Which sign gives plus to plus. Why does minus times minus give plus?

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 way to answer is: “Because these are the rules of action over 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.

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 left side, and the rest - to the right, it will work 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 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 by 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 aggregates were discovered mathematical objects, on which the following 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 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) to plus ( positive tense) gave a 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) to 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 is relative motion of 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 a number, then it is physical action, but if he denies the action itself, then this 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”???

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In order to answer such a childish 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.

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The answer is clearly not for ordinary schoolchildren!
IN junior classes I read a wonderful book - the one about Dwarfism and Al-Jebra, and maybe in a math circle 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 the 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 to 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 delicious apples+YAYA that can be gobbled 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 settlements can be reduced and not taken into account, but you can keep 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’ll reduce the rest by mutual offset) will take two rubles less from me than they will return, then my welfare 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 task!) My third-grader grandson found the problem too difficult for him yet, but understanding the answer, without a doubt, brought him closer to understanding future mathematical intricacies like “minus times 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.

More details on this topic here:

http://mnemonikon.ru/differ_pub_28.htm

What is this habit of mechanically believing textbooks? You also need to have your own brains. Especially if there are paradoxes, blind spots, obvious contradictions. All this is a consequence of errors in theory.

It is impossible to decompose the product of two negative numbers into terms, according to the current formulation of multiplication (without zero). Doesn't this bother anyone?

What kind of multiplication formulation is this that makes it impossible to perform multiplication? :)

The problem is also purely psychological. Blind trust in authorities, unwillingness to think for yourself. If the textbooks say so, if they teach so at school, then this is the ultimate truth. Everything changes, including science. Otherwise there would be no development of civilization.

Correct the wording of multiplication in all textbooks! The rules of arithmetic will not change because of this.

Moreover, as follows from the article linked above, the corrected formulation of multiplication will be similar to the formulation of raising a number to a power. There, too, they do not write down the unit when raised to a positive power. However, one is written when raising a number to a negative power.

Gentlemen of mathematics, your mother, you must always write down zero and one, even if the result does not change due to their absence.

The meaning of abbreviated entries changes (or even disappears). And schoolchildren have problems with understanding.

Answer

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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 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.

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 .

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 then be applied to all these sets of objects (this approach is typical for 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), which are subject to actions, and not the nature of the elements of the set (here it is, a new level of abstraction!). 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; there is a special element in the ring 0 (neutral addition element) 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.

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).

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.”

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 - they were 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 .

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 by 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 obey 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.

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), which are subject to actions, and not the nature of the elements of the set (here it is, a new level of abstraction!). 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 .

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 .

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) .

Evgeny Epifanov, Earth (Sol III).

"The enemy of my enemy is my friend"

Why does minus one times minus one equal plus one? Why does minus one times plus one equal minus one? 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: 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.

Let's consider the equation as an example. It can be solved this way: move the terms with the unknown to the left side, and the rest to the right, it turns out , , . 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 , . To find the unknown, you need to divide one negative number by another: . But the correct answer is known, and it remains to conclude that .

What does this simple example demonstrate? Firstly, the logic that determined the rules for actions on negative numbers becomes clear: the results of these actions must coincide with 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 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.

As a result, a new concept emerged: the ring. It's just a set of elements plus actions that can be performed on them. The fundamental ones here are precisely the rules (they are called axioms) to which actions are subject, and not the nature of the elements of the set (here it is, a new level of abstraction!). 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.

A 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:

Now let us prove that for any elements and an arbitrary ring it is true, firstly, , and secondly, . Statements about units easily follow from this: and .

To do this we will need to establish some facts. First we prove that each element can have only one opposite. In fact, let an element have two opposites: and . That is . Let's consider the amount. Using the associative and commutative laws and the property of zero, we find that, on the one hand, the sum is equal to , and on the other hand, it is equal to . Means, .

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

The first fact turns out like this: that is, it is opposite, which means it is equal.

To be mathematically rigorous, let's also explain why for any element . Indeed, . That is, adding does not change the amount. This means that this product is equal to zero.

Evgeniy Epifanov
"Elements"

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Jacques Sesiano

Over two millennia there have been three important expansions of the numerical domain. First, around 450 BC. scientists of the Pythagorean school proved the existence of rational numbers. Their initial goal was to quantify the diagonal of a unit square. Secondly, in the XIII-XV centuries, European scientists, solving systems linear equations, allowed the possibility of one negative decision. And thirdly, in 1572, the Italian algebraist Raphael Bombelli used complex numbers to obtain a real solution to a certain cubic equation.

Proskuryakov I. V.

The purpose of this book is to strictly define numbers, polynomials and algebraic fractions and justify their properties already known from school, and not to introduce the reader to new properties. Therefore, the reader will not find facts new to him here (with the possible exception of some properties, real and complex numbers), but will learn how things that are well known to him are proven, starting with “twice two is four” and ending with the rules of operations with polynomials And algebraic fractions. But the reader will get acquainted with a number of general concepts, playing a major role in algebra.

Ilya Shchurov

Mathematician Ilya Shchurov o decimals, transcendence and irrationality of the number Pi.

