When listening to a mathematics teacher, most students perceive the material as an axiom. At the same time, few people try to get to the bottom and figure out why “minus” to “plus” gives a “minus” sign, and when multiplying two negative numbers, a positive one comes out.

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, say, 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 trouble ...

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 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, it is customary to call a ring a set in which two operations with two elements are involved. But it's better to understand this with an example.

Ring axiom

There are several mathematical laws.

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

The multiplication (V x C) x D \u003d V x (C x D) also obeys them.

Nobody 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, addition-neutral element 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) \u003d 0.

Derivation of axioms for negative numbers

By accepting the above statements, we can answer the question: "Plus" on "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 must first prove that each of the elements has only one opposite "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, is equal to 0. Hence, V = V + C + D.

The value for D is derived in 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, nevertheless, the “plus” on the “minus” gives a “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 \u003d (C + (-C)) x V \u003d C x V + (-C) x V. It follows from this that C x V is opposite to (-) C x V, which means (- 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, it is possible to deduce not only how much "plus" by "minus" gives, but also what happens when negative numbers are multiplied.

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

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

Suppose that C - (-V) = D, based on 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 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) \u003d D, two identical products can be added and subtracted to the expression, which will not change its value: (-C) x (-V) + (C x V) - (C x V) \u003d 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;

It follows from this that C x V \u003d (-C) x (-V).

Similarly, we can prove that the result of dividing two negative numbers will be positive.

General mathematical rules

Of course, such an 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. After all, “plus” multiplied by “minus” gives “minus”. True, children do not try too hard 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 their teachers teach them, not at a loss to delve into all the complexities that mathematics is fraught with. “Minus” on “minus” gives “plus” - everyone knows about this without exception. This is true for both integers and fractional numbers.

Line UMK G.K. Muravina, O.V. Muravina. Mathematics (5-6)

Maths

Why does a minus times a minus always give a plus?

The benefits of natural numbers

First, let's dive into the history of arithmetic. It is quite natural that at the very beginning people used only natural numbers - one, two, three, and so on. They were used to calculate the actual number of items. Just like that, in isolation from everything, the numbers were useless, so actions began to appear, with the help of which it became possible to operate with numbers. It is absolutely logical that addition has become the most necessary for a person. This operation is simple and natural - it became easier to count the number of items, now it was not necessary to count again every time - “one, two, three”. Replacing the score is now possible using the "one plus two equals three" action. Natural numbers were added, the answer was also a natural number.

Multiplication was essentially the same addition. In practice, even now, for example, when making purchases, we also use addition and multiplication, as our ancestors did a long time ago. However, sometimes it was necessary to perform operations of subtraction and division. And the numbers were not always equivalent - sometimes the number from which they subtracted was less than the number that was subtracted. The same with division. Thus, fractional numbers appeared.

The appearance of negative numbers

Records of negative numbers appeared in Indian documents in the 7th century AD. There are older records of this mathematical "fact" in Chinese documents.

In life, we most often subtract a smaller number from a larger number. For example: I have 100 rubles, bread and milk cost 65 rubles; 100 - 65 = 35 rubles change. If I want to buy some other product, the cost of which exceeds my remaining 35 rubles, for example, one more milk, then no matter how much I want to buy it, I don’t have more money, therefore, I don’t need negative numbers.

However, continuing to talk about modern life, let's mention credit cards or the ability of a mobile operator to “go into minus” when making calls. It becomes possible to spend more money than you have, but the money that you owe does not disappear, but is written into debt. And here negative numbers already come to the rescue: there are 100 rubles on the card, bread and two milk will cost me 110 rubles; after the purchase, my balance on the card is -10 rubles.

Practically for the same purposes, they began to use negative numbers for the first time. The Chinese were the first to use them to write down debts or in intermediate solutions to equations. But the use was still only to come to a positive number (however, like our credit card repayment). The long rejection of negative numbers was facilitated by the fact that they did not express specific objects. Ten coins are ten coins, here they are, you can touch them, you can buy goods with them. What does "minus ten coins" mean? They are expected even if it is a debt. It is not known whether this debt will be returned, and whether the “recorded” coins will turn into real ones. If, when solving a problem, a negative number was obtained, it was considered that the wrong answer came out or that there was no answer at all. This distrustful attitude persisted among people for a long time, even Descartes (XVII century), who made a breakthrough in mathematics, considered negative numbers to be “false”.

The tasks of the manual allow you to prevent possible difficulties in mastering the main topics of the fourth year of teaching mathematics, help develop spatial representations, geometric observation of students, and form self-control skills.

Formation of rules for actions with negative numbers

Consider the equation 9x-12=4x-2. To solve the equation, you need to move the terms with the unknown to one side, and the known numbers to the other. This can be done in two ways.

First way.

We move the part of the equation with the unknown to the left, and the other numbers to the right. It turns out:

Answer found. For all the actions that we needed to perform, we never resorted to using negative numbers.

The second way.

