In what cases does it give a plus? How to understand why ";plus"? to ";minus"; gives ";minus";

Minus and plus are signs of negative and positive numbers in mathematics. They interact with themselves differently, so when performing any operations 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 knowing these rules, you will not be able to study not only mathematics, but also physics, chemistry, biology, and even geography.

Let's take a closer look at 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”, 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 different 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. The modulus is the value of a number, but without a sign. For example -7 and 3. Modulo -7 will simply be 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 comes out -7+3 = -4. It can be made even simpler. Just put a positive number in first place, and it will come out 3-7 = -4, perhaps this is more clear to someone. Subtraction works on exactly the same principle.

Two negatives make an affirmative- This is a rule that we learned in school and apply throughout our lives. And which of us was interested in why? Of course, it’s easier to remember this statement without asking unnecessary 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 give an explanation of this mathematical phenomenon.

Since ancient times, people have used positive natural numbers: 1, 2, 3, 4, 5,... Numbers were used to count livestock, crops, enemies, etc. When adding and multiplying two positive numbers, they always got a positive number; when dividing one quantity by another, 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 less to more and subtract less from more, and again we do not use negative numbers. It turns out that if I have 10 apples, I can only give someone less than 10 or 10. There is no way I can give 13 apples, because I don’t have them. 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 quantities that made it possible to obtain a positive number in the answer.

Let's look at an example, 6x – 30 = 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 = 30 – 9, 3x = 21, x = 7. When solving this equation, we even There were no negative numbers. We could transfer members with unknowns to right side, and without unknowns - to the left: 9 – 30 = 3x – 6x, (-21) = (-3x). When dividing a negative number by a negative number, we get a positive answer: x = 7.

What do we see?

Working with negative numbers should get us to the same answer as working with only positive numbers. We no longer have to think about the practical impossibility and meaningfulness of actions - they help us solve the problem much faster, without reducing the equation to a form with only positive numbers. In our example, we did not use complex calculations, but if there are 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 govern all numbers and operations on them (in mathematics they are called axioms). This is where it came from an axiom that states that when two negative numbers are multiplied, we get a positive number.

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Listening to a math teacher, most students perceive the material as an axiom. At the same time, few people try to get to the bottom of it and figure out why “minus” by “plus” gives a “minus” sign, and when multiplying two negative numbers, a positive result comes out.

Laws of mathematics

Most adults are unable to explain to themselves or their children why this happens. They firmly mastered this material at school, but 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 things and understand, say, why a “plus” and a “minus” gives a “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 negative and 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 have 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 it’s better to understand this with an example.

Ring axiom

There are several mathematical laws.

The first of them is commutative, according to it, C + V = V + C.
The second is called associative (V + C) + D = V + (C + D).

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

No one has canceled the rules according to which parentheses 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, when used 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

Having accepted the above statements, we can answer the question: “Plus and minus give what sign?” Knowing the axiom about multiplying 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, you will first have to prove that each element has only one “brother” opposite to it. Consider the following example of proof. Let's try to imagine that for C two numbers are opposite - V and D. From this it follows that C + V = 0 and C + D = 0, that is, C + V = 0 = C + D. Remembering the laws of commutation 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 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, is equal to 0. This means 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 “plus” to “minus” still gives “minus”, you need to understand the following. So, for the element (-C), C and (-(-C)) are opposite, 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 from this that C x V is the opposite of (-)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 = (0 + 0) x V = 0 x V + 0 x V. This means that adding the product 0 x V does not change the established amount in any way. After all, this product equals zero.

Knowing all these axioms, you can deduce not only how much “plus” and “minus” gives, but also what happens when multiplying negative numbers.

Multiplying and dividing two numbers with a “-” sign

If you don’t go deep into mathematical nuances, you can try to explain the rules for operating with negative numbers in a simpler way.

Let's assume 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 “minuses” in a row, the mentioned signs should be changed to “plus”. Now let's look at 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;

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

Similarly, you can prove that dividing two negative numbers will result in a positive number.

General mathematical rules

Of course, this explanation is not suitable for schoolchildren junior classes who are just starting to learn abstract negative numbers. It is better for them to explain on visible objects, manipulating the term behind the looking glass with which they are familiar. For example, invented but non-existent toys are located there. They can be displayed with a “-” sign. Multiplying two mirror objects transfers them to another world, which is equated to the real one, that is, as a result we have positive numbers. But multiplying an abstract negative number by a positive one only gives a result that is familiar to everyone. After all, “plus” multiplied by “minus” gives “minus”. True, children do not really try to understand all the mathematical nuances.

Although, let's face it, for many people, even with higher education Many rules remain a mystery. Everyone takes for granted what teachers teach them, without difficulty delving into all the complexities that mathematics conceals. “Minus” for “minus” gives “plus” - everyone without exception knows this. This is true for both whole and fractional numbers.

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 operating with negative numbers.” Rules that 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 based on 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, people knew only natural numbers: 1, 2, 3, ... They were used to count utensils, loot, enemies, etc. But numbers themselves are quite 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 essentially the same as 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 humanity a very long time ago. Often you have to divide some quantities by others, but here the result is not always expressed as a natural number - this is how fractional numbers appeared.

Of course, you can’t do without subtraction either. But in practice, we usually subtract the smaller number from the larger number, and there is no need to use negative numbers. (If I have 5 candies and give my sister 3, then I will have 5 - 3 = 2 candies left, but I cannot give her 7 candies even if I want to.) This can explain why people have not used negative numbers for a long time.

