Why two negatives make a positive

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 But how about If your son is clear on the concept of money and knows what a credit card is, this might be a good explanation:.

You are having dinner with your best friend when the bill arrives from the credit card company, when your friend sees the bill he generously insists on paying it for you.

Your child may have been introduced to the minus sign by means of the word opposite. This is a great term to use in your conversations. On the number line, opposite numbers are mirrored in their distances from zero, which provides a nice visual aid as well. We can use the term to describe arithmetic operations:.

The opposite of three times the opposite of five is the opposite of the opposite of fifteen Okay, so we can use language to better cultivate our understanding of negative numbers. But what about a physical example? Well, here's a cute one that a friend once told me. It's a bit contrived, but I think it gets the point across.

You can turn this into a demonstration too. One way to picture this is to imagine a number line. Now do the rotation twice. The number line is unchanged. This approach has applications with Complex numbers. What's going on? This happens whether the "something" is positive or negative. That is, multiplication by a negative is the same as two steps: multiplying by the thing as if it had no negative, then applying the negative sign.

But two flips takes anything away and then right back to itself, so two flips really does nothing. The cancel out positive numbers. They are anti-numbers.

So i positive x positive: add a bunch of positive numbers a positive number of times. Result: A big positive gain. The result is a big amount of potential cancelling.

Result: negative. Result: a loss; negative. By taking away the taking away, what's left is putting things back.

If you annihilite the annihitation the result is a net gain: Result: positive. We introduce the negative numbers and need to define multiplication with negative numbers so that we have internal consistency. In order to keep the above two properties, we're forced to define the product of two negatives as a positive.

Why not look at a multiplication table? Let's make a little one, including some negative numbers. You could of course make it bigger to make the patterns clearer. It may also be good just to look at the table - it's very symmetrical. Sign up to join this community. The best answers are voted up and rise to the top.

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And if so, how? Show 15 more comments. Active Oldest Votes. That little piece of nuance is probably a bit tricky to explain to an 8-year-old though. Add a comment. Jordan Gray 1 1 silver badge 6 6 bronze badges. Dan Christensen Dan Christensen Thank you. Separates us from the nerds who live their lives without intuition.

You cannot multiply a number by itself to get a negative number. To get a negative number, you need one negative and one positive number. The rule works the same way when you have more than two numbers to multiply or divide. An even number of negative numbers will give a positive answer.

An odd number of negative numbers will give a negative answer. But here you have one negative and one positive number, so the sign of the answer will be negative. Again, you have one positive and one negative number, so the sign of the answer will be negative. This time, you have two negative numbers, so the sign of the answer will be positive. The answer is Again, you have two negative numbers, so the answer is positive. It is To begin with, consider the first part of the calculation.

The fact that a negative number multiplied by another negative number produces a positive result can often confuse and seem counterintuitive. To explain why this is the case, think back to the number lines used earlier in this article since these help to explain this visually.

In both of these examples, you have moved forwards i. Negative signs can look a bit daunting, but the rules that govern their use are simple and straightforward.

Keep these in mind, and you will have no problems. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. 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.