Now we transfer the part of the equation with the unknown to the right, and the remaining terms to the left. We get:

To find the solution, we need to divide one negative number by another. However, we have already received the correct answer in the previous solution - this is x equal to two. Therefore, it remains to deduce that (-10)/(-5)=2.

What do these two ways of solving the same equation prove to us? The first thing that becomes clear is how the adequacy of operating with negative numbers was deduced - the answer obtained should be the same as when solving using only natural numbers. The second point is the fact that you no longer need to think about the values ​​in order to get a non-negative number without fail. You can choose the most convenient way to solve, especially for complex equations. Actions that made it possible not to think about some operations (what needs to be done so that there are only natural numbers; which number is greater in order to subtract from it, etc.) became the first steps towards the "abstraction" of mathematics.

Naturally, not all rules of action with negative numbers were formed at the same time. Solutions were accumulated, examples were generalized, on the basis of which they began to gradually “draw out” the main axioms. With the development of mathematics, with the release of new rules, new levels of abstraction appeared. For example, in the nineteenth century it became proven that integers and polynomials have much in common, although they look different. All of them can be added, subtracted and multiplied. The rules they obey affect them in one way. As for the division of some integers by others, an interesting fact “waits” here - the answer will not always be an integer. The same law applies to polynomials.

Then many other collections of mathematical objects were revealed, on which it was possible to perform such operations: formal power series, continuous functions ... Over time, mathematicians found that after studying the properties of operations, it would be possible to apply the results to all these collections of objects. The same is true in modern mathematics.

More interesting stuff:

• Features of the work of a mathematics teacher in the 2018/2019 academic year
• Typical mistakes teachers make when teaching mathematics in elementary school
• Extracurricular activities in mathematics in elementary school

Purely mathematical approach

Over time, mathematicians have identified a new term - the ring. A ring is a set of elements and operations that can be performed on them. The rules (the very axioms) to which actions are subject, and not the nature of the elements of the set, become fundamental. In order to emphasize the primacy of the structure that arises after the introduction of the axioms, the term “ring” is usually used: the ring of integers, the ring of polynomials, etc. Using the axioms and proceeding from them, one can reveal new properties of rings.

We formulate the rules of the ring, similar to the axioms of operations with integers, and prove that in any ring, multiplying a minus by a minus results in a plus.

A ring is a set with two binary operations (each operation involves two elements of the ring), traditionally called addition and multiplication, and the following axioms:

The addition of ring elements obeys commutative (A + B = B + A for any elements A and B) and combinational (A + (B + C) = (A + B) + C) laws; the ring has a special element 0 (a neutral element by addition) such that A + 0 = A, and for any element of A there is an opposite element (denoted (-A)) such that A + (-A) = 0;

Multiplication obeys the combination law: A (B C) = (A B) C;

Addition and multiplication are related by the following bracket expansion rules:

(A + B) C = A C + B C

A (B + C) = A B + A C.

Let us clarify that rings, in the most general construction, do not require multiplication to be permutable, nor its reversibility (the division operation is not always possible), nor the existence of a unit - a neutral element with respect to multiplication. If we introduce these axioms, we obtain other algebraic structures, but with all valid theorems proved for rings.

Maths. 6th grade. Workbook number 1.

The workbook contains various types of tasks for mastering and consolidating new material, tasks of a developing nature, additional tasks that allow for differentiated learning. The notebook is used in conjunction with the textbook "Mathematics. Grade 6 "(ed. A.G. Merzlyak, V.B. Polonsky, M.S. Yakir), which is included in the system of educational and methodological kits" Algorithm of Success ".

The next step is to prove that for any elements A and B of an arbitrary ring, the following is true: (-A) B = -(A B) and (-(-A)) = A.

From this we get statements about units:

(-1) 1 = -(1 1) = -1

(-1) (-1) = -((-1) 1) = -(-1) = 1.

Next, we need to prove some points. First, it is necessary to establish the existence of only one opposite for each element. Suppose element A has two opposite elements: B and C. That is, A + B \u003d 0 \u003d A + C. Let's analyze the sum A + B + C. Using the commutative and associative laws, as well as the properties of zero, we get that the sum is equal to :

B:B=B+0=B+(A+C)=A+B+C

C: A + B + C = (A + B) + C = 0 + C = C.

Therefore, B = C.

Note that both A and (-(-A)) are opposite to the element (-A). Hence we conclude that the elements A and (-(-A)) must be equal.

those. (-A) B is the opposite of A B, so it is equal to -(A B).

Note that 0 · B = 0 for any element of B.

0 B = (0 + 0) B = 0 B + 0 B,

thus adding 0 B does not change the sum. It turns out that this product is equal to zero.

Indeed, why? The easiest answer is: "Because these are the rules for working with negative numbers." The rules we learn in school and apply throughout our lives. However, the textbooks do not explain why the rules are the way they are. We remembered - that's it, and no longer ask the question.

And let's ask...