Negative numbers have appeared in Indian documents since the 7th century AD; The Chinese apparently started using them a little earlier. They were used to account for debts or in intermediate calculations to simplify the solution of equations - it was just a tool for obtaining a positive answer. The fact that negative numbers, unlike positive numbers, do not express the presence of any entity caused strong mistrust. People literally avoided negative numbers: if a problem had a negative answer, they believed that there was no answer at all. This mistrust 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 this way: 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 didn’t even encounter negative numbers.

But it was possible to 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 conclude that (–15)/(–5) = 3 .

What does this simple example demonstrate? Firstly, the logic that determined the rules for operating with negative numbers becomes clear: the results of these actions must match the answers obtained in another 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 a solution in which all actions are performed only on natural numbers. Moreover, we may no longer think every time about the meaningfulness of the transformed quantities - and this is already a step towards turning mathematics into an abstract science.

The rules for operating with 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 divided into stages: each next stage differs from the previous one with a new level of abstraction when studying objects. Thus, in the 19th century, mathematicians realized that integers and polynomials, despite all their external differences, have much in common: both can be added, subtracted and multiplied. These operations are subject to the same laws - both in the case of numbers and in the case of polynomials. But dividing integers by each other so that the result is integers again is not always possible. It's the same with polynomials.

Then other sets of mathematical objects were discovered on which such operations could 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 characteristic of all modern mathematics).

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

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

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

the addition of elements of the ring is subject to commutative ( A + B = B + A for any elements A And B) and associative ( A + (B + C) = (A + B) + C) laws; in the ring there is a special element 0 (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 ;
multiplication obeys the combinational law: A·(B·C) = (A·B)·C ;
Addition and multiplication are related by the following rules for opening parentheses: (A + B) C = A C + B C And A (B + C) = A B + A C .

Note that rings, in the most general construction, do not require either the commutability of multiplication, or its invertibility (that is, division cannot always be done), or the existence of a unit - a neutral element in multiplication. If we introduce these axioms, we get different algebraic structures, but in them all the theorems proven for rings will be true.

Now we prove that for any elements A And B of an arbitrary ring is true, firstly, (–A) B = –(A B), and secondly (–(–A)) = A. Statements about units easily follow from this: (–1) 1 = –(1 1) = –1 And (–1)·(–1) = –((–1)·1) = –(–1) = 1 .

To do this we will need to establish some facts. First we prove that each element can have only one opposite. In fact, let the element A there are two opposites: B And WITH. That is A + B = 0 = A + C. Let's consider the amount A+B+C. Using the associative and commutative laws and the property of zero, we obtain 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 C: A + B + C = (A + B) + C = 0 + C = C. Means, B=C .

Let us now note that A, And (–(–A)) are opposites of 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, which means it is equal –(A B) .

To be mathematically rigorous, let's also 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. This means that 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.

Answered: Evgeniy Epifanov

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Good answer. But for the level of a high school freshman. It seems to me that it can be explained more simply and clearly, using the example of the formula “distance = speed * time” (grade 2).

Let's say we are walking along the road, a car overtakes us and begins to move away. Time is growing - and the distance to it is growing. We will consider the speed of such a machine to be positive; it can be, for example, 10 meters per second. By the way, how many kilometers per hour is this? 10/1000(km)*60(sec)*60(min)= 10*3.6 = 36 km/h. A little. Probably the road is bad...

But the car coming towards us does not move away, but approaches. Therefore, it is convenient to consider its speed to be negative. For example -10 m/sec. The distance decreases: 30, 20, 10 meters to the oncoming car. Every second is minus 10 meters. Now it’s clear why the speed is minus? So she flew past. What is the distance to it in a second? That's right, -10 meters, i.e. "10 meters behind."

Here we have received the first statement. (-10 m/sec) * (1 sec) = -10 m.
Minus (negative speed) by plus (positive time) gave minus (negative distance, the car is behind me).

And now attention - minus to minus. Where was the oncoming car a second BEFORE it passed by? (-10 m/sec) * (- 1 sec) = 10 m.
Minus (negative speed) by minus (negative time) = plus (positive distance, the car was 10 meters in front of my nose).

Is this clear, or does anyone know an even simpler example?

Answer

Yes, it can be proven easier! 5*2 is plotted twice on the number line, in positive side, the number is 5, and then we get the number 10. if 2*(-5), then we count twice according to the number 5, but in the negative direction, and we get the number (-10), now we represent 2*(-5) as
2*5*(-1)=-10, the answer is rewritten from the previous calculation, and not obtained in this one. This means that we can say that when a number is multiplied by (-1), there is an inversion of the numerical two-polar axis, i.e. reversing polarity. What we put into the positive part became negative and vice versa. Now (-2)*(-5), we write it as (-1)*2*(-5)=(-1)*(-10), setting aside the number (-10), and changing the polarity of the axis, because . multiply by (-1), we get +10, I just don’t know if it turned out easier?

