Multiplication is an arithmetic operation in which the first number is repeated as a term as many times as the second number shows.

A number that repeats as a term is called multiplyable(it is multiplied), the number that shows how many times to repeat the term is called multiplier. The number resulting from multiplication is called work.

For example, multiplying the natural number 2 by the natural number 5 means finding the sum of five terms, each of which is equal to 2:

2 + 2 + 2 + 2 + 2 = 10

In this example, we find the sum by ordinary addition. But when the number of identical terms is large, finding the sum by adding all the terms becomes too tedious.

To write multiplication, use the sign × (slash) or · (dot). It is placed between the multiplicand and the multiplier, with the multiplicand written to the left of the multiplication sign, and the multiplier to the right. For example, the notation 2 · 5 means that the number 2 is multiplied by the number 5. To the right of the notation of multiplication, put an = (equal) sign, after which the result of the multiplication is written. Thus, the complete multiplication entry looks like this:

This entry reads like this: the product of two and five equals ten or two times five equals ten.

Thus, we see that multiplication is simply a short form of adding like terms.

Multiplication check

To check multiplication, you can divide the product by the factor. If the result of division is a number equal to the multiplicand, then the multiplication is performed correctly.

Consider the expression:

where 4 is the multiplicand, 3 is the multiplier, and 12 is the product. Now let's perform a multiplication test by dividing the product by the factor.

Explanatory dictionary of the Russian language. D.N. Ushakov

multiplication

multiplication, m.n. no, cf.

action according to verb. multiply - multiply and state according to the verb. multiply - multiply. Multiplying three by two. Income multiplication.

An arithmetic operation, repeating a given number as a term as many times as there are units in another given number (mat.). Multiplication table. Multiplying integers.

Explanatory dictionary of the Russian language. S.I.Ozhegov, N.Yu.Shvedova.

multiplication

A mathematical operation by means of which, from two numbers (or quantities), a new number (or quantity) is obtained, which (for integers) contains as a summand the first number as many times as there are units in the second. Multiplication table. Problem on y.

New explanatory dictionary of the Russian language, T. F. Efremova.

Encyclopedic Dictionary, 1998

multiplication

arithmetic operation. Indicated by a dot "." or a "?" (in literal calculations, multiplication signs are omitted). Multiplication of positive integers (natural numbers) is an action that allows, from two numbers a (the multiplicand) and b (the multiplier), to find the third number ab (the product), equal to the sum of b terms, each of which is equal to a; a and b are also called factors. Multiplication of fractional numbers a/b and c/d is determined by the equality Multiplication of two rational numbers gives a number, abs. whose value is equal to the product of the absolute values ​​of the factors and which has a plus sign (+) if both factors have the same signs, or a minus sign (-) if they have different signs. The multiplication of irrational numbers is determined using their rational approximations. Multiplying complex numbers given in the form? = a+bi and? = c+di, determined by the equality ?? = ac - bd + (a + bc)i.

Multiplication

the operation of forming, from two given objects a and b, called factors, a third object c, called a product. U. is denoted by the sign X (introduced by the English mathematician W. Oughtred in 163

or ∙ (introduced by the German scientist G. Leibniz in 1698); in the letter designation these signs are omitted and instead of a ` b or a ∙ b they write ab. U. has a different specific meaning and, accordingly, different specific definitions depending on the specific type of factors and product. The equation of positive integers is, by definition, the action that assigns to the numbers a and b a third number c, equal to the sum of b terms, each of which is equal to a, so ab = a + a +... + a (b terms). The number a is called multiplicand, b is called a multiplier. The value of fractional numbers ═ and ═ is determined by the equality ═ (see Fraction). The equation of rational numbers gives a number whose absolute value is equal to the product of the absolute values ​​of the factors, which has a plus sign (+) if both factors are of the same sign, and a minus sign (√) if they are of different signs. The value of irrational numbers is determined using the value of their rational approximations. The equation for complex numbers given in the form a = a + bi and b = c + di is determined by the equality ab = ac √ bd + (ad + bc) i. For complex numbers written in trigonometric form:

a = r1 (cosj1 + isin j1),

b = r2 (cosj2 + isin j

their modules are multiplied, and their arguments are added:

ab = r1r2(cos (j1 + j2) + i sin ((j1 + j2)).

The equation of numbers is unique and has the following properties:

1) ab = ba (commutativity, commutative law);

2) a (bc) = (ab) c (associativity, combinational law);

a (b + c) = ab + ac (distributivity, distributive law). In this case, a ×0 = 0; a×1 = a. These properties form the basis of the usual technique for calculating multi-digit numbers.

