# Why is negative x negative = positive? **URL:** \ **Category:** Factual Questions\ **Created:** [January 18, 2010, 3:44pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428 "2010-01-18T15:44:10Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![smiling\_bandit](https://avatars.discourse-cdn.com/v4/letter/s/e9a140/32.png) [@smiling\_bandit](https://boards.straightdope.com/u/smiling_bandit)\ **Post date:** [January 19, 2010, 5:34pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/21 "2010-01-19T17:34:07Z")
> [@Animastryfe](#): > > Nice. I haven’t seen a reductio ad absurdum proof of this before. yeah. You can construct a more consistent version… but sooner of later (and in most cases soner) it winds up with a completely illogical result which goes nowhere. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 19, 2010, 6:10pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/22 "2010-01-19T18:10:41Z")
> [@RealityChuck](#): > > Assume a negative number times a negative is a negative. > > Thus, -1 X -1 = -1 > > Now we know that -1 x 1 = -1 > > Then it follows -1 x -1 = -1 x 1 > > Divide by -1: -1/-1 x -1 = -1/-1 x 1 > > Since a/a = 1: 1 x -1 = 1 x 1 > > and thus -1 = 1 > > This is an impossibility, so a negative times a negative must be a positive. > [@pulykamell](#): > > Nice, but wouldn’t making the assumption that muliplying a negative by a negative makes a negative also suggest that _dividing_ a negative by a negative makes a negative, too? (if someone thinks multiplying a negative by a negative is a negative, why would they think dividing a negative by a negative is a positive? Or, more simply, division is multiplication by its reciprocal. The reciprocal of -1 is -1, so -1/-1 equals -1\*-1, which we already decided is -1.) That’s a good point, but in that case you’re breaking the “rule” that **any** nonzero number, divided by itself, yields 1. That is, in algebraic terms, every nonzero number should have a multiplicative inverse. So if _ab_ = _ac_ then _a_[sup]–1[/sup]_ab_ = _a_[sup]–1[/sup]_ac_ and so _b_ = _c_. In this case, _a_ = –1, _b_ = –1, and _c_ = 1. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 19, 2010, 6:18pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/23 "2010-01-19T18:18:55Z")
> [@pulykamell](#): > > Nice, but wouldn’t making the assumption that muliplying a negative by a negative makes a negative also suggest that _dividing_ a negative by a negative makes a negative, too? (if someone thinks multiplying a negative by a negative is a negative, why would they think dividing a negative by a negative is a positive? Or, more simply, division is multiplication by its reciprocal. The reciprocal of -1 is -1, so -1/-1 equals -1\*-1, which we already decided is -1.) Alright, but then the real absurdity becomes that division doesn’t cancel out multiplication (e.g., one wants a \* b / b = a, and thus 1 \* b / b = 1, and thus b / b = 1, in contradiction to (-1)/(-1) = 1). (I must be very slow in making these posts, because I see now that **Thudlow Boink** has made the same point) Regardless, the reduction proof is just a vaguer, more roundabout way of illustrating what’s given more constructively, I feel, by the approach of **ultrafilter** et al: -1 \* -1 = (0 - 1) \* -1 = (0 \* -1) - (1 \* -1) = 0 - (-1) = 1. As far as “algebraic” approaches go, I prefer this more constructive demonstration; it seems to illustrate more where things “come from”, without requiring any starting assumption “Well, -1 \* -1 is either 1 or -1”. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 19, 2010, 6:24pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/24 "2010-01-19T18:24:12Z")
Er, “reductio”, not “reduction” ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [January 19, 2010, 6:46pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/25 "2010-01-19T18:46:33Z")
> [@Thudlow\_Boink](#): > > That’s a good point, but in that case you’re breaking the “rule” that **any** nonzero number, divided by itself, yields 1. > That is, in algebraic terms, every nonzero number should have a multiplicative inverse. > > So if _ab_ = _ac_ > then _a_[sup]–1[/sup]_ab_ = _a_[sup]–1[/sup]_ac_ > and so _b_ = _c_. > > In this case, _a_ = –1, _b_ = –1, and _c_ = 1. Sure, I know that and you know that, but if you have someone who doesn’t believe a negative multiplied by a negative makes a positive, I’m not sure who you can convince them that -1 divided by -1 is positive one. Even if they think they know the rule a number divided by itself equals 1, the whole -1/-1 would probably be as non-intuitive as -1 \* -1 = +1. In other words, I don’t think the explanation will go far to clarify the situation. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 19, 2010, 6:52pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/26 "2010-01-19T18:52:02Z")
To such a person, the reductio argument won’t do much good either, since it makes similar use of division cancelling out multiplication. I think you’d have to either demonstrate A) here, if we want division to cancel out multiplication, it has to go like this (and then they’re no longer a problematic case), B) forget the reductio, just demonstrate a pattern (e.g., the pattern m \* (n - 1) = m \* n - m) and suggest extending the pattern, or C) you’d use the explanation based on 180 degree turns. ---