A long time ago, only natural numbers were known to people: 1, 2, 3, ... They were used to count utensils, prey, enemies, etc. But the numbers themselves are rather useless - you need to be able to handle them. Addition is clear and understandable, and besides, the sum of two natural numbers is also a natural number (a mathematician would say that the set of natural numbers is closed under the operation of addition). Multiplication is, in fact, the same addition if we are talking about natural numbers. In life, we often perform actions related to these two operations (for example, when shopping, we add and multiply), and it is strange to think that our ancestors encountered them less often - addition and multiplication were mastered by mankind a very long time ago. Often it is necessary to divide one quantity by another, but here the result is not always expressed by a natural number - this is how fractional numbers appeared.

Subtraction, of course, is also indispensable. But in practice, we tend to subtract the smaller number from the larger number, and there is no need to use negative numbers. (If I have 5 candies and I give 3 to my sister, then I will have 5 - 3 = 2 candies, but I can’t give her 7 candies with all my desire.) This can explain why people did not use negative numbers for a long time.


Negative numbers appear in Indian documents from the 7th century AD; the Chinese, apparently, began to use them a little earlier. They were used to account for debts or in intermediate calculations to simplify the solution of equations - it was only a tool to get a positive answer. The fact that negative numbers, unlike positive ones, do not express the presence of any entity, aroused strong distrust. People in the literal sense of the word avoided negative numbers: if the problem got a negative answer, they believed that there was no answer at all. This distrust persisted for a very long time, and even Descartes, one of the "founders" of modern mathematics, called them "false" (in the 17th century!).

Consider for example the equation 7x - 17 \u003d 2x - 2. It can be solved as follows: move the terms with the unknown to the left side, and the rest to the right, you get 7x - 2x \u003d 17 - 2, 5x \u003d 15, x \u003d 3. With this We didn't even encounter negative numbers in the solution.

But it could have been done in a different way: move the terms with the unknown to the right side and get 2 - 17 = 2x - 7x, (-15) = (-5)x. To find the unknown, you need to divide one negative number by another: x = (-15)/(-5). But the correct answer is known, and it remains to be concluded that (-15)/(-5) = 3.

What does this simple example demonstrate? First, it becomes clear the logic that determined the rules for actions on negative numbers: the results of these actions must match the answers that are obtained in a different way, without negative numbers. Secondly, by allowing the use of negative numbers, we get rid of the tedious (if the equation turns out to be more complicated, with a large number of terms) search for the solution path in which all actions are performed only on natural numbers. Moreover, we can no longer think every time about the meaningfulness of the quantities being converted - and this is already a step towards turning mathematics into an abstract science.

The rules for actions on negative numbers were not formed immediately, but became a generalization of numerous examples that arose when solving applied problems. In general, the development of mathematics can be conditionally divided into stages: each next stage differs from the previous one by a new level of abstraction in the study of objects. So, in the 19th century, mathematicians realized that integers and polynomials, for all their outward dissimilarity, have much in common: both can be added, subtracted, and multiplied. These operations obey the same laws - both in the case of numbers and in the case of polynomials. But the division of integers by each other, so that the result is again integers, is not always possible. The same is true for polynomials.

Then other collections of mathematical objects were discovered on which such operations can be performed: formal power series, continuous functions ... Finally, the understanding came that if you study the properties of the operations themselves, then the results can be applied to all these collections of objects (this approach is typical for all modern mathematics).

As a result, a new concept appeared: the ring. It's just a bunch of elements plus actions that can be performed on them. The fundamental rules here are just the rules (they are called axioms), which are subject to actions, and not the nature of the elements of the set (here it is, a new level of abstraction!). Wishing to emphasize that it is the structure that arises after the introduction of axioms that is important, mathematicians say: the ring of integers, the ring of polynomials, etc. Starting from the axioms, one can derive other properties of rings.

We will formulate the axioms of the ring (which are, of course, similar to the rules for operations with integers), and then we will prove that in any ring, multiplying a minus by a minus results in a plus.

A ring is a set with two binary operations (that is, two elements of the ring are involved in each operation), which are traditionally called addition and multiplication, and the following axioms:

The addition of ring elements obeys commutative (A + B = B + A for any elements A and B) and combinational (A + (B + C) = (A + B) + C) laws; the ring has a special element 0 (a neutral element by addition) such that A + 0 = A, and for any element of A there is an opposite element (denoted (-A)) such that A + (-A) = 0;
- multiplication obeys the combination law: A (B C) = (A B) C;
addition and multiplication are related by the following bracket expansion rules: (A + B) C = A C + B C and A (B + C) = A B + A C.

We note that rings, in the most general construction, do not require multiplication to be permutable, nor is it invertible (that is, it is not always possible to divide), nor does it require the existence of a unit, a neutral element with respect to multiplication. If these axioms are introduced, then other algebraic structures are obtained, but all the theorems proved for rings will be true in them.

Now let us prove that for any elements A and B of an arbitrary ring, firstly, (-A) B = -(A B), and secondly (-(-A)) = A. This easily implies statements about units: (-1) 1 = -(1 1) = -1 and (-1) (-1) = -((-1) 1) = -(-1) = 1.

