completely intuitive and so far the best I have come across is the explanation from my friend Paul over at Crossed Streams who’s fiancee asked us this very question while we were all on a night out.  Other good [...]

  4. Written by Minus times minus is plus « The Math Less Traveled
    on September 29, 2009 at 12:56 pm
    Permalink

    [...] at crossedstreams.com gives an explanation involving two quantities with real-world interpretations associated with negative values: net worth [...]

  5. Written by Ludwig
    on September 29, 2009 at 6:17 pm
    Permalink

    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”. :)

  6. Written by Paul Brabban
    on September 29, 2009 at 9:45 pm
    Permalink

    Hey @Joe, @Ludwig, thanks for taking the time! Nice alternate views.

  7. Written by Negative times negative « The Number Warrior
    on September 29, 2009 at 10:57 pm
    Permalink

    [...] Streams: When a minus times a minus equals a plus Walking Randomly: Why is a ‘minus times a minus equal to a plus’? Math Less Traveled: Minus [...]

  8. [...] At Crossed Streams [...]

  9. Written by Gavin
    on September 30, 2009 at 3:11 pm
    Permalink

    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.

  10. Written by Denise
    on October 3, 2009 at 3:38 pm
    Permalink

    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!!

  11. Written by Dave
    on October 6, 2009 at 3:12 am
    Permalink

    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.

  12. Written by maths 10/07/2009 | Dave
    on October 7, 2009 at 2:20 am
    Permalink

    [...] crossedstreams.com » When a minus times a minus equals a plus [...]

  13. Written by Jack
    on May 10, 2010 at 6:44 am
    Permalink

    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

  14. Written by Jack
    on May 23, 2010 at 11:02 am
    Permalink

    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 ‘-’
    ————————————————-

  15. Written by Abhishek
    on December 1, 2010 at 9:01 am
    Permalink

    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.

  16. Written by simon lee
    on April 6, 2011 at 3:04 am
    Permalink

    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

  17. Written by Stefan Hyltoft
    on August 18, 2011 at 6:53 pm
    Permalink

    simon lee :
    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.

  18. Written by Shantala
    on June 25, 2013 at 3:04 pm
    Permalink

    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.

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