A further generalization of the concept of control is associated with the possibility of considering numbers as operators in a set of vectors on a plane. For example, the complex number r (cosj + i sin j) corresponds to the operator of stretching all vectors by r times and rotating them by an angle j around the origin. In this case, the control of complex numbers corresponds to the control of the corresponding operators, that is, the result of the control will be an operator obtained by sequential application of two given operators. This definition of linear operators extends to other types of operators that can no longer be expressed using numbers (for example, linear transformations). This leads to the operations of control matrices, quaternions, considered as rotation and dilation operators in three-dimensional space, kernels of integral operators, etc. With such generalizations, some of the above properties of algebra may not be fulfilled, most often the property of commutativity (non-commutative algebra). The study of the general properties of the operation of U is included in the problems of general algebra, in particular the theory of groups and rings.

Wikipedia

Multiplication

Multiplication- one of the main binary mathematical operations (arithmetic operations) of two arguments. For example, for natural numbers: $c=a \cdot b = \underbrace( a+a+\cdots+a )_(b)= a_1 + a_2 + \ldots + a_b = (\displaystyle\sum_(i=1)^b a_i)$

In general form we can write: Π( a, b) = c. That is, each pair of elements ( a, b) is matched to the element c = a ⋅ b, called the product a And b.

In writing it is usually indicated using one of the “multiplication signs” - “ ⋅ ,  × ,  * ”, for example: a ⋅ b = c. Multiplication can also be defined for rational, real, complex numbers and other mathematical, physical and abstract quantities.

Multiplication has several important properties:

Commutativity: a ⋅ b = b ⋅ a; Associativity: ( a ⋅ b) ⋅ c = a ⋅ (b ⋅ c); Distributivity: x ⋅ (a + b) = (x ⋅ a) + (x ⋅ b),  ∀a, b ∈  A; Multiplying by zero (zero element) gives a number equal to zero: x⋅ 0 = 0; Multiplying by one (neutral element) gives a number equal to the original: x ⋅ 1 = x.

The figure shows an example of counting apples using the multiplication operation, 3 groups of 5 apples, resulting in 15 apples: 5 ⋅ 3 = 15.

On the set of real numbers, the range of values ​​of the multiplication function graphically has the form of a surface passing through the origin of coordinates and curved on both sides in the form of a parabola.

Examples of the use of the word multiplication in literature.

He also compares their work with leavening, with sowing seeds, and with multiplication mustard seeds.

Then there were those who did not dare to intervene at all, because their consciousness explored the events of secondary and tertiary effects as they multiplication and entanglement in all directions of the entire system.

multiplication sins and a lowering of the sinful threshold as a result of the Antichrist who has penetrated into the minds of people in the form of materialistic-atheistic teachings and the false prophet in the person of the Communist Party of Marx-Lenin.

Over the past century, it has happened again multiplication sins and a lowering of the sin threshold as a result of the Antichrist infiltrating the minds of people in the form of a materialistic-atheistic teaching and a false prophet in the person of the Communist Party of Marx-Lenin.

This is a criticism of the doctrine of mercantilism, which identified multiplication the amount of money in the country with the growth of the population's well-being.

Before describing the actions of the troops, through the unexpected multiplication who came from, so to speak, a bandit gang into the equestrian party, it would not be superfluous to introduce the reader to its private leaders.

One day on the street I heard an intricate song that rhymed the beginning of the table multiplication: One day, the gentleman arrived.

His actions and antics are meaningless, they indicate a split in Chichikov, his multiplication in the mirror 32 game of imitations, in which there is no longer an original, but only the clowning of copies.

He spoke about this at least three times later, leaving the future reteller free to montage the details: - Heisenberg's rule multiplication couldn’t get it out of my head, and after intense thought, one morning I saw the light: I remembered the algebraic theory that I had studied as a student.

Her studies show that the Earth became more and more heterogeneous as multiplication layers forming its crust, further, that it became more and more heterogeneous in relation to the composition of these layers, of which the latter, formed from fragments of old layers, became extremely complex through the mixing of the materials contained in them and, finally, that this heterogeneity was significantly enhanced by the action of still the hot core of the Earth onto its surface, which is why not only the enormous variety of plutonic mountains occurred, but also the inclination of the deposited layers at different angles, the formation of gaps, metal veins and endless irregularities and deviations. Geologists also say that the size of the elevations on the surface of the Earth changed, that the most ancient mountain systems are the least high and that the Andes and Himalayas are the newest elevations, meanwhile, in all likelihood, corresponding changes took place at the bottom of the ocean.

If it's hard to do multiplication with tension while lifting the piano, how is it possible to master the subtlest inner feelings in a complex role with the subtle psychology of Othello!