To do this, we need to establish some facts. First we prove that each element can have only one opposite. Indeed, let the element A have two opposite ones: B and C. That is, A + B = 0 = A + C. Consider the sum A + B + C. Using the associative and commutative laws and the property of zero, we get that, with on the one hand, the sum is equal to B: B = B + 0 = B + (A + C) = A + B + C, and on the other hand, it is equal to C: A + B + C = (A + B) + C = 0 + C = C. Hence, B = C.

Note now that both A and (-(-A)) are opposites of the same element (-A), so they must be equal.

The first fact is obtained as follows: 0 = 0 B = (A + (-A)) B = A B + (-A) B, that is, (-A) B is opposite to A B, so it is equal to -(A B).

To be mathematically rigorous, let's also explain why 0·B = 0 for any element of B. Indeed, 0·B = (0 + 0) B = 0·B + 0·B. That is, adding 0 B does not change the sum. So this product is equal to zero.

And the fact that there is exactly one zero in the ring (after all, the axioms say that such an element exists, but nothing is said about its uniqueness!), we will leave to the reader as a simple exercise.

Evgeny Epifanov

Minus and plus are signs of negative and positive numbers in mathematics. They interact with themselves in different ways, so when performing any actions with numbers, for example, division, multiplication, subtraction, addition, etc., it is necessary to take into account sign rules. Without these rules, you will never be able to solve even the simplest algebraic or geometric problem. Without knowledge of these rules, you will not be able to study not only mathematics, but also physics, chemistry, biology, and even geography.

Let us consider in more detail the basic rules of signs.

Division.

If we divide "plus" by "minus", we always get "minus". If we divide "minus" by "plus", we always get "minus" as well. If we divide "plus" by "plus", we get "plus". If we divide “minus” by “minus”, then, oddly enough, we also get “plus”.

Multiplication.

If we multiply "minus" by "plus", we always get "minus". If we multiply "plus" by "minus", we always get "minus" as well. If we multiply "plus" by "plus", then we get a positive number, that is, "plus". The same goes for two negative numbers. If we multiply "minus" by "minus", we get "plus".

Subtraction and addition.

They are based on other principles. If a negative number is greater in absolute value than our positive one, then the result, of course, will be negative. Surely, you are wondering what a module is and why it is here at all. Everything is very simple. Modulus is the value of a number, but without a sign. For example -7 and 3. Modulo -7 will be just 7 , and 3 will remain 3. As a result, we see that 7 is greater, that is, it turns out that our negative number is greater. So it will come out -7 + 3 \u003d -4. It can be made even easier. Just put a positive number in the first place, and 3-7 = -4 will come out, perhaps it is more understandable for someone. Subtraction works in exactly the same way.

To consolidate the ability to multiply natural numbers, ordinary and decimal fractions;

Learn to multiply positive and negative numbers;

Develop the ability to work in groups

Develop curiosity, interest in mathematics; the ability to think and speak on a topic.

Equipment: models of thermometers and houses, cards for mental counting and test work, a poster with the rules of signs for multiplication.

Motivation

Teacher . Today we are starting to explore a new topic. We are going to build a new house. Tell me, what determines the strength of the house?

Now let's check what our foundation is, that is, the strength of our knowledge. I didn't tell you the topic of the lesson. It is coded, that is, hidden in the task for oral counting. Be attentive and observant. Here are cards with examples. By solving them and matching the letter to the answer, you will find out the name of the topic of the lesson.

Teacher. So that word is multiplication. But we are already familiar with multiplication. Why do we need to study it? What numbers have you recently met?

[With positive and negative.]

Can we multiply them? Therefore, the topic of the lesson will be "Multiplication of positive and negative numbers."

You quickly and correctly solved the examples. A good foundation has been laid. ( Teacher on model house « lays» foundation.) I think that the house will be durable.

Exploring a new topic

Teacher . Now let's build walls. They connect the floor and the roof, that is, the old theme with the new one. Now you will work in groups. Each group will be given a problem to solve together and then explain the solution to the class.

1st group

The air temperature drops by 2° every hour. Now the thermometer shows zero degrees. What temperature will it show after 3 hours?

Group decision. Since the temperature is now 0 and for every hour the temperature drops by 2°, it is obvious that after 3 hours the temperature will be -6°. Let us denote the temperature decrease as –2°, and the time as +3 hours. Then we can assume that (–2) 3 = –6.

Teacher . And what happens if I rearrange the factors, that is, 3 (–2)?

Students. The answer is the same: -6, since the commutative property of multiplication is used.

The air temperature drops by 2° every hour. Now the thermometer shows zero degrees. What air temperature did the thermometer show 3 hours ago?

Group decision. Since the temperature dropped by 2° every hour, and now it is 0, it is obvious that 3 hours ago it was +6°. Let us denote the decrease in temperature by -2°, and the elapsed time by -3 hours. Then we can assume that (–2) (–3) = 6.

Teacher . You don't know how to multiply positive and negative numbers yet. But they solved problems where it was necessary to multiply such numbers. Try yourself to derive the rules for multiplying positive and negative numbers, two negative numbers. ( The students are trying to figure out the rule.) Good. Now let's open the textbooks and read the rules for multiplying positive and negative numbers. Compare your rule with what is written in the textbook.