Answer

I think you're right. I will just try to show your point of view in more detail, because... I see that not everyone understood this.
Minus means take away. If 5 apples were taken from you once, then in the end 5 apples were taken from you, which is conventionally indicated by a minus, i.e. – (+5). After all, you need to somehow indicate the action. If 1 apple was selected 5 times, then in the end the same was selected: – (+5). At the same time, the selected apples did not become imaginary, because The law of conservation of matter has not been repealed. The positive apples simply went to whoever took them. This means there are no imaginary numbers, there are relative motion matter with a + or - sign. But if so, then the notation: (-5) * (+1) = -5 or (+5) * (-1) = -5 does not accurately reflect reality, but denotes it only conditionally. Since there are no imaginary numbers, the entire product is always positive → “+” (5*1). Next, the positive product is negated, which means subtraction → “- +” (5*1). Here the minus does not compensate for the plus, but negates it and takes its place. Then in the end we get: -(5*1) = -(+5).
For two minuses, you can write: “- -” (5*1) = 5. The sign “- -” means “+”, i.e. expropriation of expropriators. First the apples were taken from you, and then you took them from your offender. As a result, all apples remained positive, but selection did not take place, because a social revolution took place.
Generally speaking, the fact that the negation of the negation eliminates the negation and everything that the negation refers to is clear to children and without explanation, because It is obvious. You only need to explain to children what adults have artificially confused, so much so that they themselves cannot figure it out. And the confusion lies in the fact that instead of negating the action, negative numbers were introduced, i.e. negative matter. So the children are perplexed why, when adding negative matter, the sum turns out to be negative, which is quite logical: (-5) + (-3) = -8, and when multiplying the same negative matter: (-5) * (-3) = 15 , it suddenly ends up being positive, which is not logical! After all, with negative matter everything should happen the same as with positive matter, only with a different sign. Therefore, it seems more logical to children that when negative matter is multiplied, it is negative matter that should multiply.
But here, too, not everything is smooth, because to multiply negative matter, it is enough for only one number to be negative. In this case, one of the factors, which denotes not the material content, but the times of repetition of the selected matter, is always positive, because times cannot be negative even if negative (selected) matter is repeated. Therefore, when multiplying (dividing), it is more correct to place signs in front of the entire product (division), which we showed above: “- +” (5*1) or “- -” (5*1).
And in order for the minus sign to be perceived not as a sign of an imaginary number, i.e. negative matter, and as an action, adults must first agree among themselves that if a minus sign is in front of a number, then it means a negative action with a number, which is always positive, and not imaginary. If the minus sign is in front of another sign, then it means a negative action with the first sign, i.e. changes it to the opposite. Then everything will fall into place naturally. Then you need to explain this to the children and they will perfectly understand and assimilate such an understandable rule of adults. After all, now all the adult participants in the discussion are actually trying to explain the inexplicable, because... There is no physical explanation for this issue, it is just a convention, a rule. But explaining abstraction by abstraction is a tautology.
If the minus sign negates the number, then it is a physical action, but if it negates the action itself, then it is simply a conditional rule. That is, adults simply agreed that if selection is denied, as in the issue under consideration, then there is no selection, no matter how many times! At the same time, everything that you had remains with you, be it just a number, be it a product of numbers, i.e. many selection attempts. That's all.
If someone disagrees, then calmly think again. After all, the example with cars, in which there is a negative speed and negative time a second before the meeting, is just a conditional rule associated with the reference system. In another frame of reference, the same speed and the same time will become positive. And the example with the looking glass is connected with the fairy tale rule, in which a minus being reflected in a mirror only conditionally, but not at all physically, becomes a plus.

Answer

Everything seems clear with the mathematical disadvantages. But in language, when a negative question is asked, how do you answer it? For example, I was always puzzled by this question: “Would you like some tea?” How can I answer this if I want tea? It seems that if you say “Yes”, then they won’t give you tea (it’s like + and -), if no, then they should give you (- and -), and if “No, I don’t want”???

Answer

In order to answer this children's question, you first need to answer a couple of adult questions: “What is a minus in mathematics?” and "What are multiplication and division?" As far as I understand, this is where the problems begin, which ultimately lead to rings and other nonsense when answering such a simple childish question.

Answer

The answer is clearly not for ordinary schoolchildren!
In elementary school, I read a wonderful book - the one about Dwarfism and Al-Jebra, and maybe in a math club they gave an example - they put two people with apples on opposite sides of an equal sign different colors and offered to give each other apples. Then other signs were placed between the participants in the game - plus, minus, more, less.

Answer

Childish answer, huh??))
It may sound cruel, but the author himself does not understand why a minus on a minus gives a plus :-)
Everything in the world can be explained visually, because abstractions are needed only to explain the world. They are tied to reality, and do not live by themselves in delusional textbooks.
Although for an explanation you need to at least know physics and sometimes biology, coupled with the basics of human neurophysiology.

But nevertheless, the first part gave hope to understand, and very clearly explained the need for negative numbers.
But the second traditionally slipped into schizophrenia. A and B - these must be real objects! so why call them with these letters when you can take, for example, loaves of bread or apples
If.. if it were possible... yes?))))))

And... even using the right basis from the first part (that multiplication is the same as addition) - with minuses we get a contradiction))
-2 + -2 = -4
But
-2 * -2 =+4))))
and even if we consider that this is minus two, taken minus two times, it will turn out
-2 -(-2) -(-2) = +2

It was worth simply admitting that since the numbers are virtual, then for relatively correct accounting we had to come up with virtual rules.
And this would be the TRUTH, and not ringed nonsense.

Answer

In his example, Academon made a mistake:
In fact, (-2)+(-2) = (-4) is 2 times (-2), i.e. (-2) * 2 = (-4).
As for multiplying two negative numbers, without contradiction, this is the same addition, only on the other side of “0” on the number line. Namely:
(-2) * (-2) = 0 –(-2) –(-2) = 2 + 2 = 4. So it all adds up.
Well, regarding the reality of negative numbers, how do you like this example?
If I have, say, $1000 in my pocket, my mood can be called “positive.”
If $0, then the state will be “none”.
What if (-1000)$ is a debt that needs to be repaid, but there is no money...?