We are specialists in research, analysis and measurement, we are the custodians and constant checkers of all alphabets, tables multiplication and methods, we are branders of spiritual weights and measures.

He did not read books, our captain Trotta, and secretly pitied his growing son, who was soon to be faced with a pencil, a board and a sponge, paper, a ruler and a table multiplication and for which the inevitable textbooks were already waiting.

The new manager - a strong, salty guy - quickly brought Uzhik to clean water, discovered that he had not even mastered the tables multiplication, and thunderously kicked him out of school.

These operations may include addition, subtraction, and multiplication functions, comparison of functions, similar operations on a function and a number, finding the maximum of functions, calculating an indefinite integral, calculating a definite integral of the derivative of two functions, shifting a function along the abscissa, etc.

MULTIPLY meaning

T.F. Efremova New dictionary of the Russian language. Explanatory and word-formative

multiplication

Meaning:

multiply e knowledge

1) The process of action according to meaning. verb: to multiply (1), to multiply.

Meaning:

arithmetic operation. Indicated by a dot "." or a "?" (in literal calculations, multiplication signs are omitted). Multiplication of positive integers (natural numbers) is an action that allows, from two numbers a (the multiplicand) and b (the multiplier), to find the third number ab (the product), equal to the sum of b terms, each of which is equal to a; a and b are also called factors. Multiplication of fractional numbers a/b and c/d is determined by the equality Multiplication of two rational numbers gives a number, abs. whose value is equal to the product of the absolute values ​​of the factors and which has a plus sign (+) if both factors have the same signs, or a minus sign (-) if they have different signs. The multiplication of irrational numbers is determined using their rational approximations. Multiplying complex numbers given in the form? = a+bi and? = c+di, determined by the equality ?? = ac - bd + (a + bc)i.

Small Academic Dictionary of the Russian Language

multiplication

Meaning:

I, Wed

Action according to verb. multiply - multiply (by 2); action and state by value. verb multiply - multiply.

As the family multiplied, supervision became more difficult. Pomyalovsky, Danilushka.

- We need an increase in human pleasures and an alleviation of human suffering. Sun. Ivanov, Blue Sands.

The inverse of division is a mathematical operation by which from two numbers (or quantities) a new number (or quantity) is obtained, which (for integers) contains as a term the first number as many times as there are units in the second.

Multiplication table.

Several rules apply when multiplying and dividing integers. In this lesson we will look at each of them.

When multiplying and dividing integers, pay attention to the signs of the numbers. It will depend on them which rule to apply. Also, it is necessary to study several laws of multiplication and division. Studying these rules allows you to avoid some annoying mistakes in the future.

Multiplication laws

We looked at some of the laws of mathematics in the lesson. But we have not considered all the laws. There are many laws in mathematics, and it would be wiser to study them sequentially as needed.

First, let's remember what multiplication consists of. Multiplication consists of three parameters: multiplicand, multiplier And works. For example, in the expression 3 × 2 = 6, the number 3 is the multiplicand, the number 2 is the multiplier, and the number 6 is the product.

Multiplicand shows what exactly we are increasing. In our example we increase the number 3.

Factor shows how many times you need to increase the multiplicand. In our example, the multiplier is the number 2. This multiplier shows how many times the multiplicand 3 needs to be increased. That is, during the multiplication operation, the number 3 will be doubled.

Work This is the actual result of the multiplication operation. In our example, the product is the number 6. This product is the result of multiplying 3 by 2.

The expression 3 × 2 can also be understood as the sum of two triplets. Multiplier 2 in this case will show how many times you need to repeat the number 3:

Thus, if the number 3 is repeated twice in a row, the number 6 will be obtained.

Commutative law of multiplication

The multiplicand and the multiplier are called by one common word - factors. The commutative multiplication law is as follows:

Rearranging the places of the factors does not change the product.

Let's check if this is true. For example, let's multiply 3 by 5. Here 3 and 5 are factors.

3 × 5 = 15

Now let's swap the factors:

5 × 3 = 15

In both cases, we get the answer 15, which means we can put an equal sign between the expressions 3 × 5 and 5 × 3, since they are equal to the same value:

3 × 5 = 5 × 3

15 = 15

And with the help of variables, the commutative law of multiplication can be written as follows:

a × b = b × a

Where a And b- factors

Combinative law of multiplication

This law says that if an expression consists of several factors, then the product will not depend on the order of actions.