Rule 1 To multiply two numbers with different signs, you need to multiply the modules of these numbers and put a “-” sign in front of the resulting product.

Rule 2. To multiply two numbers with the same signs, you need to multiply the modules of these numbers and put a “+” sign in front of the resulting product.

Teacher. As you saw when building the foundation, you have no problem multiplying natural and fractional numbers. Problems can arise when multiplying positive and negative numbers. Why?

Remember! When multiplying positive and negative numbers:

1) determine the sign;
2) find the product of modules.

Teacher . For multiplication signs, there are mnemonic rules that are very easy to remember. Briefly they are formulated as follows:

"+" "+" \u003d "+" - a plus on a plus gives a plus;
“–” “+” = “–” - minus plus gives minus;
"+" "-" \u003d "-" - plus a minus gives a minus;
“–” · “–” = “+” - minus times minus gives plus.

(In notebooks, students write down the rule of signs.)

Teacher . If we consider ourselves and our friends positive, and our enemies negative, then we can say this:

My friend's friend is my friend.
My friend's enemy is my enemy.
A friend of my enemy is my enemy.
The enemy of my enemy is my friend.

Primary comprehension and application of the studied

Examples for oral solution on the board. Students say the rule:

Teacher . All clear? No questions? So the walls are built. ( The teacher puts up walls.) Now what are we building?

(Four students are called to the board.)

Teacher. Is the roof ready?

(The teacher puts a roof on a model house.)

Pupils complete the work in one version.

After completing the work, they exchange notebooks with their neighbor. The teacher reports the correct answers, and the students give marks to each other.

Summary of the lesson. Reflection

Teacher. What was our goal at the beginning of the lesson? Have you learned how to multiply positive and negative numbers? ( They repeat the rules.) As you saw in this lesson, each new topic is a house that needs to be built capitally, for years. Otherwise, all your buildings will collapse after a short time. Therefore, everything depends on you. I wish, guys, that luck always smiles at you, success in mastering knowledge.

Sign rules

sign rules

Let us consider in more detail the basic rules of signs.

If we divide "plus" by "minus", we always get "minus". If we divide "minus" by "plus", we always get "minus" as well. If we divide "plus" by "plus", we get "plus". If we divide “minus” by “minus”, then, oddly enough, we also get “plus”.

If we multiply "minus" by "plus", we always get "minus". If we multiply "plus" by "minus", we always get "minus" as well. If we multiply "plus" by "plus", then we get a positive number, that is, "plus". The same goes for two negative numbers. If we multiply "minus" by "minus", we get "plus".

They are based on other principles. If a negative number is greater in absolute value than our positive one, then the result, of course, will be negative. Surely, you are wondering what a module is and why it is here at all. Everything is very simple. Modulus is the value of a number, but without a sign. For example -7 and 3. Modulo -7 will be just 7 , and 3 will remain 3. As a result, we see that 7 is greater, that is, it turns out that our negative number is greater. So it will come out -7 + 3 \u003d -4. It can be made even easier. Just put a positive number in the first place, and 3-7 = -4 will come out, perhaps it is more understandable for someone. Subtraction works in exactly the same way.

Why does a minus times a minus equal a plus?

"The enemy of my enemy is my friend."

A long time ago, only natural numbers were known to people: 1, 2, 3, . They were used to count utensils, loot, enemies, etc. But the numbers themselves are pretty useless - you need to be able to handle them. Addition is clear and understandable, besides, the sum of two natural numbers is also a natural number (a mathematician would say that the set of natural numbers is closed under the addition operation). Multiplication is, in fact, the same addition if we are talking about natural numbers. In life, we often perform actions related to these two operations (for example, when shopping, we add and multiply), and it is strange to think that our ancestors encountered them less often - addition and multiplication were mastered by mankind a very long time ago. Often it is necessary to divide one quantity by another, but here the result is not always expressed as a natural number - this is how fractional numbers appeared.

Negative numbers appear in Indian documents from the 7th century AD; the Chinese, apparently, began to use them a little earlier. They were used to account for debts or in intermediate calculations to simplify the solution of equations - it was only a tool to get a positive answer. The fact that negative numbers, unlike positive ones, do not express the presence of any entity, aroused strong distrust. People in the literal sense of the word avoided negative numbers: if the problem got a negative answer, they believed that there was no answer at all. This distrust persisted for a very long time, and even Descartes - one of the "founders" of modern mathematics - called them "false" (in the 17th century!).

7x - 17 = 2x - 2. It can be solved like this: move the terms with the unknown to the left side, and the rest to the right, it will turn out 7x - 2x = 17 - 2 , 5x = 15 , x=3

But one could accidentally do it differently: move the terms with the unknown to the right side and get 2 - 17 = 2x - 7x , (–15) ​​= (–5)x. To find the unknown, you need to divide one negative number by another: x = (–15)/(–5). But the correct answer is known, and it remains to be concluded that (–15)/(–5) = 3 .

