Why is two negatives multiplied a positive

This proof of the Law of Signs uses well-known laws of positive integers esp. These fundamental laws of "numbers" are axiomatized by the algebraic structure known as a ring , and various specializations thereof. Since the above proof uses only ring laws most notably the distributive law , the Law of Signs holds true in every ring, e.

In fact every nontrivial ring theorem i. Without the distributive law a ring degenerates to a set with two completely unrelated additive and multiplicative structures. So, in a sense, the distributive law is a keystone of the ring structure. Indeed we have. Generally such functions enjoy only a weaker near -ring structure. In the above case of rings, distributivity implies that multiplication is linear hence odd viewing the ring in Cayley-style as the ring of endormorphisms of its abelian additive group, i.

I think a lot of answers are either too simple or stray away from mathematics too much. Just remember that multiplication is repeated addition. When dealing with negative numbers, it becomes repeated subtraction. This is really one of those important questions that leads many people to say "Math sucks!

In fact, for many students, mathematics stopped making sense somewhere along the way. In an article in the Journal of Mathematical Behavior, she described how it happened:. What did me in was the idea that a negative number times a negative number comes out to a positive number. This seemed and still seems inherently unlikely — counter intuitive, as mathematicians say. I wrestled with the idea for what I imagine to be several weeks, trying to get a sensible explanation from my teacher, my classmates, my parents, any- body.

Whatever explanations they offered could not overcome my strong sense that multiplying intensifies something, and thus two negative numbers multiplied together should properly produce a very negative result. I have since been offered a moderately convincing explanation that features a film of a swimming pool being drained that gets run back- wards through the projector. At the time, however, nothing convinced me. The most commonsense of all school subjects had abandoned common sense; I was indignant and baffled.

I would have to pay attention to the next topic, and the only practical course open to me was to pretend to agree that negative times nega- tive equals positive. The book and the teacher and the general consensus of the algebra survivors of so- ciety were clearly more powerful than I was. I capitu- lated. Underneath, however, a kind of resentment and betrayal lurked, and I was not surprised or dismayed by any further foolishness my math teachers had up their sleeves Intellectually, I was disengaged, and when math was no longer required, I took German instead.

There are many ways that we can use to show that result, but I'd like to show my personal way of thinking about the latter. Let's imagine we're sitting near a road, and there is a car that is moving with a constant speed. We also have a clock, and so we can measure time. Before going any further, we should first specify some assumptions like if the car is moving in the right, then its velocity will be positive, and if it's moving in the left direction, then its velocity will be negative.

Imagine now that you have a video of the above scene, and time is positive if you play the video normally but is negative if you play it backwards. Thus it moves "a positive distance". The elementary intuition behind the product of two negatives can be thought of as follows.

You have a bank account. This value should be positive since it results in you receiving money. Here's a proof. When you apply the two flips it gets you back to where you started because you flip to negative and then flip back. I think the x and y get in the way a bit; you can see the crucial steps using just 1 and You can film someone walking forwards positive rate or walking backwards negative rate.

Now, play the film back, but in reverse another negative rate. What do you see if you play a film backwards of someone walking backwards? Extend reals to the complex plane. One thing that must be understood is that this law cannot be proven in the same way that the laws of positive rational and integral arithmetic can be.

The reason for this is that negatives lack any "external" external to mathematics, ie. For example. Similar informal but entirely convincing, reasonable, and I would say irrefutable reasoning can be used to demonstrate the rules for manipulating positive fractions, say. Notice that in the above paragraph I used the fact that both positive integers and positive integer multiplication have pre-axiomatic, "physical" definitions. Ask someone why the product of two negatives is positive, and the best they can do is explain , not prove.

Another common one begins with "we would like the usual properties of arithmetic to hold, so assume they do Euler himself, in an early chapter of his textbook on algebra , gave the following supremely questionable justification. It remains to resolve the case in which - is multiplied by -; or, for example, -a by -b. With no disrespect to Euler especially consdiering this was intended as an introductory textbook , I think we can agree that this is a pretty philosophically dubious argument.

The reason it is impossible is because there is no pre-axiomatic definition for what a negative number or negative multiplication really is. Oh, you could probably come up with one involving opposite "directions", and notions of symmetry, but it would be quite artificial and not at all obviously "the best" definition.

In my opinion, negatives are ultimately best understood as purely abstract objects. It so happens - and this is quite myseterious - that these utterly abstract laws of calculation lead to physically meaningful results.

