Listening to a math teacher, most students perceive the material as an axiom. At the same time, few people are trying to get to the bottom and figure out why the “minus” by the “plus” gives the minus sign, and when you multiply two negative numbers, a positive one comes out.
Laws of mathematics
Most adults cannot explain to themselves or their children why this happens. They firmly learned this material at school, but did not even try to figure out where such rules came from. But in vain. Often, modern children are not so gullible, they need to get to the bottom of the essence and understand, say, why a plus to a minus gives a minus. And sometimes tomboys 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 mischief ...
By the way, it should be noted that the rule mentioned above is effective for both multiplication and division. The product of a negative and a positive number will give only a minus. If we are talking about two digits with a “-” sign, then the result will be a positive number. The same goes for division. If one of the numbers is negative, then the quotient will also be with a “-” sign.
To explain the correctness of this law of mathematics, it is necessary to formulate the axioms of the ring. But first you need to understand what it is. In mathematics, a ring is usually called a set in which two operations with two elements are involved. But to deal with this is better by example.
Axiom rings
There are several mathematical laws.
- The first of them is relocatable, according to it, C + V = V + C.
- The second is called combinational (V + C) + D = V + (C + D).
Multiplication (V x C) x D = V x (C x D) also obeys them.
No one has canceled the rules by which the brackets are opened (V + C) x D = V x D + C x D, it is also true that C x (V + D) = C x V + C x D.
In addition, it was found that a special element neutral in addition can be introduced into the ring, using which the following will be true: C + 0 = C. In addition, for each C there is an opposite element, which can be denoted as (-C). Moreover, C + (-C) = 0.
Derivation of axioms for negative numbers
By accepting the above statements, you 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 really (-C) x V = - (C x V). And also, that such equality is true: (- (- C)) = C.
To do this, you will first have to prove that for each of the elements there is only one opposite "brother" to it. Consider the following example of the proof. Let's try to imagine that two numbers are opposite for C - V and D. It follows from this that C + V = 0 and C + D = 0, that is, C + V = 0 = C + D. Remembering the transitive laws and about the properties of the number 0, we can consider the sum of all three numbers: C, V and D. Let's try to find 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 adopted above, is 0. Therefore, V = V + C + D.
In the same way, the value for D is also derived: D = V + C + D = (V + C) + D = 0 + D = D. Based on this, it becomes clear that V = D.
In order to understand why, nevertheless, a “plus” to a “minus” gives a “minus”, it is necessary to deal with 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. It follows that C x V is opposite to (-) C x V, which means that (- 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 work is zero.
Knowing all these axioms, one can derive not only how much “plus” by “minus” gives, but also what happens when multiplying negative numbers.
Multiplication and division of two numbers with a "-" sign
If you do not delve into the mathematical nuances, then you can try to explain in a simpler way the rules of action with negative numbers.
Suppose that C - (-V) = D, on the basis of 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 “minuses” in a row, the above signs should be changed to “plus”. Now let's deal with multiplication.
(-C) x (-V) = D, you can add and subtract two identical products into the expression that will not change its value: (-C) x (-V) + (C x V) - (C x V) = D.
Remembering the rules for working with brackets, 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, we can prove that as a result of division of two negative numbers, a positive one will come out.
General math rules
Of course, such an explanation is not suitable for elementary school students who are just beginning to learn abstract negative numbers. It is better for them to explain on visible objects, manipulating the familiar term for them through the looking glass. For example, invented but not existing toys are located there. They can be displayed with a “-" sign. The multiplication of two behind-the-mirror objects takes them to yet another world, which is equated to the present, that is, as a result, we have positive numbers. But multiplying an abstract negative number by a positive only gives a result familiar to everyone. After all, the "plus" multiplied by the "minus" gives a "minus". True, at primary school age, children are not too trying to understand all the mathematical nuances.
Although, if you face the truth, for many people, even with higher education, many rules remain a mystery. Everyone takes for granted what teachers teach them, not bothering to delve into all the difficulties that mathematics conceals. “Minus” to “minus” gives a “plus” - everyone knows about it without exception. This is true for both integer and fractional numbers.