For example, the expression 3 × 2 × 4 consists of several factors. To calculate it, you can multiply 3 and 2, then multiply the resulting product by the remaining number 4. It will look like this:

3 × 2 × 4 = (3 × 2) × 4 = 6 × 4 = 24

This was the first solution. The second option is to multiply 2 and 4, then multiply the resulting product by the remaining number 3. It will look like this:

3 × 2 × 4 = 3 × (2 × 4) = 3 × 8 = 24

In both cases, we get the answer 24. Therefore, we can put an equal sign between the expressions (3 × 2) × 4 and 3 × (2 × 4), since they are equal to the same value:

(3 × 2) × 4 = 3 × (2 × 4)

and with the help of variables the associative law of multiplication can be written as follows:

a × b × c = (a × b) × c = a × (b × c)

where instead of a, b,c Any numbers can be.

Distributive law of multiplication

The distributive law of multiplication allows you to multiply a sum by a number. To do this, each term of this sum is multiplied by this number, then the resulting results are added.

For example, let's find the value of the expression (2 + 3) × 5

The expression in parentheses is the sum. This sum must be multiplied by the number 5. To do this, each term of this sum, that is, the numbers 2 and 3, must be multiplied by the number 5, then the resulting results must be added:

(2 + 3) × 5 = 2 × 5 + 3 × 5 = 10 + 15 = 25

This means that the value of the expression (2 + 3) × 5 is 25.

Using variables, the distribution law of multiplication is written as follows:

(a + b) × c = a × c + b × c

where instead of a, b, c Any numbers can be.

Law of multiplication by zero

This law says that if there is at least one zero in any multiplication, then the answer will be zero.

The product is equal to zero if at least one of the factors is equal to zero.

For example, the expression 0 × 2 is equal to zero

In this case, the number 2 is a multiplier and shows how many times the multiplicand needs to be increased. That is, how many times to increase zero. Literally this expression reads like this: "double zero" . But how can you double a zero if it is zero? The answer is no.

In other words, if “nothing” is doubled or even a million times, it will still turn out to be “nothing.”

And if you swap the factors in the expression 0 × 2, you will again get zero. We know this from the previous displacement law:

Examples of applying the law of multiplication by zero:

5 × 5 × 5 × 0 = 0

2 × 5 × 0 × 9 × 1 = 0

In the last two examples there are several factors. Having seen a zero in them, we immediately put a zero in the answer, applying the law of multiplication by zero.

We looked at the basic laws of multiplication. Next, we'll look at multiplying integers.

Multiplying Integers

Example 1. Find the value of the expression −5 × 2

This is the multiplication of numbers with different signs. −5 is a negative number and 2 is a positive number. For such cases, the following rule should be applied:

To multiply numbers with different signs, you need to multiply their modules and put a minus in front of the resulting answer.

−5 × 2 = − (|−5| × |2|) = − (5 × 2) = − (10) = −10

Usually written shorter: −5 × 2 = −10

Any multiplication can be represented as a sum of numbers. For example, consider the expression 2 × 3. It equals 6.

The multiplier in this expression is the number 3. This multiplier shows how many times you need to increase the two. But the expression 2 × 3 can also be understood as the sum of three twos:

The same thing happens with the expression −5 × 2. This expression can be represented as the sum

And the expression (−5) + (−5) is equal to −10. We know this from . This is the addition of negative numbers. Recall that the result of adding negative numbers is a negative number.

Example 2. Find the value of the expression 12 × (−5)

This is the multiplication of numbers with different signs. 12 is a positive number, (−5) is negative. Again we apply the previous rule. We multiply the modules of numbers and put a minus in front of the resulting answer:

12 × (−5) = − (|12| × |−5|) = − (12 × 5) = − (60) = −60

Usually the solution is written shorter:

12 × (−5) = −60

Example 3. Find the value of the expression 10 × (−4) × 2

This expression consists of several factors. First, multiply 10 and (−4), then multiply the resulting number by 2. Along the way, apply the previously learned rules:

First action:

10 × (−4) = −(|10| × |−4|) = −(10 × 4) = (−40) = −40

Second action:

−40 × 2 = −(|−40 | × | 2|) = −(40 × 2) = −(80) = −80

So the value of the expression 10 × (−4) × 2 is −80

Let's write down the solution briefly:

10 × (−4) × 2 = −40 × 2 = −80

Example 4. Find the value of the expression (−4) × (−2)

This is the multiplication of negative numbers. In such cases, the following rule must be applied:

To multiply negative numbers, you need to multiply their modules and put a plus in front of the resulting answer.

(−4) × (−2) = |−4| × |−2| = 4 × 2 = 8

Traditionally, we don’t write down the plus, so we just write down the answer 8.