**Author:** ![hooahcrackers](https://avatars.discourse-cdn.com/v4/letter/h/94ad74/32.png) [@hooahcrackers](https://boards.straightdope.com/u/hooahcrackers)\ **Post date:** [December 30, 2011, 12:56am UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/27 "2011-12-30T00:56:03Z")
the mind metaphorically conceptualizes numbers based upon our bodily experiences. Math is a metaphor for creation. Therefore, what bodily experience is a negative number times a negative number? The left side of the math represents a prior experience of a collection of objects. The operator (multiplication) tells us that we are behaving to add a series of collections. The right side of the math tells us to execute this behavior X number of times. The left side, as a negative number, is the experience of debt. Adding a collection of debts multiple times gives you a larger collection of debt. Debt requires you to understand the concept of possession and therefore, the lack of possession. The right side, as a negative number, adds a series of collections opposite of the identity of the left side. Therefore, a debt of a collection of objects (a negative number) times -the opposite of- adding a series of collections of this debt equals, or -creates-, what would be a positive collection. This is like thinking, “each month I owe 5 dollars, I paid this debt for 12 months, what money is created if I traveled backwards in time collecting 5 dollars each month?” ---
**Author:** ![Spectre\_of\_Pithecanthropus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spectre_of_pithecanthropus/32/12343_2.png) [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)\ **Post date:** [December 30, 2011, 7:44am UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/28 "2011-12-30T07:44:29Z")
> [@Finagle](#): > > Multiplying a “hole in the ground” by a negative 1 is, in your metaphor. the equivalent of replacing the hole with the bucket of sand that you originally dug out of the hole. Wouldn’t multiplying a “hole in the ground” by negative one not only refill the original hole, but additionally result in a new pile of dirt where the hole used to be, and equal to it in volume? I think a better analogy is to accept as given that scalar numbers have signs denoting directionality on the line of real numbers. The number itself is only an absolute value of distance while the sign tells us the direction. If we pick an ordinary positive integer, say 9, then it’s easy to think of it as nine units of positive magnitude. Multiplying by -1 inverts that and gives us nine units of negative magnitude. For me at least, the rules of how positive and negative numbers interact make a lot more sense when I start with my simple example. ---
**Author:** ![MikeF](https://avatars.discourse-cdn.com/v4/letter/m/45deac/32.png) [@MikeF](https://boards.straightdope.com/u/MikeF)\ **Post date:** [December 30, 2011, 3:44pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/29 "2011-12-30T15:44:37Z")
Simple. Something that’s not not, is. ---
**Author:** ![Indian](https://avatars.discourse-cdn.com/v4/letter/i/c0e974/32.png) [@Indian](https://boards.straightdope.com/u/Indian)\ **Post date:** [December 30, 2011, 3:54pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/30 "2011-12-30T15:54:12Z")
But zombies do exist! ---
**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\ **Post date:** [December 30, 2011, 4:05pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/31 "2011-12-30T16:05:37Z")
> [@Spectre\_of\_Pithecanthropus](#): > > Wouldn’t multiplying a “hole in the ground” by negative one not only refill the original hole, but additionally result in a new pile of dirt where the hole used to be, and equal to it in volume? Depends on if the hole was already there to start with or if its digging was a “pending instruction”. That is, were we adding (-1 x -1) to a flat ground (0) or an already-present hole (-1)? If the latter, then we’d just be filling in a hole, resulting in a total of 0, as **Finagle** said. If the former, then that hole never existed and doesn’t need to be filled in, but would result in a pile of dirt like you describe. Glad that’s cleared. Now let’s put the zombie down. ---
**Author:** ![KneadToKnow](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kneadtoknow/32/3999_2.png) [@KneadToKnow](https://boards.straightdope.com/u/KneadToKnow)\ **Post date:** [December 30, 2011, 4:08pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/32 "2011-12-30T16:08:28Z")
> [@Pedro](#): > > I think the most elementary way to look at this is that multiplying by -1 rotates you by 180º on the real line. So (-1)\*(-1) equals 360º and gets you back where you started (and so on and so forth). I believe this is exactly how it was explained to me in elementary school. Granted, if you heard the whole story of my elementary school mathematics experience, you’d shudder and run away, but I’m almost certain I remember this demonstration on the chalkboard. ---
**Author:** ![BlinkingDuck](https://avatars.discourse-cdn.com/v4/letter/b/eada6e/32.png) [@BlinkingDuck](https://boards.straightdope.com/u/BlinkingDuck)\ **Post date:** [December 30, 2011, 4:09pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/33 "2011-12-30T16:09:36Z")
I answered this question many times as a math teacher. The debt example worked ok. The hole in the ground not so much. The one I had the best luck is this… If the tempature has been dropping 3 degrees an hours and it is 45 degrees now…what was the tempature 5 hours ago? I know, sounds more complicated but for some reason people seemed to ‘get it’ better. ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [December 30, 2011, 4:47pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/34 "2011-12-30T16:47:28Z")
nm ---
