When adding minus by minus what gives. Sign rules for multiplication and addition

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

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

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

Let's take the equation as an example. It can be solved like this: move the terms with the unknown to the left side, and the rest to the right, it will turn out , , . With this solution, we did not even meet negative numbers.

But it could be done in a different way by chance: 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 be concluded that .

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

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

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

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

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

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

Note that rings, in the most general construction, do not require multiplication to be permutable or invertible (that is, it is not always possible to divide), or the existence of a unit - a neutral element with respect to multiplication. If these axioms are introduced, then other algebraic structures are obtained, but all the theorems proved for rings will be true in them.

Now we prove that for any elements and an arbitrary ring, firstly, and secondly, . From this, statements about units easily follow: and .

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

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

The first fact is obtained as follows: , that is, opposite to , which means it is equal to .

To be mathematically rigorous, let's also explain why for any element . Indeed, . That is, the addition does not change the sum. So this product is equal to zero.

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

Evgeny Epifanov
"Elements"

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Why does a minus times a minus make a plus?

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

A minus times a minus gives a plus because it's a school rule. At the moment, there is no exact answer why, in my opinion. This is the rule and it has been around for many years. You just need to remember a sliver for a sliver gives a clothespin.

From the school mathematics course, we know that a minus times a minus gives a plus. There is also a simplified, playful explanation of this rule: minus is one line, two minuses are two lines, plus just consists of 2 lines. Therefore, minus times minus gives a plus sign.

I think so: minus is a stick - add one more minus stick - then you get two sticks, and if you connect them crosswise, then the sign + quot ; will learn, this is how I said my opinion on the question: minus minus dates plus.

A minus times a minus does not always give a plus, even in mathematics. But basically, I compare this statement with mathematics, where it is most often found. They also say they knock out scrap with a crowbar - this is also somehow associated with minuses.

Imagine that you borrowed 100 rubles. Now your account: -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

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

As in a joke:

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

2nd - on the other side

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

Imagine a scale with two bowls. The fact that on the right bowl always has a plus sign, on the left bowl - minus. 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, we get 10 apples on the left bowl, that is -10. Now multiply -5 by -2. This means 5 apples on the left bowl multiplied by 2 and transferred to the right bowl, that is, the answer is 10. Interestingly, multiplying plus by minus, that is, apples on the right bowl, has a negative result, that is, apples go to the left. And multiplying minus left apples by plus leaves them in the minus, on the left bowl.

I think this can be demonstrated in the following way. If you put five apples into 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, baskets go as a minus.

You can also demonstrate this very well with the following example. If your house is on fire, that's a minus. But if you forgot to turn off the faucet in the bath, and you started to flood, then this is also a minus. But this is separate. But if it all happened at the same time, then minus by minus gives a plus, and your apartment has a chance to survive.

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

"The enemy of my enemy is my friend."

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Evgeny Epifanov, Earth (Sol III).

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

And let's ask...

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

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

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

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

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

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

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

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

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

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

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

Evgeny Epifanov

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

Laws of Mathematics

Most adults are unable to explain to themselves or their children why this happens. They had thoroughly learned this material in school, but they did not even try to find out where such rules came from. But in vain. Often, modern children are not so gullible, they need to get to the bottom of the matter and understand, say, why "plus" on "minus" gives "minus". And sometimes tomboys deliberately ask tricky questions in order to enjoy the moment when adults cannot give an intelligible answer. And it’s really a disaster if a young teacher gets into trouble ...

By the way, it should be noted that the rule mentioned above is valid for both multiplication and division. The product of a negative and a positive number will only give a "minus". If we are talking about two digits with a "-" sign, then the result will be a positive number. The same applies to division. If one of the numbers is negative, then the quotient will also be with the sign "-" ".

To explain the correctness of this law of mathematics, it is necessary to formulate the axioms of the ring. But first you need to understand what it is. In mathematics, it is customary to call a ring a set in which two operations with two elements are involved. But it's better to understand this with an example.