Let's write the solution shorter (−4) × (−2) = 8

The question arises: why does multiplying negative numbers suddenly produce a positive number? Let's try to prove that (−4) × (−2) equals 8 and nothing else.

First we write the following expression:

Let's enclose it in brackets:

(4 × (−2))

Let's add to this expression our expression (−4) × (−2). Let's put it in brackets too:

(4 × (−2) ) + ((−4) × (−2) )

Let's equate all this to zero:

(4 × (−2)) + ((−4) × (−2)) = 0

Now the fun begins. The point is that we must evaluate the left side of this expression and get 0 as a result.

So the first product (4 × (−2)) is −8. Let's write the number −8 in our expression instead of the product (4 × (−2))

−8 + ((−4) × (−2)) = 0

Now instead of the second work we will temporarily put an ellipsis

Now let's look carefully at the expression −8 + ... = 0. What number should stand in place of the ellipsis for equality to be maintained? The answer suggests itself. Instead of an ellipsis there should be a positive number 8 and nothing else. This is the only way equality will be maintained. After all, −8 + 8 equals 0.

We return to the expression −8 + ((−4) × (−2)) = 0 and instead of the product ((−4) × (−2)) we write the number 8

Example 5. Find the value of the expression −2 × (6 + 4)

Let's apply the distributive law of multiplication, that is, multiply the number −2 by each term of the sum (6 + 4)

−2 × (6 + 4) = −2 × 6 + (−2) × 4

Now let's do the multiplication and add up the results. Along the way, we apply the previously learned rules. The entry with modules can be skipped so as not to clutter the expression

First action:

−2 × 6 = −12

Second action:

−2 × 4 = −8

Third action:

−12 + (−8) = −20

So the value of the expression −2 × (6 + 4) is −20

Let's write down the solution briefly:

−2 × (6 + 4) = (−12) + (−8) = −20

Example 6. Find the value of the expression (−2) × (−3) × (−4)

The expression consists of several factors. First, multiply the numbers −2 and −3, and multiply the resulting product by the remaining number −4. Let's skip the entry with modules so as not to clutter the expression

First action:

(−2) × (−3) = 6

Second action:

6 × (−4) = −(6 × 4) = −24

So the value of the expression (−2) × (−3) × (−4) is equal to −24

Let's write down the solution briefly:

(−2) × (−3) × (−4) = 6 × (−4) = −24

Laws of division

Before dividing integers, you need to learn the two laws of division.

First of all, let’s remember what division consists of. The division consists of three parameters: divisible, divisor And private. For example, in expression 8: 2 = 4, 8 is the dividend, 2 is the divisor, 4 is the quotient.

Dividend shows what exactly we are sharing. In our example we are dividing the number 8.

Divider shows how many parts the dividend must be divided into. In our example, the divisor is the number 2. This divisor shows how many parts the dividend 8 needs to be divided into. That is, during the division operation, the number 8 will be divided into two parts.

Private- This is the actual result of the division operation. In our example, the quotient is 4. This quotient is the result of dividing 8 by 2.

You can't divide by zero

Any number cannot be divided by zero.

The fact is that division is the inverse action of multiplication. This phrase can be understood in its literal sense. For example, if 2 × 5 = 10, then 10:5 = 2.

It can be seen that the second expression is written in reverse order. If, for example, we have two apples and we want to increase them five times, then we will write 2 × 5 = 10. The result will be ten apples. Then, if we want to reduce those ten apples back down to two, we write 10: 5 = 2

You can do the same with other expressions. If, for example, 2 × 6 = 12, then we can return back to the original number 2. To do this, just write the expression 2 × 6 = 12 in reverse order, dividing 12 by 6

Now consider the expression 5 × 0. We know from the laws of multiplication that the product is equal to zero if at least one of the factors is equal to zero. This means that the expression 5 × 0 is equal to zero

If we write this expression in reverse order, we get:

The answer that immediately catches your eye is 5, which is obtained by dividing zero by zero. This is impossible.

In reverse order, you can write another similar expression, for example 2 × 0 = 0

In the first case, dividing zero by zero we got 5, and in the second case 2. That is, each time dividing zero by zero, we can get different values, and this is unacceptable.

The second explanation is that dividing the dividend by the divisor means finding a number that, when multiplied by the divisor, gives the dividend.

For example, the expression 8: 2 means finding a number that, when multiplied by 2, gives 8

Here, instead of an ellipsis, there should be a number that, when multiplied by 2, will give the answer 8. To find this number, just write this expression in reverse order:

We got the number 4. Let's write it instead of the ellipsis:

Now imagine that you need to find the value of the expression 5: 0. In this case, 5 is the dividend, 0 is the divisor. Dividing 5 by 0 means finding a number that when multiplied by 0 gives 5

Here, instead of an ellipsis, there should be a number that, when multiplied by 0, will give the answer 5. But there is no number that, when multiplied by zero, gives 5.

