The minus by minus table gives a plus. Actions with a minus

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, you always got a positive number; when dividing one quantity by another, you didn’t always get integers- 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?

Actions using negative numbers should lead us to the same answer as acting only with 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 when large quantities Addends 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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Instructions

There are four types of mathematical operations: addition, subtraction, multiplication and division. Therefore, there will be four types of examples. Negative numbers within the example are highlighted so as not to confuse the mathematical operation. For example, 6-(-7), 5+(-9), -4*(-3) or 34:(-17).

Addition. This action can look like: 1) 3+(-6)=3-6=-3. Replacement action: first, the parentheses are opened, the “+” sign is changed to the opposite, then from the larger (modulo) number “6” the smaller one, “3,” is subtracted, after which the answer is assigned the larger sign, that is, “-”.
2) -3+6=3. This can be written according to the principle ("6-3") or according to the principle "subtract the smaller from the larger and assign the sign of the larger to the answer."
3) -3+(-6)=-3-6=-9. When opening, the action of addition is replaced by subtraction, then the modules are summed up and the result is given a minus sign.

Subtraction.1) 8-(-5)=8+5=13. The parentheses are opened, the sign of the action is reversed, and an example of addition is obtained.
2) -9-3=-12. The elements of the example are added and get general sign "-".
3) -10-(-5)=-10+5=-5. When opening the brackets, the sign changes again to “+”, then the smaller number is subtracted from the larger number and the sign of the larger number is taken away from the answer.

Multiplication and division: When performing multiplication or division, the sign does not affect the operation itself. When multiplying or dividing numbers with the answer, a “minus” sign is assigned; if the numbers have the same signs, the result always has a “plus” sign. 1) -4*9=-36; -6:2=-3.
2)6*(-5)=-30; 45:(-5)=-9.
3)-7*(-8)=56; -44:(-11)=4.

Sources:

table with cons

How to decide examples? Children often turn to their parents with this question if homework needs to be done at home. How to correctly explain to a child the solution to examples of adding and subtracting multi-digit numbers? Let's try to figure this out.

You will need

1. Textbook on mathematics.
2. Paper.
3. Handle.

Instructions

Read the example. To do this, divide each multivalued into classes. Starting from the end of the number, count three digits at a time and put a dot (23.867.567). Let us remind you that the first three digits from the end of the number are to units, the next three are to class, then millions come. We read the number: twenty-three eight hundred sixty-seven thousand sixty-seven.

Write down an example. Please note that the units of each digit are written strictly below each other: units under units, tens under tens, hundreds under hundreds, etc.

Perform addition or subtraction. Start performing the action with units. Write down the result under the category with which you performed the action. If the result is number(), then we write the units in place of the answer, and add the number of tens to the units of the digit. If the number of units of any digit in the minuend is less than in the subtrahend, we take 10 units of the next digit and perform the action.

Read the answer.

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note

Prohibit your child from using a calculator even to check the solution to an example. Addition is tested by subtraction, and subtraction is tested by addition.

Helpful advice

If a child has a good grasp of the techniques of written calculations within 1000, then operations with multi-digit numbers, performed in an analogous manner, will not cause any difficulties.
Give your child a competition to see how many examples he can solve in 10 minutes. Such training will help automate computational techniques.

Multiplication is one of the four basic mathematical operations that underlies many more complex functions. In fact, multiplication is based on the operation of addition: knowledge of this allows you to correctly solve any example.

To understand the essence of the multiplication operation, it is necessary to take into account that there are three main components involved in it. One of them is called the first factor and is a number that is subject to the multiplication operation. For this reason, it has a second, somewhat less common name - “multiplicable”. The second component of the multiplication operation is usually called the second factor: it represents the number by which the multiplicand is multiplied. Thus, both of these components are called multipliers, which emphasizes their equal status, as well as the fact that they can be swapped: the result of the multiplication will not change. Finally, the third component of the multiplication operation, resulting from its result, is called the product.

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 solving a large number of similar examples in this way 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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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 a negative and a positive number will only give “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 more in a simple way Explain the rules for dealing with negative numbers.

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.

"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.

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 candy and I give it to my sister, then I will have some candy left, but I cannot give her candy 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!).

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.

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: the ring. It's just a set of elements plus actions that can be performed on them. The fundamental rules 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, 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.

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:

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 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.

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.

Evgeniy Epifanov
"Elements"

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Jacques Sesiano

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Proskuryakov I. V.

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Leon Takhtajyan

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Vladimir Arnold

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Reed Miles

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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 move terms with unknowns to the 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.

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