Answer

Minus for minus - there will always be a plus,
Why this happens, I cannot say.

Why -na-=+ puzzled me back in school, in 7th grade (1961). I tried to come up with another, more “fair” algebra, where +na+=+, and -na-=-. It seemed to me that it would be more honest. But what then to do with +na- and -na+? I didn’t want to lose the commutativity of xy=yx, but there’s no other way.
What if you take not 2 signs but three, for example +, - and *. Equal and symmetrical.

ADDITION
(+a)+(-a),(+a)+(*a),(*a)+(-a) do not add up(!), like the real and imaginary parts of a complex number.
But for that (+a)+(-a)+(*a)=0.

For example, what is (+6)+(-4)+(*2) equal to?

(+6)=(+2)+(+2)+(+2)
(-4)=(-2)+(-2)
(*2)=(*2)
(+2)+(-2)+(*2)=0
(+6)+(-4)+(*2)=(+2)+(+2)+(+2)+(-2)+(-2)+(*2)=(+2)+(+2)+(-2)= (+4)+(-2)
It's not easy, but you can get used to it.

Now MULTIPLICATION.
Let us postulate:
+na+=+ -na-=- *na*=* (fair?)
+na-=-na+=* +na*=*na+=- -na*=*na-=+ (fair!)
It would seem that everything is fine, but multiplication is not associative, i.e.
a(bc) is not equal to (ab)c.

And if so
+on+=+ -on-=* *on*=-
+na-=-na+=- +na*=*na+=* -na*=*na-=+
Again unfair, + singled out as special. BUT A NEW ALGEBRA with three signs was born. Commutative, associative and distributive. It has a geometric interpretation. It is isomorphic Complex numbers. It can be expanded further: four characters, five...
This have not happened before. Take it, people, use it.

Answer

A child's question is generally a child's answer.
There is our world, where everything is “plus”: apples, toys, cats and dogs, they are real. You can eat an apple, you can pet a cat. And there is also an imaginary world, a looking glass. There are also apples and toys there, through the looking glass, we can imagine them, but we cannot touch them - they are made up. We can get from one world to another using the minus sign. If we have two real apples (2 apples), and we put a minus sign (-2 apples), we will get two imaginary apples in the looking glass. The minus sign takes us from one world to another, back and forth. There are no mirror apples in our world. We can imagine a whole bunch of them, even a million (minus a million apples). But you won’t be able to eat them, because we don’t have minus apples, all the apples in our stores are plus apples.
To multiply means to arrange some objects in the form of a rectangle. Let's take two dots ":" and multiply them by three, we get: ": : :" - six dots in total. You can take a real apple (+I) and multiply it by three, we get: “+YAYA” - three real apples.
Now let's multiply the apple by minus three. We will again get three apples "+YAYA", but the minus sign will take us to the looking glass, and we will have three looking-glass apples (minus three apples -YAYA).
Now let's multiply minus apple (-I) by minus three. That is, we take an apple, and if there is a minus in front of it, we transfer it to the looking glass. There we multiply it by three. Now we have three looking glass apples! But there is one more drawback. He will move the received apples back to our world. As a result, we get three real tasty apples +YAYA that you can gobble up.

Answer

Everything is fine until last step. When multiplied by minus one of three mirror apples, we must reflect these apples in another mirror. Their location will coincide with the real ones, but they will be as imaginary as the first mirror ones and just as inedible. That is (-1)*(-1)= --1<> 1.
In fact, I am confused by another point related to multiplying negative numbers, namely:
Is the equality true:
((-1)^1,5)^2 = ((-1)^2)^1,5 = (-1)^3 ?
This question arose from an attempt to understand the behavior of the graph of the function y=x^n, where x and n are real numbers.
It turns out that the graph of the function will always be located in the 1st and 3rd quarters, except for those cases when n is even. In this case, only the curvature of the graph changes. But parity n is a relative value, because we can take another reference system, in which n = 1.1*k, then we get
y = x^(1,1*k) = (x^1,1)^k
and the parity here will be different...
And in addition, I propose to add to the argument what happens to the graph of the function y = x^(1/n). I assume, not without reason, that the graph of the function should be symmetrical to the graph of y = x^n relative to the graph of the function y = x.

Answer

There are several ways to explain the rule “minus for minus gives plus.” Here is the simplest. Multiplication by naturals. the number n is the stretching of the segment (located on the number axis) n times. Multiplication by -1 is a reflection of the segment relative to the origin. As a shortest explanation of why (-1)*(-1) = +1, this method is suitable. The bottleneck of this approach is that it is necessary to separately determine the sum of such operators.

Answer

You can go when explaining from complex numbers
as a more general form of representing numbers
Trigonometric form of a complex number
Euler's formula
The sign in this case is just an argument (angle of rotation)
When multiplying, angles are added
0 degrees corresponds to +
180 degrees corresponds to -
Multiplying - by - is equivalent to 180+180=360=0

Answer

Will this work?

Denials are the opposite. For simplicity, in order to temporarily move away from the minuses, we will replace the statements and make the starting point larger. Let's start counting not from zero, but from 1000.