**Author:** ![simster](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/simster/32/1047_2.png) [@simster](https://boards.straightdope.com/u/simster)\ **Post date:** [December 30, 2011, 5:29pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/35 "2011-12-30T17:29:13Z")
> [@MikeF](#): > > Simple. Something that’s not not, is. jethro bodine and his fancy 6th grade education rises again! ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [December 30, 2011, 5:40pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/36 "2011-12-30T17:40:35Z")
Another way of approaching this is from the [fundamental axioms of arithmetic](http://en.wikipedia.org/wiki/Peano_axioms) a_b = (2a-a)_(2b-b) \ = (2a_2b) + (2a)_(-b)+ (-a)_(2b) +(-a)_(-b) \ = (2a \* (2b+ (-b)) + (-a)_(2b) +(-a)_(-b) \ = (2a b) + (-a)(2b) +(-a)_(-b) \ = (a_(2b) ) + (-a)_(2b) +(-a)_(-b) \ = ((a+(-a))_2b) +(-a)_(-b) \ = (0_2b) +(-a)_(-b) \ = (-a)\*(-b) \ so for the axioms to hold (a_b ) = (-a)_(-b) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 30, 2011, 7:09pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/37 "2011-12-30T19:09:21Z")
Here’s what, I think, is a cleaner way to look at it from “fundamental axioms” (though what counts as “fundamental axioms” depends on what one is interested in): [There’s this slight notational messiness of using - both to indicate subtraction (as in 5 - 7) and negation (as in -7). I will use N(X) for the negation of X instead to make clearer the idea.] 0 = A \* 0 = A \* (B + N(B)) = A \* B + A \* N(B), no matter what A and B are. Accordingly, by subtracting A \* B from both sides, we have that N(A \* B) = A \* N(B). Thus, negating one of the factors in a product is as good as negating the whole product. This is the key realization from which everything follows. Doing this twice, once to each factor, we have that negating both of the factors in a product is as good as double-negating the whole product. [In symbols, N(N(A \* B)) = N(A) \* N(B)] Finally, double-negation is as good as doing nothing [if N(X) is X’s additive inverse, then X is N(X)'s additive inverse]. So negating both of the factors in a product is as good as keeping the product the same. ---
**Author:** ![shijinn](https://avatars.discourse-cdn.com/v4/letter/s/cc9497/32.png) [@shijinn](https://boards.straightdope.com/u/shijinn)\ **Post date:** [December 31, 2011, 1:59am UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/38 "2011-12-31T01:59:43Z")
would explaining it like this be helpful at all? 3 x 3 is 3 groups of 3 equals 9 3 x -3 is 3 groups of _the opposite of_ 3 (which is -3) equals -9 -3 x -3 is _the opposite of_ 3 groups of -3 equals 9 i like this one - > [@Trans Fat Og](#): > > First, take a photograph of **me**. (This is much easier to do these days because they’ve been building cameras sturdier since my youth. Ah, the wonders of modern materials. I can even have mirrors at my place nowadays.) > > Now, take the photo and hag it upside-down. It looks wrong right? Well, of course, it’s a photo of the ugliest guy on the boards, but I mean that it looks wrong because you are seeing my face upside-down. > > Now, without changing anything about the photo, look at it again, **while hanging upside-down**. > > That is your -1 times -1! ---
**Author:** ![TATG](https://avatars.discourse-cdn.com/v4/letter/t/50afbb/32.png) [@TATG](https://boards.straightdope.com/u/TATG)\ **Post date:** [December 31, 2011, 2:41am UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/39 "2011-12-31T02:41:37Z")
One could take division as basic, and explain multiplication through division. Divide a negative number by a positive and you have a negative number; this seems intuitively reasonable. Given that we want, if a/b = c, then a \* c = b, we have that multipling a negative number by a negative number gives you a positive number. A better expanation might explain the relationship between multiplication and divison. ---
**Author:** ![Spectre\_of\_Pithecanthropus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/spectre_of_pithecanthropus/32/12343_2.png) [@Spectre\_of\_Pithecanthropus](https://boards.straightdope.com/u/Spectre_of_Pithecanthropus)\ **Post date:** [January 1, 2012, 11:50pm UTC](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428/40 "2012-01-01T23:50:24Z")
> [@Chessic\_Sense](#): > > Depends on if the hole was already there to start with or if its digging was a “pending instruction”. That is, were we adding (-1 x -1) to a flat ground (0) or an already-present hole (-1)? If the latter, then we’d just be filling in a hole, resulting in a total of 0, as **Finagle** said. If the former, then that hole never existed and doesn’t need to be filled in, but would result in a pile of dirt like you describe. I assumed we start with a hole equivalent, as it were, to -9, and also that whoever dug the hole got rid of the dirt, so it doesn’t enter into the analogy. So if we merely _add_ +9 we just get the original flat ground. But multiply by -1 and we end up with a mound of dirt equal to the original hole. Again, we’re magically creating new dirt just as we didn’t worry about the other dirt that originally filled the hole. Come to think of it, I don’t think the hole-in-the-dirt analogy works very well when it comes to multiplication of negative numbers. For that it’s probably better just to stick with a more abstract model like a number line. [Previous page](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428.md?page=1) [Next page](https://boards.straightdope.com/t/why-is-negative-x-negative-positive/525428.md?page=3)