How to understand why the "plus" to "minus" gives a "minus"?

Listening to the teacher of mathematics, most students perceive the material as an axiom. At the same time, few people are trying to get to the bottom and understand why the "minus" to "plus" is given by the minus sign, and with the multiplication of two negative numbers, there is a positive sign.

The laws of mathematics

Most adults can not explain to themselves or their children why it happens. They firmly grasped this material at school, but they did not even try to find out where the rules came from. But in vain. Often, modern children are not so trusting, they need to get to the bottom and understand, let's say, why the "plus" to "minus" gives a "minus". And sometimes, forebears specifically ask tricky questions, in order to enjoy the moment when adults can not give an intelligible answer. And it's really a disaster if a young teacher gets trapped ...

By the way, it should be noted that the rule mentioned above is effective both for multiplication and for division. The product of a negative and a positive number will give only "minus. If it is a question of two digits with the sign "-", then the result is a positive number. The same concerns division. If one of the numbers is negative, the quotient will also have a "-" sign.

To explain the correctness of this law of mathematics, it is necessary to formulate the axioms of a ring. But first you need to understand what it is. In mathematics, a ring is called a ring, in which two operations with two elements are involved. But to understand this better by example.

Axiom of a ring

There are several mathematical laws.

• The first of them is movable, according to him, C + V = V + C.
• The second is called the combination (V + C) + D = V + (C + D).

It also obeys multiplication (V x C) x D = V x (C x D).

Nobody canceled the rules by which the brackets (V + C) x D = V x D + C x D are opened, it is also true that C x (V + D) = C x V + C x D.

In addition, it is established that a special, element-neutral element can be introduced into the ring, with the use of which the following will be true: C + 0 = C. In addition, for each C there is an opposite element, which can be designated as (-C). In this case, C + (-C) = 0.

Derivation of axioms for negative numbers

Accepting the above statements, one can answer the question: "Plus" to "minus" gives which sign? "Knowing the axiom about the multiplication of negative numbers, it is necessary to confirm that indeed (-C) x V = - (C x V). And also, that the following equality is true: (- (- C)) = C.

To do this, we must first prove that each of the elements has only one opposite "colleague". Consider the following example of proof. Let's try to imagine that for C the two numbers V and D are opposite. It follows that C + V = 0 and C + D = 0, that is C + V = 0 = C + D. Remembering the relocatable laws and On the properties of the number 0, we can consider the sum of all three numbers: C, V and D. Let's try to find out the value of V. It is logical that V = V + 0 = V + (C + D) = V + C + D, because the value of C + D, as was assumed above, equals 0. Hence, V = V + C + D.

In the same way, the value for D: D = V + C + D = (V + C) + D = 0 + D = D is also output. Proceeding from this, it becomes clear that V = D.

In order to understand why all the same "plus" to "minus" gives a "minus", it is necessary to understand the following. So, for the element (-C) the opposite are C and (- (- C)), that is, they are equal to each other.

Then it is obvious that 0 x V = (C + (-C)) x V = C x V + (-C) x V. From this it follows that C x V is opposite (-) C x V, hence (- C) x V = - (C x V).

For complete mathematical rigor, it is still necessary to confirm that 0 x V = 0 for any element. If you follow the logic, then 0 x V = (0 + 0) x V = 0 x V + 0 x V. And this means that adding the product 0 x V does not change the set amount. After all, this product is zero.

Knowing all these axioms, one can deduce not only how much "plus" and "minus" gives, but what happens when multiplying negative numbers.

Multiplication and division of two numbers with the sign "-"

If you do not delve into the mathematical nuances, you can try a simpler way to explain the rules of action with negative numbers.

Assume that C - (-V) = D, starting from this, C = D + (-V), that is, C = D - V. We transfer V and we get that C + V = D. That is, C + V = C - (-V). This example explains why in the expression, where there are two "minus" in a row, the mentioned signs should be changed to "plus". Now let's look at the multiplication.

(-C) x (-V) = D, you can add and subtract two identical products in the expression that do not change its values: (-C) x (-V) + (C x V) - (C x V) = D.

Remembering the rules of working with parentheses, we get:

1) (-C) x (-V) + (C x V) + (-C) x V = D;

2) (-C) x ((-V) + V) + C x V = D;

3) (-C) x 0 + C x V = D;

4) C x V = D.

It follows that C x V = (-C) x (-V).

Similarly, it can be shown that, as a result of dividing two negative numbers, a positive result will appear.

General mathematical rules

Of course, such an explanation is not suitable for schoolchildren who are only beginning to learn abstract negative numbers. It is better for them to explain on visible objects, manipulating the familiar term of the looking-glass. For example, invented, but not existing toys are there. They can be displayed with a "-" sign. Multiplication of two mirror-like objects transfers them to another world, which is equated to the present, that is, as a result, we have positive numbers. But the multiplication of the abstract negative number by the positive only gives the result known to all. After all, the "plus" multiply by "minus" gives a "minus". However, at the younger school age, children do not try to understand all the mathematical nuances.

Although, if you look at the truth in your eyes, for many people, even with higher education, many rules remain a mystery. Everyone takes for granted what teachers teach them, without difficulty in delving into all the difficulties that mathematics entails. "Minus" to "minus" gives a "plus" - everyone knows about it without exception. This is true for integers and fractional numbers.