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This question explains better what this one tried: Understanding the intuition behind math

In the history mathematics we always see how the numbers were created and for what purpose. Like ordering, calculation of areas and taxes. This is the most common approach on documentaries and so in the math books.

It's always easy to understand the basics: Add, subtract, multiply and divide. The reason it is easy is that it is intuitive. We can imagine those things happening in our minds through visual representations.

I'm digging these documentaries looking for the part where we jumped from these basic operations to the complex ones. At least to me, the complexity starts when we add negative numbers. For example, in this simple operation:

$1 + (-1) = 0$

It's easy to understand how to apply and use the provided logic of signs. But to me it is not easy to imagine what is really happening here. And when I say "happening", I mean that I can't create any object visualization of that operations, like the ones we learn on high school:

"Imagine that you have two breads. You have eat one. Now how many do you have?"

On that kind of visualization you can imagine the two objects and then you can take off one and see how it looks like. This can be done for all the basic four operations on small scales.

You could say that the operation above is the same as $1 - 1 = 0$ But that's not the case. Doing so you are omitting the signs part and using it just as predefined formula.

The main question is about when in the history and for what reason the man came out with the signs and that kind of operation. In other words: Which was the first step given after these basic and very intuitive operations? And which was the necessities that leverage these discoveries? Like the need to have a convention where plus and minus become minus.

It is easy to think of debts. In that case, can you provide a simple visualization like the mentioned above for the product of two negative numbers?

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Keyne Viana
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    Have you seen this? – J. M. ain't a mathematician Dec 17 '11 at 08:53
  • $5-3$ is quite clearly $2$, yes. But I presume eventually some "crazy" person would come along and ask what $3 - 7$ is... – Srivatsan Dec 17 '11 at 08:55
  • @Srivatsan This is also simple to visualize as a deficit. That's not the point of the question. – Keyne Viana Dec 17 '11 at 10:41
  • @J.M. I did not see this. But after read, my question still remains. I can perform all this operations easily. But my question is about the psychology facts and types of visualizations. I'm not sure if I made my self clear enough. – Keyne Viana Dec 17 '11 at 10:43
  • If you have -2 dollars, you owe somebody 2 dollars. – J. M. ain't a mathematician Dec 17 '11 at 10:55
  • @Srivatsan I'm assuming that the idea behind negative numbers it's not just about debts. I'm looking for ways to express my self better or to discover that actually my question is wrong... I will come back to edit it. – Keyne Viana Dec 17 '11 at 11:01
  • @J.M. Yes. That's simple. Read what I've wrote to Srivatsan... Can you give the same example for the product of two negative numbers? – Keyne Viana Dec 17 '11 at 11:02
  • It's not unintuitive to think about the product of two negative numbers. – J. M. ain't a mathematician Dec 17 '11 at 11:30
  • @J.M. So how did we came with that? In which step we have changed from intuitive concepts two the non intuitive ones? – Keyne Viana Dec 17 '11 at 11:32
  • Clearly you didn't read my last comment carefully. Again: it's never not easy to demonstrate why the product of two negative numbers is positive. – J. M. ain't a mathematician Dec 17 '11 at 11:37
  • @J.M. Sorry. Actually a read it. But I'm not getting it. Maybe because I'm to dumb =\ I was hoping at least a kind of visualization for that. Just it. Probably you're fully right, the problem is with my mental model. – Keyne Viana Dec 17 '11 at 12:36
  • Maybe I should highlight the section you should have seen: "it's never not easy to demonstrate why the product of two negative numbers is positive." – J. M. ain't a mathematician Dec 17 '11 at 12:38
  • @J.M. Ok. Thanks for the highlight... Despite the fact the it can be or not be easy to demonstrate, my hope was to discover the steps that lead our minds to that direction, it must be the simple ones, then a combination of that simple ones that turn into a more complex concept. If you consider that I still do not read what you have wrote. You can simply ignore this comment. Thanks anyway! – Keyne Viana Dec 17 '11 at 13:02
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    I see. So you aren't familiar with the concept of a double negative... – J. M. ain't a mathematician Dec 17 '11 at 13:09
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    @J.M. Dear J.M., I would guess that the OP's native language is not English, and I've been told that the concept of double negative doesn't always operate in other languages the way it does in English. Regards, – Matt E Dec 17 '11 at 13:16
  • I didn't consider that. Thanks @Matt. Things will be harder to discuss, then... – J. M. ain't a mathematician Dec 17 '11 at 13:17
  • @J.M. But are you sure that if I understand that it's never not easy to demonstrate... Then the question is answered or maybe I will realized that it was not properly formulated? This mean that I will never find visual demonstration of the product of two negative numbers? Can you give me a final comment about this? Are you trying to explain how it works or how I can not imagine it? – Keyne Viana Dec 17 '11 at 13:38
  • I don't wish to be definitive. I'm only saying how I see "two negatives make a positive". Probably there is a way that might be more satisfactory to you. If Matt's supposition is correct, then I don't know how I'll get the point across you. – J. M. ain't a mathematician Dec 17 '11 at 14:01

2 Answers2

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Intuitively, think of it like this: the term $-x$ does not imply that $-x$ is negative, but simply that it is of the opposite sign of $x$. So when one writes the equation $3+(-x)=5$, we have that $x=-2$. So in this instance, even though you are "adding a negative " (-x) you are actually adding a positive, which it is easy to visualize. These sign operations make generalizations about the above operation similar. A similar process describes negative multiplication; $-x=(-1)*x$. If $x=-y$, then $y=(-1)*(-y)=(-1)*(-1)*y$. Dividing both sides by y yields $1=(-1)*(-1)$. The sign multiplication then makes sense.

analysisj
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This is more a question of pragmatism. What negative numbers 'are' has confounded great mathematicians and philosophers, to no definite conclusion I am aware of. Have a look at this article.

http://www.bookrags.com/research/positive-and-negative-numbers-wsd/

To say they are 'debt' doesn't really address what you are after I believe. Numbers should count something, and debt is countable, even if your net worth is reduced by it. It's only negative when you balance your portfolio; not by itself - although we all view it this way nowadays, obviously.

They just turned out to be useful - as in the earlier wiki reference to solutions of simple linear equations that otherwise have none. You can visualize them maybe as a drift in the opposite direction of the increase of natural numbers. Girard, quoted in the above, produced some phenomenal solutions of cubic equations, say, by accepting interim results that were imaginary ("let's assume for a moment this equation had a solution, in some space"), kept further transforming, and eventually received real results because the imaginary terms cancelled.

gnometorule
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