Leon Takhtajyan

These will be four short stories. We will start with numbers, then we will talk about movement, about change, then we will discuss shapes and sizes, and then beginning and ending. In this somewhat encrypted style, we will try to look at mathematics from the inside and outside, and precisely as a subject. What mathematicians think about and live by - we can talk about this later.

Vladlen Timorin

Mathematician Vladlen Timorin on the advantages of complex numbers, Hamilton's quaternions, eight-dimensional Cayley numbers and the variety of numbers in geometry.

Jacques Sesiano

We know little about Diophantus. I think he lived in Alexandria. None of the Greek mathematicians mention him before the 4th century, so he probably lived in the middle of the 3rd century. The most main job Diophanta, “Arithmetic” (Ἀριθμητικά), took place in the beginning of 13 “books” (βιβλία), i.e. chapters. Today we have 10 of them, namely: 6 in the Greek text and 4 others in the medieval Arabic translation, whose place is in the middle of the Greek books: books I-III in Greek, IV-VII in Arabic, VIII-X in Greek. Diophantus' "Arithmetic" is primarily a collection of problems, about 260 in total. To tell the truth, there is no theory; there are only general instructions in the introduction of the book, and private comments in some problems, when necessary. "Arithmetic" already has the features of an algebraic treatise. First Diophantus uses different signs to express the unknown and its powers, also some calculations; like all algebraic symbolism of the Middle Ages, its symbolism comes from mathematical words. Then, Diophantus explains how to solve the problem algebraically. But Diophantus's problems are not algebraic in the usual sense, because almost all of them boil down to solving an indeterminate equation or systems of such equations.

The world of mathematics is unthinkable without them - without prime numbers. What's happened prime numbers what is special about them and what significance do they have for Everyday life? In this film, British mathematics professor Marcus du Sautoy will reveal the secret of prime numbers.

Georgy Shabat

At school, we are all instilled with the erroneous idea that on the set of rational numbers Q there is a unique natural distance (the modulus of the difference), with respect to which all arithmetic operations are continuous. However, there is also an infinite number of distances, the so-called p-adic, one for each number p. According to Ostrovsky's theorem, the “ordinary” distance, together with all p-adic ones, already really exhausts all reasonable distances Q. The term adelic democracy was introduced by Yu. I. Manin. According to the principle of adelic democracy, all reasonable distances on Q are equal before the laws of mathematics (maybe only the traditional “a little=slightly equal...”). The course will introduce the adelic ring, which allows you to work with all these distances at the same time.

Vladimir Arnold

J.L. Lagrange proved that a sequence of incomplete quotients (starting from a certain place) is periodic if and only if the number x is a quadratic irrationality. R. O. Kuzmin proved that in the sequence of incomplete quotients of almost any real number, the fraction d_m equal to m incomplete quotients is the same (for typical real numbers). The fraction d_m decreases as m→∞ as 1/m^2 and its value was predicted by Gauss (who proved nothing). V.I. Arnol'd expressed (about 20 years ago) the hypothesis that the Gauss–Kuzmin statistics d_m also holds for periods of continued fractions of roots quadratic equations x^2+px+q=0 (with integer p and q): if we write down together the incomplete quotients that make up the periods of all continued fractions of the roots of such equations with p^2+q^2≤R^2, then the share of the incomplete quotient m among they will tend to the number d_m as R→∞. V. A. Bykovsky and his Khabarovsk students recently proved this long-standing hypothesis. Despite this, the question of statistics not of letters, but of words composed of them, which are the periods of continued fractions of any roots x of the equations x^2+px+q=0, is far from resolved.

Reed Miles

I leave the title and abstract as vague as possible, so that I can talk about whatever I feel like on the day. Many varieties of interest in the classification of varieties are obtained as Spec or Proj of a Gorenstein ring. In codimension ⩽3, the well known structure theory provides explicit methods of calculating with Gorenstein rings. In contrast, there is no useable structure theory for rings of codimension ⩾4. Nevertheless, in many cases, Gorenstein projection (and its inverse, Kustin–Miller unprojection) provide methods of attacking these rings. These methods apply to sporadic classes of canonical rings of regular algebraic surfaces, and to more systematic constructions of Q-Fano 3-folds, Sarkisov links between these, and the 3-folds flips of Type A of Mori theory.

Indeed, why? 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 remember that this is exactly how it is and we no longer ask the question.

Let's ask ourselves...

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 - they were 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, you get 7x - 2x = 17 - 2, 5x = 15, x = 3. With this In our 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.

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 by 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 obey 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.

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.

A 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 ring elements obeys commutative (A + B = B + A for any elements A and B) and combinational (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)) such that 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.

Now let us prove that for any elements A and B of an arbitrary ring it is true, firstly, (-A) B = -(A B), and secondly (-(-A)) = A. This easily follows statements about units: (-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 have two opposites: 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 obtain 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. So B = C.

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

The first fact turns out like this: 0 = 0 B = (A + (-A)) B = A B + (-A) B, that is, (-A) B is opposite to 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, adding 0·B does not change the amount. This means that this product is equal to zero.

Evgeniy Epifanov