The expression ... × 0 = 5 contradicts the law of multiplication by zero, which states that the product is equal to zero when at least one of the factors is equal to zero.

This means that writing the expression... × 0 = 5 in reverse order, dividing 5 by 0 makes no sense. That's why they say you can't divide by zero.

Using variables, this law is written as follows:

At b ≠ 0

Number a can be divided by a number b, provided that b not equal to zero.

Property of private

This law says that if the dividend and the divisor are multiplied or divided by the same number, the quotient will not change.

For example, consider expression 12: 4. The value of this expression is 3

Let's try to multiply the dividend and divisor by the same number, for example by the number 4. If we believe the property of the quotient, we should again get the number 3 in the answer

(12 × 4) : (4 × 4)

(12 × 4) : (4 × 4) = 48: 16 = 3

We received answer 3.

Now let's try not to multiply, but to divide the dividend and divisor by the number 4

(12: 4 ) : (4: 4 )

(12: 4 ) : (4: 4 ) = 3: 1 = 3

We received answer 3.

We see that if the dividend and divisor are multiplied or divided by the same number, then the quotient does not change.

Integer division

Example 1. Find the value of expression 12: (−2)

This is the division of numbers with different signs. 12 is a positive number, (−2) is negative. To solve this example, you need Divide the module of the dividend by the module of the divisor, and put a minus before the resulting answer.

12: (−2) = −(|12| : |−2|) = −(12: 2) = −(6) = −6

Usually written shorter:

12: (−2) = −6

Example 2. Find the value of the expression −24: 6

This is the division of numbers with different signs. −24 is a negative number, 6 is a positive number. Yet again Divide the module of the dividend by the module of the divisor, and put a minus in front of the resulting answer.

−24: 6 = −(|−24| : |6|) = −(24: 6) = −(4) = −4

Let's write down the solution briefly:

Example 3. Find the value of the expression −45: (−5)

This is division of negative numbers. To solve this example, you need Divide the module of the dividend by the module of the divisor, and put a plus sign in front of the resulting answer.

−45: (−5) = |−45| : |−5| = 45: 5 = 9

Let's write down the solution briefly:

−45: (−5) = 9

Example 4. Find the value of the expression −36: (−4) : (−3)

According to, if the expression contains only multiplication or division, then all actions must be performed from left to right in the order they appear.

Divide −36 by (−4), and divide the resulting number by −3

First action:

−36: (−4) = |−36| : |−4| = 36: 4 = 9

Second action:

9: (−3) = −(|9| : |−3|) = −(9: 3) = −(3) = −3

Let's write down the solution briefly:

−36: (−4) : (−3) = 9: (−3) = −3

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Multiplying one integer by another means repeating one number as many times as the other contains units. To repeat a number means to take it as an addend several times and determine the sum.

Definition of multiplication

Multiplication of integers is an operation in which you need to take one number as addends as many times as another number contains units, and find the sum of these addends.

Multiplying 7 by 3 means taking the number 7 as its addend three times and finding the sum. The required amount is 21.

Multiplication is the addition of equal terms.

The data in multiplication is called multiplicand and multiplier, and the required - work.

In the proposed example, the data will be the multiplicand 7, the multiplier 3, and the desired product 21.

Multiplicand. A multiplicand is a number that is multiplied or repeated by a addend. A multiplicand expresses the magnitude of equal terms.

Factor. The multiplier shows how many times the multiplicand is repeated by the addend. The multiplier shows the number of equal terms.

Work. A product is a number that is obtained from multiplication. It is the sum of equal terms.

The multiplicand and the multiplier together are called manufacturers.

When multiplying integers, one number increases by as many times as the other number contains units.

Multiplication sign. The action of multiplication is denoted by the sign × (indirect cross) or. (dot). The multiplication sign is placed between the multiplicand and the multiplier.

Repeating the number 7 three times as a summand and finding the sum means 7 multiplied by 3. Instead of writing

write using the multiplication sign in short:

7 × 3 or 7 3

Multiplication is a shortened addition of equal terms.

Sign ( × ) was introduced by Oughtred (1631), and the sign. Christian Wolf (1752).

The relationship between the data and the desired number is expressed in multiplication

in writing:

7 × 3 = 21 or 7 3 = 21

verbally:

seven multiplied by three is 21.

To make a product of 21, you need to repeat 7 three times

To make a factor of 3, you need to repeat the unit three times

From here we have another definition of multiplication: Multiplication is an action in which a product is made up of the multiplicand in the same way as a factor is made up of a unit.

