Listening to a math teacher, most students take the material as an axiom. At the same time, few people try to get to the bottom and figure out why the “minus” on the “plus” gives a “minus” sign, and when multiplying two negative numbers, it comes out positive.
Laws of mathematics
Most adults are unable to explain to themselves or their children why this happens. They had thoroughly learned this material in school, but they did not even try to find 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 matter and understand, for example, why “plus” on “minus” gives “minus”. And sometimes tomboys deliberately ask tricky questions in order to enjoy the moment when adults cannot give an intelligible answer. And it’s really a disaster if a young teacher gets into a mess…
By the way, it should be noted that the rule mentioned above is valid for both multiplication and division. The product of a negative and a positive number will only give 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 aone 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, it is customary to call a ring a set in which two operations with two elements are involved. But it is better to deal with this with an example.
Axiom of the Ring
There are several mathematical laws.
- The first one is commutative, according to him, C + V=V + C.
- The second one is called associative (V + C) + D=V + (C + D).
They also obey the multiplication (V x C) x D=V x (C x D).
No one has canceled the rules by which 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 has been established that a special element, neutral in terms of 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). In this case, C + (-C)=0.
Derivation of axioms for negative numbers
Accepting the above statements, we can answer the question: ""Plus" to "minus" gives what 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 will first have to prove that each of the elements has only oneopposite brother. Consider the following proof example. Let's try to imagine that two numbers are opposite for C - V and D. From this it follows that C + V=0 and C + D=0, that is, C + V=0=C + D. Remembering the displacement 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 figure 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 accepted above, equals 0. Hence, V=V + C + D.
The value for D is derived in exactly the same way: D=V + C + D=(V + C) + D=0 + D=D. Based on this, it becomes clear that V=D.
In order to understand why "plus" on "minus" gives "minus", you need 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 to (-)C x V, so (-C) x V=-(C x V).
For complete mathematical rigor, it is also necessary to confirm that 0 x V=0 for any element. If you follow the logic, then 0 x V \u003d (0 + 0) x V \u003d 0 x V + 0 x V. This means that adding the product 0 x V does not change the set amount in any way. After all, this product is equal to zero.
Knowing all these axioms, you can deduce not only how much "plus" by "minus" gives, but also what happens when you multiply negative numbers.
Multiplication and division of two numbers with "-" sign
If you do not go deep into mathematicalnuances, you can try to explain the rules of operations with negative numbers in a simpler way.
Let's assume that C - (-V)=D, so C=D + (-V), i.e. C=D - V. Transfer V and get C + V=D. That is, C + V=C - (-V). This example explains why in an expression where there are two "minus" in a row, the mentioned signs should be changed to "plus". Now let's deal with multiplication.
(-C) x (-V)=D, you can add and subtract two identical products to the expression, which 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 dividing two negative numbers will result in a positive one.
General math rules
Of course, this explanation is not suitable for elementary school students who are just starting to learn abstract negative numbers. It is better for them to explain on visible objects, manipulating the familiar term 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 looking-glass 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 an abstract negative number by a positive one only gives the result familiar to everyone. Because "plus"multiply by "minus" gives "minus". True, at primary school age, children do not really try to delve into 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 the teachers teach them, not at a loss to delve into all the complexities that mathematics is fraught with. "Minus" on "minus" gives a "plus" - everyone knows about this without exception. This is true for both integers and fractional numbers.