. Secondly, by allowing the use of negative numbers, we get rid of the tedious (if the equation turns out to be more complicated, with a large number of terms) search for the solution path in which all actions are performed only on natural numbers. Moreover, we can no longer think every time about the meaningfulness of the quantities being converted - and this is already a step towards turning mathematics into an abstract science.

The rules for actions on negative numbers were not formed immediately, but became a generalization of numerous examples that arose when solving applied problems. In general, the development of mathematics can be conditionally divided into stages: each next stage differs from the previous one by a new level of abstraction in the study of objects. So, in the 19th century, mathematicians realized that integers and polynomials, for all their outward dissimilarity, have much in common: both can be added, subtracted, and multiplied. These operations obey the same laws - both in the case of numbers and in the case of polynomials. But the division of integers by each other, so that the result is again integers, is not always possible. The same is true for polynomials.

ring axioms

ring

• A + B = B + A for any elements A and B) and associative ( A + (B + C) = (A + B) + C A + 0 = A, and for any element A (–A)), what A + (–A) = 0 ;
• multiplication obeys the combination law: A (B C) = (A B) C ;

Note that rings, in the most general construction, do not require multiplication to be permutable, nor is it invertible (that is, it is not always possible to divide), nor does it require the existence of a unit - a neutral element with respect to multiplication. If these axioms are introduced, then other algebraic structures are obtained, but all the theorems proved for rings will be true in them.

A there are two opposites: B and FROM. That is A + B = 0 = A + C. Consider the sum A+B+C B: C: . Means, B=C .

Let us now note that A, and (–(–A)) (–A)

The first fact is obtained as follows: that is, (–A) B opposite A B, so it is equal to –(A B) .

0 B = 0 for any element B. Indeed, 0 B = (0 + 0) B = 0 B + 0 B. That is, the addition 0 B

Rules for multiplying minus by minus





With some stretch, the same explanation is suitable for the product 1-5, if we assume that the "sum" of a single

term is equal to this term. But the product 0 5 or (-3) 5 cannot be explained in this way: what does the sum of zero or minus three terms mean?

It is possible, however, to rearrange the factors

If we want the product not to change when the factors are rearranged - as it was for positive numbers - then we must thereby assume that

Now let's move on to the product (-3) (-5). What is it equal to: -15 or +15? Both options make sense. On the one hand, a minus in one factor already makes the product negative - all the more it should be negative if both factors are negative. On the other hand, in Table. 7 already has two minuses, but only one plus, and "fairly" (-3)-(-5) should be equal to +15. So what do you prefer?



Of course, you will not be confused by such conversations: from a school mathematics course, you firmly learned that a minus by a minus gives a plus. But imagine that your younger brother or sister asks you: why? What is it - a teacher's whim, an indication of higher authorities, or a theorem that can be proven?

Usually, the rule for multiplying negative numbers is explained using examples like the one presented in Table. eight.



It can be explained in another way. Let's write numbers in a row

Now let's write the same numbers multiplied by 3:

It is easy to see that each number is 3 more than the previous one. Now let's write the same numbers in reverse order (starting, for example, with 5 and 15):



At the same time, the number -15 turned out to be under the number -5, so 3 (-5) \u003d -15: plus by minus gives minus.

Now let's repeat the same procedure, multiplying the numbers 1,2,3,4,5. by -3 (we already know that plus times minus equals minus):

Each next number of the bottom row is less than the previous one by 3. Let's write the numbers in reverse order



The number -5 turned out to be 15, so (-3) (-5) = 15.

Perhaps these explanations would satisfy your younger brother or sister. But you have the right to ask how things really are and is it possible to prove that (-3) (-5) = 15?

The answer here is that it can be proven that (-3) (-5) must equal 15, if only we want the usual properties of addition, subtraction, and multiplication to remain true for all numbers, including negative ones. The outline of this proof is as follows.

Let us first prove that 3 (-5) = -15. What is -15? This is the opposite of 15, i.e. the number that adds up to 15 to 0. So we need to prove that

(By parenthesizing 3, we have used the distributive law ab + ac = a(b + c) for - after all, we assume that it remains true for all numbers, including negative ones.) So, (The meticulous reader will ask us why. We honestly admit : the proof of this fact - like the discussion of what zero is in general - we skip.)

Let us now prove that (-3) (-5) = 15. To do this, we write



and multiply both sides of the equation by -5:

Let's open the brackets on the left side:

i.e. (-3) (-5) + (-15) = 0. Thus, the number is opposite to the number -15, i.e. equal to 15. (There are also gaps in this reasoning: it would be necessary to prove that and that there is only one number opposite to -15.)

Negative rule. Why minus times minus equals plus

When listening to a mathematics teacher, most students perceive the material as an axiom. At the same time, few people try to get to the bottom and figure out why “minus” to “plus” gives a “minus” sign, and when multiplying two negative numbers, a positive one comes out.

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, say, 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 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, it is customary to call a ring a set in which two operations with two elements are involved. But it's better to understand this with an example.