This was nicely expressed in by the mathematician John Playfair, when addressing the then controversial issues of negative and complex numbers:. Here then is a paradox which remains to be explained. If the operations of this imaginary arithmetic are unintelligible, why are they not altogether useless? Is investigation an art so mechanical, that it may be conducted by certain manual operations? Or is truth so easily discovered, that intelligence is not necessary to give success to our researches?

One way of approching the problem is with the idea that negative numbers are a different name for subtraction. The idea of negatives could be described as the insight that rather than having two operations and one type of number, we can have one operation and two types of number. But even that explanation doesn't altogether satisfy me. I've become convinced that my education cheated me on how deep an idea negative numbers are, and I expect to remain puzzled by them for many years.

Anyway, I hope some of the above is useful to someone. As for the product of two negatives being a positive, simply consider the multiplicative inverse:. I prefer the explanation by my favorite mathematician , V.

Arnold physicist really, since in his own words, "mathematics is a part of physics" and "an experimental science". Repeated addition allows us to multiply a positive number and a negative number. Using piles and holes this looks like:. Interpreting negative times a positive and negative times negative through repeated addition, however, is problematic. The truth is that multiplication has no meaning here in context of repeated addition. We have entered new territory and if we want to open up our world to new types of numbers it is not surprising that previously concrete, literal definitions begin to flail.

So we have to engage in a sophisticated shift of thinking, letting go of the question What is multiplication? How would we like multiplication to behave?

Comment: Let me stress this point. To approach this we first have to be clear on what features of arithmetic we feel should still be true. Such heartwarming nostalgia. I actually want to go back to school now. Thank you for this post. We start out learning about numbers, whole numbers, by adding and subtracting them.

We can picture and hold representations of them. Then we learn to multiply these positive numbers. Multiplication is a short-hand way of adding numbers. So far so good. You could also picture this, as recommended above, to think of this as a matter of directions. I cannot find a way to express this as an addition question. Multiplication IS addition. Just as division IS subtraction. One teacher tried to explain it thus: two bad people leave town three times.

Lots of mathematicians throughout history quibbled with it or resisted it. A little late, but not fifty years late…thanks, Ben. My mind is slightly less boggled. That makes some sense, if we accept those rules.

My inner 15 year-old is still balking somewhat. That seems like magic. Positive three, I can hold that in my hand. Is there any other world, other than directions vectors? Maybe that would help. It definitely resonates for some people though and is good to have in the repertoire.

Thanks Ben, that is an interesting quote! Great examples of real world negative values. I know you mathematicians must be shaking your heads slowly, sadly. In your example, your debt of 25 units is It seems to me that the entire debt is a negative, so why would I suddenly classify any part of it as a positive? Make a large circle on a piece of paper. When the circle is completely empty, it represents 0.

Keep in mind that there are multiple ways to represent positive and negative integers with this model. But the net effect of these is identical. This suggests that subtracting a negative removing red chips is equivalent to adding a positive adding black chips. AND, because multiplication is basically shorthand for addition, i.

Late yesterday, I found a Reddit discussion on this topic. There was a link to a module on negatives at purple math. They had an interesting suggestion for understanding this, which got me a little further down the road:. Instead, you control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes. If you add a hot cube add a positive number to the pot, the temperature of the stew goes up. If you add a cold cube add a negative number , the temperature goes down.

If you remove a hot cube subtract a positive number , the temperature goes down. And if you remove a cold cube subtract a negative number , the temperature goes UP! That is, subtracting a negative is the same as adding a positive. Video transcript Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers But now you are faced with a conundrum What happens when you multiply negative numbers?

Either when you multiply a positive number times a negative number Or when you multiply two negative numbers So, for example You aren't quite sure what should happen if you were to multiply and im just picking two numbers where one is positive and one is negative What would happen if you were to multiply 5 times negative 3 You're not quite sure about this just yet You're also not quite sure what would happen if you multiply two negative numbers.

So lets say negative two times negative 6 This is also unclear to you What you do know, because you are a mathematician, is however you define this or whatever this should be It should hopefully be consistant with all of the other properties of mathematics that you already know And preferably all of the other properties of multiplication That would make you feel comfortable that you are getting this right.

That's also consistent with the intuition of adding negative three repeatedly five times, now look above above us slightly higher so you can see ideas of multiplying two negatives, but we can do the exact same product experiment. We want whatever this answer to be consistent with the rest of mathematics that we know so we can do the same product experiment. What would negative two times six plus negative six to be equal to.