The main property of the work

The product does not change due to a change in the order of producers.

Proof. Multiplying 7 by 3 means repeating 7 three times. Replacing 7 with the sum of 7 units and inserting them in vertical order, we have:

Thus, when multiplying two numbers, we can consider either of the two producers to be the multiplier. On this basis, manufacturers are called factors or simply multipliers.

The most common method of multiplication is to add equal terms; but if the producers are large, this technique leads to long calculations, so the calculation itself is arranged differently.

Multiplying single digit numbers. Pythagorean table

To multiply two single-digit numbers, you need to repeat one number as a addend as many times as the other contains units, and find their sum. Since multiplying integers leads to multiplying single-digit numbers, they create a table of products of all single-digit numbers in pairs. Such a table of all products of single-digit numbers in pairs is called multiplication table.

Its invention is attributed to the Greek philosopher Pythagoras, after whom it is called Pythagorean table. (Pythagoras was born around 569 BC).

To create this table, you need to write the first 9 numbers in a horizontal row:

1, 2, 3, 4, 5, 6, 7, 8, 9.

Then under this line you need to sign a series of numbers expressing the product of these numbers by 2. This series of numbers will be obtained when in the first line we add each number to itself. From the second line of numbers we move sequentially to 3, 4, etc. Each subsequent line is obtained from the previous one by adding the numbers of the first line to it.

Continuing to do this until line 9, we get the Pythagorean table in the following form

To use this table to find the product of two single-digit numbers, you need to find one manufacturer in the first horizontal row, and the other in the first vertical column; then the required product will be at the intersection of the corresponding column and row. Thus, the product 6 × 7 = 42 is at the intersection of the 6th row and 7th column. The product of zero and a number and a number and zero always produces zero.

Since multiplying a number by 1 gives the number itself and changing the order of the factors does not change the product, all the different products of two single-digit numbers that you should pay attention to are contained in the following table:

Products of single-digit numbers not contained in this table are obtained from the data if only the order of the factor in them is changed; thus 9 × 4 = 4 × 9 = 36.

Multiplying a multi-digit number by a single-digit number

Multiplying the number 8094 by 3 is indicated by signing the multiplier under the multiplicand, placing a multiplication sign on the left and drawing a line to separate the product.

Multiplying the multi-digit number 8094 by 3 means finding the sum of three equal terms

therefore, to multiply, you need to repeat all orders of a multi-digit number three times, that is, multiply by 3 units, tens, hundreds, etc. Addition begins with one, therefore, multiplication must begin with one, and then move from the right hand to left to higher order units.

In this case, the progress of calculations is expressed verbally:

We start multiplication with units: 3 × 4 equals 12, we sign 2 under the units, and apply the unit (1 ten) to the product of the next order by the factor (or remember it in our minds).

Multiplying tens: 3 × 9 equals 27, but 1 in your head equals 28; We sign the tens 8 and 2 in our heads.

Multiplying hundreds: Zero multiplied by 3 gives zero, but 2 in your head equals 2, we sign 2 under the hundreds.

Multiplying thousands: 3 × 8 = 24, we sign completely 24, because we do not have the following orders.

This action will be expressed in writing:

From the previous example we derive the following rule. To multiply a multi-digit number by a single-digit number, you need:

Sign the multiplier under the units of the multiplicand, put a multiplication sign on the left and draw a line.

Start multiplication with simple units, then, moving from the right hand to the left, sequentially multiply tens, hundreds, thousands, etc.

If, during multiplication, the product is expressed as a single-digit number, then it is signed under the multiplied digit of the multiplicand.

If the product is expressed as a two-digit number, then the units digit is signed under the same column, and the tens digit is added to the product of the next order by the factor.

Multiplication continues until the full product is obtained.

Multiplying numbers by 10, 100, 1000...

Multiplying numbers by 10 means turning simple units into tens, tens into hundreds, etc., that is, increasing the order of all numbers by one. This is achieved by adding one zero to the right. Multiplying by 100 means increasing all orders of magnitude of what is being multiplied by two units, that is, turning units into hundreds, tens into thousands, etc.

This is achieved by adding two zeros to the number.

From here we conclude:

To multiply an integer by 10, 100, 1000, and generally by 1 with zeros, you need to assign as many zeros to the right as there are in the factor.

Multiplying the number 6035 by 1000 can be expressed in writing:

When the multiplier is a number ending in zeros, only the significant digits are signed under the multiplicand, and the zeros of the multiplier are added to the right.