Ring axiom

There are several mathematical laws.

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

The multiplication (V x C) x D \u003d V x (C x D) also obeys them.

Nobody 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, addition-neutral element 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) \u003d 0.

Derivation of axioms for negative numbers

By accepting the above statements, we can answer the question: "" Plus "on" 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 must first prove that each of the elements has only one opposite "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, is equal to 0. Hence, V = V + C + D.

The value for D is derived in 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, nevertheless, the “plus” on the “minus” gives a “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 \u003d (C + (-C)) x V \u003d C x V + (-C) x V. It follows from this that C x V is opposite to (-) C x V, which means (- 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, it is possible to deduce not only how much "plus" by "minus" gives, but also what happens when negative numbers are multiplied.

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

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

Suppose that C - (-V) = D, based on 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 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) \u003d D, two identical products can be added and subtracted to the expression, which will not change its value: (-C) x (-V) + (C x V) - (C x V) \u003d 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;

It follows from this that C x V \u003d (-C) x (-V).

Similarly, we can prove that the result of dividing two negative numbers will be positive.

General mathematical rules

Of course, such an 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. After all, “plus” multiplied by “minus” gives “minus”. True, children do not try too hard 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 their 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 this without exception. This is true for both integers and fractional numbers.

Minus and plus are signs of negative and positive numbers in mathematics. They interact with themselves in different ways, so when performing any actions with numbers, for example, division, multiplication, subtraction, addition, etc., it is necessary to take into account sign rules. Without these rules, you will never be able to solve even the simplest algebraic or geometric problem. Without knowledge of these rules, you will not be able to study not only mathematics, but also physics, chemistry, biology, and even geography.

Subtraction and addition.

Two negatives make an affirmative- this is a rule that we learned in school and apply all our lives. Who among us wondered why? Of course, it is easier to memorize this statement without further questions and not delve deeply into the essence of the issue. Now there is already enough information that needs to be “digested”. But for those who are still interested in this question, we will try to explain this mathematical phenomenon.

Since ancient times, people have been using positive natural numbers: 1, 2, 3, 4, 5, ... Cattle, crops, enemies, etc. were counted with the help of numbers. When adding and multiplying two positive numbers, they always got a positive number, when dividing some quantities by others, they did not always get natural numbers - this is how fractional numbers appeared. What about subtraction? From childhood, we know that it is better to add the smaller to the larger and subtract the smaller from the larger, while again we do not use negative numbers. It turns out that if I have 10 apples, I can only give less than 10 or 10 to someone. There is no way I can give 13 apples, because I don’t have any. There was no need for negative numbers for a long time.

Only from the 7th century AD. negative numbers were used in some counting systems as auxiliary values, which made it possible to obtain a positive number in the answer.

Consider an example, 6x - 30 \u003d 3x - 9. To find the answer, it is necessary to leave the terms with unknowns on the left side, and the rest on the right: 6x - 3x \u003d 30 - 9, 3x \u003d 21, x \u003d 7. When solving this equation, we even there are no negative numbers. We could transfer terms with unknowns to the right side, and without unknowns - to the left: 9 - 30 \u003d 3x - 6x, (-21) \u003d (-3x). When dividing a negative number by a negative one, we get a positive answer: x \u003d 7.

Actions with negative numbers should lead us to the same answer as actions with only positive numbers. We can no longer think about the practical unsuitability and meaningfulness of actions - they help us solve the problem much faster, without reducing the equation to the form only with positive numbers. In our example, we did not use complex calculations, but with a large number of terms, calculations with negative numbers can make our work easier.

Over time, after lengthy experiments and calculations, it was possible to identify the rules that all numbers and actions on them obey (in mathematics they are called axioms). That's where it came from an axiom that states that when you multiply two negative numbers, you get a positive number.

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1) Why does minus one times minus one equal plus one?
2) Why does minus one times plus one equal minus one?

"The enemy of my enemy is my friend."

The easiest answer is: "Because these are the rules for working with negative numbers." The rules we learn in school and apply throughout our lives. However, the textbooks do not explain why the rules are the way they are. We will first try to understand this from the history of the development of arithmetic, and then we will answer this question from the point of view of modern mathematics.

A long time ago, only natural numbers were known to people: 1, 2, 3, . They were used to count utensils, loot, enemies, etc. But the numbers themselves are pretty useless - you need to know how to handle them. Addition is clear and understandable, and besides, the sum of two natural numbers is also a natural number (a mathematician would say that the set of natural numbers is closed under the operation of addition). Multiplication is, in fact, the same addition if we are talking about natural numbers. In life, we often perform actions related to these two operations (for example, when shopping, we add and multiply), and it is strange to think that our ancestors encountered them less often - addition and multiplication were mastered by mankind a very long time ago. Often it is necessary to divide one quantity by another, but here the result is not always expressed by a natural number - this is how fractional numbers appeared.

