Why multiplication of two negative numbers is positive

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Asked 7 years, 3 months ago. Active 5 years, 5 months ago. Viewed 15k times. Or exponentiation is repeated multiplication. All of these operation are only defined that way on natural number, but there is no reasons to expect that they are still defined the same way once it is extended, nor even the same property. See also: exponentiation with exponent being matrices; fractional derivative. We can start with multiplication as repeted addition.

Add a comment. 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".

I believe it's the most natural yet totally mathematical explanation of a basic notion like multiplication of negative numbers.

This is an excerpt from Arnold's wonderful memoir "Yesterday and Long Ago" 3d ed. The translation into English, I believe, is not of best quality, but it's the only one so far.

He, being a student of great algebraists, S. Shatunovsky and E. One can study any axioms! Since that day I have preserved the healthy aversion of a naturalist to the axiomatic method with its non-motivated axioms. Contrary to the deductive theories of my father and Descartes, as a ten year old, I started thinking about a naturally-scientific sense of the rule of signs , and I have come to the following conclusion.

A real positive or negative number is a vector on the axis of coordinates if a number is positive the corresponding vector is positively directed along this axis. A product of two numbers is an area of a rectangle whose sides correspond to these numbers one vector is along one axis and the other is along a perpendicular axis in the plane. A rectangle, given by an ordered pair of vectors, possesses, as a part of the plane, a definite orientation rotation from one vector to another can be clockwise or anti-clockwise.

The anti-clockwise rotation is customarily considered positive and the clockwise rotation is then negative. Thus, the rule of signs is not an axiom taken out of the blue, but becomes a natural property of orientation which is easily verified experimentally. My first trouble in school was caused by the rule for multiplication of negative numbers, and I asked my father to explain this peculiar rule.

My father did not say a word either about the oriented area of a rectangular or about any non-mathematicai interpretation of signs and products. But since that time I have disliked the axiomatic method with its non-motivated definitions. Probably it was for this reason that by this time I got used to talking with non-algebraists like L.

Tamm, P. Novikov, E. Feinberg, M. Leontovich, and A. Gurvich who treated an ignorant interlocutor with full respect and tried to explain non-trivial ideas and facts of various sciences such as physics and biology, astronomy and radiolocation. It is not possible to explain to algebraists that their axiomatic method is mostly useless for students. One should ask children: at what time will high tide be tomorrow if today it is at 3 pm?

This is a feasible problem, and it helps children to understand negative numbers better than algebraic rules do. Once I read from an ancient author probably from Herodotus that the tides "always occur three and nine o'clock". To understand that the monthly rotation of the Moon about the Earth affects the tide timetable, there is no need to live near an ocean. Here, not in axioms, is laid true mathematics.

Why don't we tell a story! The dastardly Dalton gang is on the loose but Al Catchem is hot on their trail and nearly catches them at their latest heist. When pulling out of the parking lot, there are a few possibilities:. Perhaps some intuition can be gained by plotting each number's position on the number line. Taking the inverse of any number is visualized by taking the mirror-image of the original plot.

So the inverse of a positive number a point to the right of zero is a negative number a point to the left of zero, at the same distance from zero.

Likewise, the inverse of a negative number is a positive number. If we agree that multiplying a number by -1 is the same as finding the inverse, then we can see that the product of two negatives must be a positive, because the mirror-image of a mirror-image is the original image. Notice in each case, as we reduce the second factor by 1, the product is being reduced by 3. Notice in each case, as we reduce the first multiplier by 1, the product is being increased by 2.

I would go for the flipping explanation of the negative numbers: multiplying with a negative number flips from positive to negative and from negative to positive.

It might be easiest to explain using whole numbers. So negative times positive is positive. Same idea for positive times negative. When it comes to negative times negative, it's a little harder In which case we can write:. So what mathematics could guide us in our thinking here? As the area model is just a representation our belief in expanding brackets, the area model should hold for negative numbers too! Even though geometrically it makes no sense to have a negative side length in a geometric figure, we see that the mathematics each diagram represents is still correct mathematics.

People try to give concrete meaning to all this with models of soldiers marching on number lines turning different directions, systems of profit and debt, working with temperatures above and below freezing, and so on. Each model is good for illustrating SOME aspect of the arithmetic of negative numbers, but not all.

We must start instead with a discussion on what we think should be true about multiplication in general and how it behaves. And for students ready for it, the axiomatic approach clinches it. I just know it is algebraically consistent. Your support is so much appreciated and enables the continued creation of great course content.

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