Let's say two people owe me two rubles: 2_people*2_rubles=4_rubles owe me in total. (my balance is 1004)

Now the inverses (negative numbers, but inverse/positive statements):

minus 2 people = it means they don’t owe me, but I owe (I owe more people than they owe me). For example, I owe 10 people, but I only owe 8. Mutual calculations can be reduced and not taken into account, but you can keep this in mind if it is more convenient to work with positive numbers. That is, everyone gives money to each other.

minus 2 rubles = a similar principle - you must take more than you give. So I owe everyone two rubles.

-(2_people)*2_rubles=I_ow_2_to each_=-4 from me. My balance is 996 rubles.

2_people*(-2_rubles) = two_should_take_2_rubles_from_me=- 4 from me. My balance is 996 rubles.

-(2_people)*(-2_rubles) = everyone_should_take_from_me_less_than_should_give_by_2_rubles

In general, if you imagine that everything is spinning not around 0, but around, for example, 1000, and they give out money in 10 increments, taking away 8 in increments. Then you can consistently perform all the operations of giving someone money or taking it away, and come to the conclusion that if the extra two (we will reduce the rest by mutual offset) will take two rubles less from me than they will return, then my well-being will increase by a positive figure of 4.

Answer

In search of a SIMPLE (understandable to a child) answer to the question posed (“Why does a minus on a minus give a plus”), I carefully read both the article proposed by the author and all the comments. I consider the most successful answer to be the one included in the epigraph: “The enemy of my enemy is my friend.” Much clearer! Simple and brilliant!

A certain traveler arrives on an island, about the inhabitants of which he knows only one thing: some of them tell only the truth, others only lies. Outwardly it is impossible to distinguish them. The traveler lands on the shore and sees the road. He wants to find out if this road leads to the city. Seeing a local resident on the road, he asks him ONLY ONE question, allowing him to find out that the road leads to the city. How did he ask this?

The solution is three lines below (just to pause and give you adults a chance to pause and think about this wonderful problem!) My third-grader grandson found the problem a bit too much for him yet, but understanding the answer no doubt brought him closer to understanding future math wisdom like “minus for minus gives plus.”

So the answer is:

“If I asked you if this road leads to the city, what would you tell me?”

The “algebraic” explanation could not shake either my ardent love for my father or my deep respect for his science. But I forever hated the axiomatic method with its unmotivated definitions.

It is interesting that this answer by I.V. Arnold to a child’s question practically coincided with the publication of his book “Negative Numbers in an Algebra Course.” There (in Chapter 7) a completely different answer is given, in my opinion, very clear. The book is available in in electronic format http://ilib.mccme.ru/djvu/klassik/neg_numbers.htm

Answer

If there is a paradox, you need to look for errors in the basics. There are three errors in the formulation of multiplication. This is where the “paradox” comes from. You just need to add a zero.

(-3) x (-4) = 0 - (-3) - (-3) - (-3) - (-3) = 0 + 3 + 3 + 3 + 3 = 12

Multiplication is adding to (or subtracting from) zero over and over again.

Multiplier (4) shows the number of addition or subtraction operations (the number of minus or plus signs when decomposing multiplication into addition).

The minus and plus signs for the multiplier (4) indicate either subtracting the multiplicand from zero or adding the multiplicand to zero.

In this particular example, (-4) indicates that you need to subtract ("-") from zero the multiplicand (-3) four times (4).

Correct the wording (three logical errors). Just add a zero. The rules of arithmetic will not change because of this.

Order of multiplication operation

The essence of the multiplication operation is based on a simpler arithmetic operation- . In fact, multiplication is the sum of the first factor, or multiplicand, a number of times that corresponds to the second factor. For example, in order to multiply 8 by 4, you need to add the number 8 4 times, resulting in 32. This method, in addition to providing an understanding of the essence of the multiplication operation, can be used to check the result obtained when calculating the desired product. It should be borne in mind that the verification necessarily assumes that the terms involved in the summation are identical and correspond to the first factor.

Solving multiplication examples

Thus, in order to solve the problem associated with the need to carry out multiplication, it may be sufficient to add a given number of times required number first multipliers. This method can be convenient for carrying out almost any calculations related to this operation. At the same time, in mathematics there are quite often standard numbers that involve standard single-digit integers. In order to facilitate their calculation, the so-called multiplication system was created, which includes a complete list of products of positive integer single-digit numbers, that is, numbers from 1 to 9. Thus, once you have learned, you can significantly facilitate the process of solving multiplication examples, based on the use of such numbers. However, for more complex options it will be necessary to implement this mathematical operation on one's own.

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Sources:

Multiplication in 2019

Multiplication is one of the four basic arithmetic operations, which is often used both in school and in Everyday life. How can you quickly multiply two numbers?

The basis of the most complex mathematical calculations are the four basic arithmetic operations: subtraction, addition, multiplication and division. Moreover, despite their independence, these operations, upon closer examination, turn out to be interconnected. Such a connection exists, for example, between addition and multiplication.

Number multiplication operation

There are three main elements involved in the multiplication operation. The first of these, usually called the first factor or multiplicand, is the number that will be subject to the multiplication operation. The second, called the second factor, is the number by which the first factor will be multiplied. Finally, the result of the multiplication operation performed is most often called a product.

It should be remembered that the essence of the multiplication operation is actually based on addition: to carry it out, it is necessary to add together a certain number of the first factors, and the number of terms of this sum must be equal to the second factor. In addition to calculating the product of the two factors in question, this algorithm can also be used to check the resulting result.