To multiply 2039 by 300, you need to take the number 2029 by adding it 300 times. Taking 300 terms is the same as taking three times 100 terms or 100 times three terms. To do this, multiply the number by 3, and then by 100, or multiply first by 3, and then add two zeros to the right.

The progress of the calculation will be expressed in writing:

Rule. To multiply one number by another, represented by a digit with zeros, you must first multiply the multiplicand by the number expressed by the significant digit, and then add as many zeros as there are in the multiplier.

Multiplying a multi-digit number by a multi-digit number

To multiply a multi-digit number 3029 by a multi-digit 429, or find the product 3029 * 429, you need to repeat the 3029 addend 429 times and find the sum. Repeating 3029 with terms 429 times means repeating it with terms first 9, then 20 and finally 400 times. Therefore, to multiply 3029 by 429, you need to multiply 3029 first by 9, then by 20 and finally by 400 and find the sum of these three products.

Three works

are called private works.

The total product 3029 × 429 is equal to the sum of three quotients:

3029 × 429 = 3029 × 9 + 3029 × 20 + 3029 × 400.

Let us find the values ​​of these three partial products.

Multiplying 3029 by 9, we find:

3029 × 9 27261 first private work

Multiplying 3029 by 20, we find:

3029 × 20 60580 second particular work

Multiplying 3026 by 400, we find:

3029 × 400 1211600 third partial work

Adding these partial products, we get the product 3029 × 429:

It is not difficult to notice that all these partial products are products of the number 3029 by the single-digit numbers 9, 2, 4, and one zero is added to the second product, resulting from multiplication by tens, and two zeros to the third.

Zeros assigned to partial products are omitted during multiplication and the progress of the calculation is expressed in writing:

In this case, when multiplying by 2 (the tens digit of the multiplier), sign 8 under the tens, or move to the left by one digit; when multiplying by the hundreds digit 4, sign 6 in the third column, or move to the left by 2 digits. In general, each particular work begins to be signed from the right hand to the left, according to the order to which the multiplier digit belongs.

Looking for the product of 3247 by 209, we have:

Here we begin to sign the second quotient product under the third column, because it expresses the product of 3247 by 2, the third digit of the multiplier.

Here we have omitted only two zeros, which should have appeared in the second partial product, as it expresses the product of a number by 2 hundreds or by 200.

From all that has been said, we derive a rule. To multiply a multi-digit number by a multi-digit number,

you need to sign the multiplier under the multiplicand so that the numbers of the same orders are in the same vertical column, put a multiplication sign on the left and draw a line.

Multiplication begins with simple units, then moves from the right hand to the left, multiplying the sequential multiplicand by the digit of tens, hundreds, etc. and creating as many partial products as there are significant digits in the multiplier.

The units of each partial product are signed under the column to which the digit of the multiplier belongs.

All partial products found in this way are added together and the total product is obtained.

To multiply a multi-digit number by a factor ending in zeros, you need to discard the zeros in the factor, multiply by the remaining number, and then add as many zeros to the product as there are in the factor.

Example. Find the product of 342 by 2700.

If the multiplicand and the multiplier both end in zeros, during multiplication they are discarded and then as many zeros are added to the product as are contained in both producers.

Example. Calculating the product of 2700 by 35000, we multiply 27 by 35

By adding five zeros to 945, we get the desired product:

2700 × 35000 = 94500000.

Number of digits of the product. The number of digits of the product 3728 × 496 can be determined as follows. This product is more than 3728 × 100 and less than 3728 × 1000. The number of digits of the first product 6 is equal to the number of digits in the multiplicand 3728 and in the multiplier 496 without one. The number of digits of the second product 7 is equal to the number of digits in the multiplicand and in the multiplier. A given product of 3728 × 496 cannot have digits less than 6 (the number of digits of the product is 3728 × 100, and more than 7 (the number of digits of the product is 3728 × 1000).

Where we conclude: the number of digits of any product is either equal to the number of digits in the multiplicand and in the factor, or equal to this number without a unit.

Our product may contain either 7 or 6 digits.

Degrees

Among different works, those in which the producers are equal deserve special attention. For example:

2 × 2 = 4, 3 × 3 = 9.

Squares. The product of two equal factors is called the square of a number.

In our examples, 4 is square 2, 9 is square 3.

cubes. The product of three equal factors is called the cube of a number.

So, in the examples 2 × 2 × 2 = 8, 3 × 3 × 3 = 27, the number 8 is the cube of 2, 27 is the cube of 3.

At all the product of several equal factors is calledpower of number . The powers get their names from the number of equal factors.

Products of two equal factors or squares are called second degrees.

Products of three equal factors or cubes are called third degrees, etc.