Minus times minus is plus

I’ve seen several recent posts wondering about an intuitive explanation for the fact that “a minus times a minus equal to a plus”. I understand why people would wonder about this—the way it’s often taught, it seems to be just an arbitrary rule with no “why”! But in fact, it’s quite logical.

Paul at crossedstreams.com gives an explanation involving two quantities with real-world interpretations associated with negative values: net worth (negative means a decrease in net worth), and time (the past is negative). It’s fairly intuitive, but in some sense it only shows why it is nice that a minus times a minus is a plus, since it allows us to model this real-world situation; it doesn’t really show why it is true in a deep sense. Mike at Walking Randomly gives a proof essentially from the field axioms, but (as he himself admits) it is not particularly intuitive.

Here’s how I would explain it. It isn’t rigorous at all, and I’m not even completely satisfied with it, but I hope it is helpful for building some intuition.

Think about the familiar “number line”: positive numbers go off to the right, and negative numbers are to the left. Adding a positive number corresponds to moving right along the number line. Adding a negative number (that is, subtracting a positive one) corresponds to moving left along the number line. So with addition we already see this idea of negative corresponding to doing something “in the opposite direction”.

So, what does multiplication correspond to on the number line? Of course, multiplication corresponds to scaling, or stretching: for example, if we start at a point on the number line and multiply by 3, we will end up at a point three times as far from zero as we started. And what about multiplying by a negative number? It corresponds to a scale in the other direction: for example, if we start at a point on the number line and multiply by -3, we end up at a point on the other side of zero, and three times as far. That is, multiplying by a negative number means that we flip from one side of zero to the other. So, of course if we start from the left of zero (a negative number) and multiply by a negative, we end up on the right of zero (a positive number)!

[As an afterthought, I think trying to explain it any more deeply than this really does require bringing in the distributive property in some way---it is the only thing that formally connects addition (remember, negative numbers are defined as additive inverses) and multiplication. I really like the idea posted by Eric as a comment on Mike's post regarding an illustration of the distributive property.

Also, I meant to put nice pictures in this post, but for now if I wait to make nice pictures it will never happen. So I'll just post it for now, and maybe I'll come back and add some pictures later.]

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14 Responses to Minus times minus is plus

  1. I used to explain the fact “- x – = +” using the construction of Z out of N, at the same time explaining why this construction (N x N modulo equivalence) captures pretty natural our sense of negative numbers.

    I think the next question to your explanation would be: why is multiplication with -1 a reflection on the number line?

  2. Pingback: Negative times negative « The Number Warrior

  3. JM says:

    I learned it like this:

    When a factor decreases by 1, the product decreases by the amount of the other factor. Here all the products will decrease by 3.

    3×3=9
    3×2=6 (6 is 3 less than 9)
    3×1=3 (3 is 3 less than 6)
    3×0=0 (0 is 3 less than 3)
    3x-1=-3 (-3 is 3 less than 0)
    3x-2=-6 (-6 is 3 less than -3)
    3x-3=-9 (-9 is 3 less than -6)

    Now we can apply the same pattern, only to the other factor. The product will now decrease by -3, or increase by 3.

    3x-3=-9
    2x-3=-6
    1x-3=-3
    0x-3=0
    -1x-3=3
    -2x-3=6
    -3x-3=9

    I like this because it is a pattern that is both easy to see and obviously true. I had never seen your reasoning before, and I like it too.

  4. Pingback: Why a negative times a negative is a positive « Hans Gilde’s weblog

  5. Trouble with explaining it as a scaling operation is that while multiplying m by +n moves you by m(n-1), multiplying by -n moves you by m(n+1). Or alternatively, people might think that if multiplying by +n puts you n times further away from the origin, maybe multiplying by -n should put you n times *closer* – that is, it should represent division. Shades of raising something to a -ve power here.

  6. Brent says:

    JM: I like it, thanks!

    David: all good points. This is why I think ultimately an explanation in terms of distributivity is the only really satisfying answer.

  7. Fergal Daly says:

    If our sceptic accepts that – * + = – then to get the sign rule, distributivity is not needed.

    If you don’t have – * – = + then you can’t have inverses. The inverse of a -ve must be -ve otherwise X * inv(X) would be – * + which gives -ve but we know it’s +1. So inv(X) is -ve if X is -ve and so X * inv(X) is a collection of example of – * – = +. If that doesn’t satisfy our sceptic then the general case looks like

    -X * -Y
    = X * inverse(-X) * (some +ve scaling factor to get from inverse(-X) to -Y)
    = 1 * (the +ve scaling factor)
    = the +ve scaling factor

    so -X * -Y is +ve for any X and Y.

    This seems way too complex but the basic argument “if -*- wasn’t + inverses would be impossible” works intuitively.

    I don’t think you can get rid of the scaling factor without distributivity or a rule like |a*b| = |a|*|b| or even just -1 * 1 = -1 (are all of these equivalent to distributivity given all the other axioms, they seem a little weaker) Otherwise it could be that -1 * -1 = 16. This could be true if the the object you write as “-1″ is the object usually called “-4″. I don’t think you run into any contradiction if you do that (and you leave out the distributive axiom), just confusion.