An example of solving a multiplication problem

Let's look at solutions to multiplication problems. Suppose, according to the conditions of the task, it is necessary to calculate the product of two numbers, among which the first factor is 8, and the second is 4. In accordance with the definition of the multiplication operation, this actually means that you need to add the number 8 4 times. The result is 32 - this is the product of the numbers in question, that is, the result of their multiplication.

In addition, it must be remembered that the so-called commutative law applies to the multiplication operation, which states that changing the places of the factors in the original example will not change its result. Thus, you can add the number 4 8 times, resulting in the same product - 32.

Multiplication table

It is clear that to solve this way a large number of drawing examples of the same type is a rather tedious task. In order to facilitate this task, the so-called multiplication was invented. In fact, it is a list of products of positive single-digit integers. Simply put, a multiplication table is a set of results of multiplying with each other from 1 to 9. Once you have learned this table, you no longer have to resort to multiplication every time you need to solve an example for such prime numbers, but just remember its result.

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Do we understand multiplication correctly?

"- A and B were sitting on the pipe. A fell, B disappeared, what was left on the pipe?
“Your letter I remains.”

(From the film "Youths in the Universe")

Why does multiplying a number by zero result in zero?

7 * 0 = 0

Why does multiplying two negative numbers produce a positive number?

7 * (-3) = + 21

Teachers come up with everything they can to give answers to these two questions.

But no one has the courage to admit that there are three semantic errors in the formulation of multiplication!

Is it possible to make mistakes in basic arithmetic? After all, mathematics positions itself as an exact science...

School mathematics textbooks do not provide answers to these questions, replacing explanations with a set of rules that need to be memorized. Perhaps this topic is considered difficult to explain in middle school? Let's try to understand these issues.

7 is the multiplicand. 3 is a multiplier. 21-work.

According to the official wording:

to multiply a number by another number means to add as many multiplicands as the multiplier prescribes.

According to the accepted formulation, the factor 3 tells us that there should be three sevens on the right side of the equality.

7 * 3 = 7 + 7 + 7 = 21

But this formulation of multiplication cannot explain the questions posed above.

Let's correct the wording of multiplication

Usually in mathematics there is a lot that is meant, but it is not talked about or written down.

This refers to the plus sign before the first seven on the right side of the equation. Let's write down this plus.

7 * 3 = + 7 + 7 + 7 = 21

But what is the first seven added to? This means to zero, of course. Let's write down zero.

7 * 3 = 0 + 7 + 7 + 7 = 21

What if we multiply by three minus seven?

7 * 3 = 0 + (-7) + (-7) + (-7) = - 21

We write the addition of the multiplicand -7, but in fact we are subtracting from zero multiple times. Let's open the brackets.

7 * 3 = 0 - 7 - 7 - 7 = - 21

Now we can give a refined formulation of multiplication.

Multiplication is the process of repeatedly adding to (or subtracting from zero) the multiplicand (-7) as many times as the multiplier indicates. The multiplier (3) and its sign (+ or -) indicate the number of operations that are added to or subtracted from zero.

Using this refined and slightly modified formulation of multiplication, the “sign rules” for multiplication when the multiplier is negative are easily explained.

7 * (-3) - there must be three minus signs after the zero = 0 - (+7) - (+7) - (+7) = - 21

7 * (-3) - again there should be three minus signs after the zero =

0 - (-7) - (-7) - (-7) = 0 + 7 + 7 + 7 = + 21

Multiply by zero

7 * 0 = 0 + ... no addition to zero operations.

If multiplication is an addition to zero, and the multiplier shows the number of operations of addition to zero, then the multiplier zero shows that nothing is added to zero. That's why it remains zero.

So, in the existing formulation of multiplication, we found three semantic errors that block the understanding of the two “sign rules” (when the multiplier is negative) and the multiplication of a number by zero.

You don't need to add the multiplicand, but add it to zero.
Multiplication is not only adding to zero, but also subtracting from zero.
The multiplier and its sign do not show the number of terms, but the number of plus or minus signs when decomposing the multiplication into terms (or subtracted ones).

Having somewhat clarified the formulation, we were able to explain the rules of signs for multiplication and the multiplication of a number by zero without the help of the commutative law of multiplication, without the distributive law, without involving analogies with the number line, without equations, without proof from the inverse, etc.

The sign rules for the refined formulation of multiplication are derived very simply.

7 * (+3) = 0 + (+7) + (+7) + (+7) = +21 (++ = +)

7 * (+3) = 0 + (-7) + (-7) + (-7) = 0 - 7 - 7 - 7 = -21 (- + = -)

7 * (-3) = 0 - (+7) - (+7) - (+7) = 0 - 7 - 7 - 7 = -21 (+ - = -)

7 * (-3) = 0 - (-7) - (-7) - (-7) = 0 + 7 + 7 + 7 = +21 (- - = +)

The multiplier and its sign (+3 or -3) indicate the number of "+" or "-" signs on the right side of the equation.

The modified formulation of multiplication corresponds to the operation of raising a number to a power.

2^3 = 1*2*2*2 = 8

2^0 = 1 (one is not multiplied or divided by anything, so it remains one)

2^-1 = 1: 2 = 1/2

2^-2 = 1: 2: 2 = 1/4

2^-3 = 1: 2: 2: 2 = 1/8

Mathematicians agree that raising a number to a positive power is multiplying one many times. And raising a number to a negative power is dividing one multiple times.

The operation of multiplication should be similar to the operation of exponentiation.