Ring axiom

There are several mathematical laws.

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

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

Nobody canceled the rules by which brackets are opened (V + C) x D = V x D + C x D, it is also true that C x (V + D) = C x V + C x D.

In addition, it has been established that a special, addition-neutral element can be introduced into the ring, using which the following will be true: C + 0 = C. In addition, for each C there is an opposite element, which can be denoted as (-C). In this case, C + (-C) \u003d 0.

Derivation of axioms for negative numbers

Having accepted the above statements, we can answer the question: ""Plus" on "minus" gives what sign? Knowing the axiom about the multiplication of negative numbers, it is necessary to confirm that indeed (-C) x V = -(C x V). And also that the following equality is true: (-(-C)) = C.

To do this, we must first prove that each of the elements has only one opposite "brother". Consider the following proof example. Let's try to imagine that two numbers are opposite for C - V and D. From this it follows that C + V = 0 and C + D = 0, that is, C + V = 0 = C + D. Recalling the displacement laws and about the properties of the number 0, we can consider the sum of all three numbers: C, V and D. Let's try to figure out the value of V. It is logical that V = V + 0 = V + (C + D) = V + C + D, because the value of C + D, as was accepted above, is equal to 0. Hence, V = V + C + D.

The value for D is derived in the same way: D = V + C + D = (V + C) + D = 0 + D = D. Based on this, it becomes clear that V = D.

In order to understand why, nevertheless, the "plus" on the "minus" gives a "minus", you need to understand the following. So, for the element (-C), the opposite are C and (-(-C)), that is, they are equal to each other.

Then it is obvious that 0 x V \u003d (C + (-C)) x V \u003d C x V + (-C) x V. It follows from this that C x V is opposite to (-) C x V, which means (- C) x V = -(C x V).

For complete mathematical rigor, it is also necessary to confirm that 0 x V = 0 for any element. If you follow the logic, then 0 x V \u003d (0 + 0) x V \u003d 0 x V + 0 x V. This means that adding the product 0 x V does not change the set amount in any way. After all, this product is equal to zero.

Knowing all these axioms, one can deduce not only how much "plus" by "minus" gives, but also what happens when negative numbers are multiplied.

Multiplication and division of two numbers with "-" sign

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

Suppose that C - (-V) = D, based on this, C = D + (-V), that is, C = D - V. We transfer V and we get that C + V = D. That is, C + V = C - (-V). This example explains why in an expression where there are two "minus" in a row, the mentioned signs should be changed to "plus". Now let's deal with multiplication.

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

Remembering the rules for working with brackets, we get:

1) (-C) x (-V) + (C x V) + (-C) x V = D;

2) (-C) x ((-V) + V) + C x V = D;

3) (-C) x 0 + C x V = D;

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

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

General mathematical rules

Of course, such an explanation is not suitable for elementary school students who are just starting to learn abstract negative numbers. It is better for them to explain on visible objects, manipulating the familiar term through the looking glass. For example, invented, but not existing toys are located there. They can be displayed with a "-" sign. The multiplication of two looking-glass objects transfers them to another world, which is equated to the present, that is, as a result, we have positive numbers. But the multiplication of an abstract negative number by a positive one only gives the result familiar to everyone. After all, "plus" multiplied by "minus" gives "minus". True, children do not try too hard to delve into all the mathematical nuances.

Although, if you face the truth, for many people, even with higher education, many rules remain a mystery. Everyone takes for granted what their teachers teach them, not at a loss to delve into all the complexities that mathematics is fraught with. "Minus" on "minus" gives "plus" - everyone knows about it without exception. This is true for both integers and fractional numbers.

Attention, only TODAY!

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Laws of Mathematics

Ring axiom

Derivation of axioms for negative numbers

Multiplication and division of two numbers with "-" sign

General mathematical rules

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