  8. Gisele from Brazil says:

    It’s the first time I’ve read this blog, and I’ve really like this explanation using the numbers line – I’ve never though this way about the theme, and after reading this, it seems so obvious hehehe :P

    Congratulations for the blog! :)

  9. advolo says:

    1.) 1/2 times -4 2.)4/5 times -5 3.)-1/4j=2/3
    1.) -2
    2.) -1
    3.) j = -2 2/3
    ————————–

    [ (1/2) X (-4) ]
    [ (+0.5) X (-4) ]
    [ - { (+0.5) X (4) } ]
    [ - { (+0.5) X 4 } ]
    [ - { (+2) } ]
    [ - (+2) ]
    [ + (-2) ]
    [ (-2) ]

    [ (4/5) X (-5) ]
    [ (4/5) X (-5)/(+1) ]
    [ { (4/5) } X (-5)/(+1) ]
    [ { (+4)/(+5) } X (-5)/(+1) ]
    [ { (+4) X (-5) }/{ (+5) X (+1) } ]
    [ { (+4) X (-5) }/{ (+5) X (+1) } ]
    [ { - }/{ (+5) X (+1) } ]
    [ { - }/{ (+5) X (+1) } ]
    [ { - }/{ (+5) X (+1) } ]
    [ { - (+20) }/{ (+5) X (+1) } ]
    [ { + (-20) }/{ (+5) X (+1) } ]
    [ { (-20) }/{ (+5) X (+1) } ]
    [ (-20)/{ (+5) X (+1) } ]
    [ (-20)/{ + } ]
    [ (-20)/{ + } ]
    [ (-20)/{ + } ]
    [ (-20)/{ } ]
    [ (-20)/{ (+5) } ]
    [ (-20)/(+5) ]
    [ + { (-20)/(5) } ]
    [ + { (-20)/5 } ]
    [ { (-20)/5 } ]
    [ (-20)/5 ]
    [ (-4) ]

    EX :
    [ (Debt $20) / (5 Persons) = (Debt $4) ]
    [ (-20) / 5 = (-4) ]
    [ (-20) / (5) = (-4) ]

    [ (-20) / (+5) = (-4) ]
    [ + { (-20) / (5) } = (-4) ]

    [ + { (-20) / 5 } = (-4) ]
    [ { (-20) / 5 } = (-4) ]
    [ (-20) / 5 = (-4) ]

    [ - (+N) ] = [ + (-N) ] = [ (-N) ]
    [ - (-N) ] = [ + (+N) ] = [ (+N) ] = [ (N) ] = [ N ]

    Law of Liuhui Brahmagupta
    [ N X (+N) ] = [ + ( N X N ) ] = [ ( N X N ) ]
    [ N X (-N) ] = [ - ( N X N ) ]

    http://www.webanswers.com/education/homework-questions/one-half-times-negative-four-ccf3ce#reply

    Construction of theory in Negative number and Positive number, on Number line

    4 Basic Arithmetic

    5. Understanding of Yinyang, … ( YinYang = Harmony )

    First application of YinYang -> [ Negative number (Debt), Positive number (Fortune, Credit) ]
    First application of [ Negative number (Debt), Positive number (Fortune, Credit) ] -> [ Number Line ]

    YinYang is not (ONE dimension) but (THREE dimensions).
    YinYang can not be explained and can not be proved, by Number Line.
    It is a clear contradiction that YinYang can be explained and be proved, by Number Line.

    http://www.flickr.com/photos/trapassing/4247621320/sizes/o/

    http://www.flickr.com/photos/trapassing/4246850551/sizes/o/

    http://www.flickr.com/photos/trapassing/4246852489/sizes/o/

    http://knol.google.com/k/coupdetat-net/024-english-construction-of-theory-in/g7m8xnrw3ss1/25#

  10. Jagjit (Jack) says:

    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, for instance when -1 + x = 0, that means x=1
    From above, (-1) * (-1) = – (-1) = negative of (-1) = 1

    That shows (minus * minus ) yields plus

  11. Jagjit (Jack) says:

    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 ‘-’
    ————————————————-

  12. Nullius in Verba says:

    Draw a graph, for x in the range -5 to +5, of the functions
    y = 3x
    y = 2x
    y = 1x
    y = 1/2 x
    y = 0x
    and finally
    y = -1x

    The first few build up intuition that you get a straight line through the origin, and that y = kx passes through the point (1,k). The extension of the line to negative x is both visually plausible, and justified on the basis of taking a negative step a positive number of times. (Strictly, that’s multiplying an ordinal by a cardinal – two different types – but it’s highly unlikely that anyone will point that out.)

    Then as soon as you continue the progression to k negative, the result is obvious.

  13. sivaganesh.p says:

    little confusion about the concept

  14. Leona says:

    I’m confuffled!

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