2*3 = 0 + 2 + 2 + 2 = 6

2*2 = 0 + 2 + 2 = 4

2*0 = 0 (nothing is added to zero and nothing is subtracted from zero)

2*-1 = 0 - 2 = -2

2*-2 = 0 - 2 - 2 = -4

2*-3 = 0 - 2 - 2 - 2 = -6

The modified formulation of multiplication does not change anything in mathematics, but returns the original meaning of the multiplication operation, explains the “rules of signs”, multiplying a number by zero, and reconciles multiplication with exponentiation.

Let's check whether our formulation of multiplication is consistent with the division operation.

15: 5 = 3 (inverse of multiplication 5 * 3 = 15)

The quotient (3) corresponds to the number of operations of addition to zero (+3) during multiplication.

Dividing the number 15 by 5 means finding how many times you need to subtract 5 from 15. This is done by sequential subtraction until a zero result is obtained.

To find the result of division, you need to count the number of minus signs. There are three of them.

15: 5 = 3 operations of subtracting five from 15 to get zero.

15 - 5 - 5 - 5 = 0 (15:5 division)

0 + 5 + 5 + 5 = 15 (multiplying 5 * 3)

Division with remainder.

17 - 5 - 5 - 5 - 2 = 0

17: 5 = 3 and 2 remainder

If there is division with a remainder, why not multiplication with an appendage?

2 + 5 * 3 = 0 + 2 + 5 + 5 + 5 = 17

Let's look at the difference in wording on the calculator

Existing formulation of multiplication (three terms).

10 + 10 + 10 = 30

Corrected multiplication formulation (three additions to zero operations).

0 + 10 = = = 30

(Press “equals” three times.)

10 * 3 = 0 + 10 + 10 + 10 = 30

A multiplier of 3 indicates that the multiplicand 10 must be added to zero three times.

Try multiplying (-10) * (-3) by adding the term (-10) minus three times!

(-10) * (-3) = (-10) + (-10) + (-10) = -10 - 10 - 10 = -30 ?

What does the minus sign for three mean? Maybe so?

(-10) * (-3) = (-10) - (-10) - (-10) = - 10 + 10 + 10 = 10?

Ops... I can’t decompose the product into the sum (or difference) of terms (-10).

The revised wording does this correctly.

0 - (-10) = = = +30

(-10) * (-3) = 0 - (-10) - (-10) - (-10) = 0 + 10 + 10 + 10 = 30

The multiplier (-3) indicates that the multiplicand (-10) must be subtracted from zero three times.

Sign rules for addition and subtraction

Above we showed a simple way to derive the rules of signs for multiplication by changing the meaning of the wording of multiplication.

But for the conclusion we used the rules of signs for addition and subtraction. They are almost the same as for multiplication. Let's create a visualization of the rules of signs for addition and subtraction, so that even a first-grader can understand it.

What is "minus", "negative"?

There is nothing negative in nature. No negative temperature, no negative direction, no negative mass, no negative charges... Even sine by its nature can only be positive.

But mathematicians came up with negative numbers. For what? What does "minus" mean?

A minus sign means the opposite direction. Left - right. Top bottom. Clockwise - counterclockwise. Back and forth. Cold - hot. Light heavy. Slow - fast. If you think about it, you can give many other examples where it is convenient to use negative values.

In the world we know, infinity starts from zero and goes to plus infinity.

"Minus infinity" in real world does not exist. This is the same mathematical convention as the concept of “minus”.

So, “minus” denotes the opposite direction: movement, rotation, process, multiplication, addition. Let's analyze different directions when adding and subtracting positive and negative (increasing in the other direction) numbers.

The difficulty in understanding the rules of signs for addition and subtraction is due to the fact that these rules are usually explained on the number line. On the number line, three different components are mixed, from which rules are derived. And due to confusion, due to lumping different concepts into one heap, difficulties of understanding are created.

To understand the rules, we need to divide:

the first term and the sum (they will be on the horizontal axis);
the second term (it will be on the vertical axis);
direction of addition and subtraction operations.

This division is clearly shown in the figure. Mentally imagine that the vertical axis can rotate, superimposing on the horizontal axis.

The addition operation is always performed by rotating the vertical axis clockwise (plus sign). The subtraction operation is always performed by rotating the vertical axis counterclockwise (minus sign).

Example. Diagram in lower right corner.

It can be seen that two adjacent minus signs (the sign of the subtraction operation and the sign of the number 3) have different meanings. The first minus shows the direction of subtraction. The second minus is the sign of the number on the vertical axis.

Find the first term (-2) on the horizontal axis. Find the second term (-3) on the vertical axis. Mentally rotate the vertical axis counterclockwise until (-3) aligns with the number (+1) on the horizontal axis. The number (+1) is the result of addition.

Subtraction operation

gives the same result as the addition operation in the diagram in the upper right corner.

Therefore, two adjacent minus signs can be replaced by one plus sign.

We are all accustomed to using ready-made rules of arithmetic without thinking about their meaning. Therefore, we often don’t even notice how the rules of signs for addition (subtraction) differ from the rules of signs for multiplication (division). Do they seem the same? Almost... A slight difference can be seen in the following illustration.

Now we have everything we need to derive the sign rules for multiplication. The output sequence is as follows.

We clearly show how the rules of signs for addition and subtraction are obtained.
We make semantic changes to the existing formulation of multiplication.
Based on the modified formulation of multiplication and the rules of signs for addition, we derive the rules of signs for multiplication.