Subtraction, of course, is also indispensable. But in practice, we tend to subtract the smaller number from the larger number, and there is no need to use negative numbers. (If I have 5 candies and I give 3 to my sister, then I will have 5 - 3 = 2 candies, but I can’t give her 7 candies with all my desire.) This can explain why people did not use negative numbers for a long time.

Negative numbers appear in Indian documents from the 7th century AD; the Chinese, apparently, began to use them a little earlier. They were used to account for debts or in intermediate calculations to simplify the solution of equations - it was only a tool to get a positive answer. The fact that negative numbers, unlike positive ones, do not express the presence of any entity, aroused strong distrust. People in the literal sense of the word avoided negative numbers: if the problem got a negative answer, they believed that there was no answer at all. This distrust persisted for a very long time, and even Descartes, one of the "founders" of modern mathematics, called them "false" (in the 17th century!).

Consider, for example, the equation 7x - 17 = 2x - 2. It can be solved like this: move the terms with the unknown to the left side, and the rest to the right, it will turn out 7x - 2x = 17 - 2 , 5x = 15 , x=3. With this solution, we did not even meet negative numbers.

What does this simple example demonstrate? First, it becomes clear the logic that determined the rules for actions on negative numbers: the results of these actions must match the answers that are obtained in a different way, without negative numbers. Secondly, by allowing the use of negative numbers, we get rid of the tedious (if the equation turns out to be more complicated, with a large number of terms) search for the solution path in which all actions are performed only on natural numbers. Moreover, we can no longer think every time about the meaningfulness of the quantities being converted - and this is already a step towards turning mathematics into an abstract science.

The rules for actions on negative numbers were not formed immediately, but became a generalization of numerous examples that arose when solving applied problems. In general, the development of mathematics can be conditionally divided into stages: each next stage differs from the previous one by a new level of abstraction in the study of objects. So, in the 19th century, mathematicians realized that integers and polynomials, for all their outward dissimilarity, have much in common: both can be added, subtracted, and multiplied. These operations obey the same laws - both in the case of numbers and in the case of polynomials. But the division of integers by each other, so that the result is again integers, is not always possible. The same is true for polynomials.

Then other collections of mathematical objects were discovered on which such operations can be performed: formal power series, continuous functions. Finally, the understanding came that if you study the properties of the operations themselves, then the results can then be applied to all these collections of objects (this approach is typical for all modern mathematics).

As a result, a new concept appeared: ring. It's just a bunch of elements plus actions that can be performed on them. The fundamental rules here are just the rules (they are called axioms) to which actions are subject, not the nature of the elements of the set (here it is, a new level of abstraction!). Wishing to emphasize that it is the structure that arises after the introduction of axioms that is important, mathematicians say: the ring of integers, the ring of polynomials, etc. Starting from the axioms, one can derive other properties of rings.

We will formulate the axioms of the ring (which are, of course, similar to the rules for operations with integers), and then we will prove that in any ring, multiplying a minus by a minus results in a plus.

ring is called a set with two binary operations (that is, two elements of the ring are involved in each operation), which are traditionally called addition and multiplication, and the following axioms:

• addition of ring elements obeys commutative ( A + B = B + A for any elements A and B) and associative ( A + (B + C) = (A + B) + C) laws; the ring contains a special element 0 (a neutral element by addition) such that A + 0 = A, and for any element A there is an opposite element (denoted (–A)), what A + (–A) = 0 ;
• addition and multiplication are related by the following parentheses expansion rules: (A + B) C = A C + B C and A (B + C) = A B + A C .

We note that rings, in the most general construction, do not require multiplication to be permutable, nor is it invertible (that is, it is not always possible to divide), nor does it require the existence of a unit, a neutral element with respect to multiplication. If these axioms are introduced, then other algebraic structures are obtained, but all the theorems proved for rings will be true in them.

We now prove that for any elements A and B arbitrary ring is true, firstly, (–A) B = –(A B), and secondly (–(–A)) = A. From this, statements about units easily follow: (–1) 1 = –(1 1) = –1 and (–1) (–1) = –((–1) 1) = –(–1) = 1 .

To do this, we need to establish some facts. First we prove that each element can have only one opposite. Indeed, let the element A there are two opposites: B and FROM. That is A + B = 0 = A + C. Consider the sum A+B+C. Using the associative and commutative laws and the property of zero, we get that, on the one hand, the sum is equal to B : B = B + 0 = B + (A + C) = A + B + C, and on the other hand, it is equal to C : A + B + C = (A + B) + C = 0 + C = C. Means, B=C .

Let us now note that A, and (–(–A)) are opposite to the same element (–A), so they must be equal.

The first fact goes like this: 0 = 0 B = (A + (–A)) B = A B + (–A) B, that is (–A) B opposite A B, so it is equal to –(A B) .

To be mathematically rigorous, let's explain why 0 B = 0 for any element B. Indeed, 0 B = (0 + 0) B = 0 B + 0 B. That is, the addition 0 B does not change the amount. So this product is equal to zero.

And the fact that there is exactly one zero in the ring (after all, the axioms say that such an element exists, but nothing is said about its uniqueness!), we will leave to the reader as a simple exercise.