Note.

Below are written Sign rules for addition and subtraction,obtained from the visualization. And in red, for comparison, the same rules of signs from the mathematics textbook. The gray plus in brackets is an invisible plus, which is not written for a positive number.

There are always two signs between the terms: the operation sign and the number sign (we don’t write plus, but we mean it). The rules of signs prescribe the replacement of one pair of characters with another pair without changing the result of addition (subtraction). In fact, there are only two rules.

Rules 1 and 3 (for visualization) - duplicate rules 4 and 2.. Rules 1 and 3 in the school interpretation do not coincide with the visual scheme, therefore, they do not apply to the rules of signs for addition. These are some other rules...

1. +(+) = -- ......... + (+) = + ???

2. +- = -(+).......... + - = - (+) ok

3. -(+) = +- ......... - (+) = - ???

4. -- = +(+) ......... - - = + (+) ok

School rule 1. (red) allows you to replace two pluses in a row with one plus. The rule does not apply to the replacement of signs in addition and subtraction.

School rule 3. (red) allows you not to write a plus sign for a positive number after a subtraction operation. The rule does not apply to the replacement of signs in addition and subtraction.

The meaning of the rules of signs for addition is the replacement of one PAIR of signs with another PAIR of signs without changing the result of the addition.

School methodologists mixed two rules in one rule:

Two rules of signs when adding and subtracting positive and negative numbers (replacing one pair of signs with another pair of signs);

Two rules for not writing a plus sign for a positive number.

Two different rules, mixed into one, are similar to the rules of signs in multiplication, where two signs result in a third. They look exactly alike.

Great confusion! The same thing again, for better detangling. Let us highlight the operation signs in red to distinguish them from the number signs.

1. Addition and subtraction. Two rules of signs according to which pairs of signs between terms are interchanged. Operation sign and number sign.

+ + = - - |||||||||| 2 + (+2) = 2 - (-2)

+ - = - + |||||||||| 2 + (-2) = 2 - (+2)

2. Two rules according to which the plus sign for a positive number is allowed not to be written. These are the rules for the entry form. Does not apply to addition. For a positive number, only the sign of the operation is written.

- + = - |||||||||| - (+2) = - 2

+ + = + |||||||||| + (+2) = + 2

3. Four rules of signs for multiplication. When two signs of factors result in a third sign of the product. The multiplication sign rules contain only number signs.

+ * + = + |||||||||| 2 * 2 = 2

+ * - = - |||||||||| 2 * (-2) = -2

- * + = - |||||||||| -2 * 2 = - 2

- * - = + |||||||||| -2 * -2 = 2

Now that we have separated the form rules, it should be clear that the sign rules for addition and subtraction are not at all similar to the sign rules for multiplication.

V. Kozarenko

Why does minus times minus give plus?

- (1 stick) - (2 sticks) = ((1 stick)+(2 sticks))= 2 sticks (And two sticks are equal + because there are 2 sticks at a pole)))

Minus on minus gives a plus because this is a school rule. At the moment, in my opinion, there is no exact answer why. This is the rule and it has been around for many years. You just have to remember that sliver for sliver gives a clothespin.

From the school mathematics course we know that minus times minus gives plus. There is also a simplified, humorous explanation of this rule: a minus is one line, two minuses are two lines, a plus consists of two lines. Therefore, minus by minus gives a plus sign.

I think like this: minus is a stick - add another minus stick - then you get two sticks, and if you connect them crosswise, you get the + sign, this is what I said about my opinion on the question: minus by minus plus.

Minus for minus does not always give a plus, even in mathematics. But basically I compare this statement with mathematics, where it occurs most often. They also say they knock it out with a crowbar - I also somehow associate this with disadvantages.

Imagine that you borrowed 100 rubles. Now your score: -100 rubles. Then you repaid this debt. So it turns out that you have reduced (-) your debt (-100) by the same amount of money. We get: -100-(-100)=0

A minus sign indicates the opposite: the opposite number of 5 is -5. But -(-5) is the opposite number of the opposite, i.e. 5.

As in the joke:

1st -Where is the opposite side of the street?

2nd - on the other side

1st - and they said that on this...

Let's imagine a scale with two bowls. What always has a plus sign on the right bowl, always has a minus sign on the left bowl. Now, multiplying by a number with a plus sign will mean that it occurs on the same bowl, and multiplying by a number with a minus sign will mean that the result is carried over to another bowl. Examples. We multiply 5 apples by 2. We get 10 apples on the right bowl. We multiply - 5 apples by 2, and get 10 apples on the left bowl, that is -10. Now multiply -5 by -2. This means that 5 apples on the left bowl were multiplied by 2 and transferred to the right bowl, that is, the answer is 10. Interestingly, multiplying a plus by a minus, that is, apples on the right bowl, has a negative result, that is, the apples move to the left. And multiplying the minus left apples by a plus leaves them in the minus, on the left bowl.

I think this can be demonstrated as follows. If you put five apples in five baskets, then there will be 25 apples in total. In baskets. And minus five apples means that I did not report them, but took them out of each of the five baskets. and it turned out the same 25 apples, but not in baskets. Therefore, the baskets go as a minus.

This can also be perfectly demonstrated with the following example. If a fire starts in your home, this is a minus. But if you also forgot to turn off the tap in the bathtub, and you have a flood, then this is also a minus. But this is separate. But if it all happened at the same time, then a minus for a minus gives a plus, and your apartment has a chance to survive.