Multiplying natural numbers and its properties. Work (mathematics)

If the concert hall is illuminated with 3 chandeliers 25 light bulbs each, then the light bulbs in these chandeliers will be 25 + 25 + 25, that is 75.

The amount in which all the components are equal to each other are written in short: instead of 25 + 25 + 25 write 25 3. So, 25 3 \u003d 75 (Fig. 43). The number 75 is called work numbers 25 and 3, and numbers 25 and 3 called multipliers.

Fig. 43. The product of numbers 25 and 3

Multiply multiply M to the natural number N is to find the amount of N of the terms, each of which is M.

The expression M n and the value of this expression is called work numbersm.andn.. Numbers that change call multipliers. Those. M and N - multipliers.

Works 7 4 and 4 7 are equal to the same number 28 (Fig. 44).

Fig. 44. Production 7 4 \u003d 4 7

1. The product of two numbers does not change when the multipliers are permuting.

movement

a. × b. = b. × a. .

Produces (5 3) 2 \u003d 15 2 and 5 (3 2) \u003d 5 6 have the same value 30. So, 5 (3 2) \u003d (5 3) 2 (Fig. 45).

Fig. 45. Work (5 3) 2 \u003d 5 (3 2)

2. To multiply the number on the work of two numbers, you can first multiply it on the first multiplier, and then the resulting product is multiplied by the second factor.

This property of multiplication is called combining. With the help of letters, it is written so:

but (b. c) \u003d (andb. from).

The sum N of the terms, each of which is 1, equal to n. Therefore, the equality 1 n \u003d n is true.

The sum N of the terms, each of which is zero, is zero. Therefore, the equality is 0 n \u003d 0.

In order for the multiplication program to be correct when n \u003d 1 and n \u003d 0, it was agreed that M 1 \u003d M and M 0 = 0.

In front of the lettering multiplies, multiplication sign usually: instead of 8 h. Write 8. h., instead butb. write butb..

Lower the sign of multiplication and in front of the brackets. For example, instead of 2 ( a +.b.) write 2. (A +.b.) , and instead ( h. + 2) (y + 3) write (x + 2) (y + 3).

Instead of aB) C write aBC.

When there are no brackets in the recordings, multiplication is performed in order from left to right.

Works read, calling every multiplier in the parental case. For example:

1) 175 60 - work one hundred seventy five and sixty;

2) 80 (h. + 1 7) - Production R.P. R.P.

eighty and amount of X and seventeen

We will solve the task.

How many three-digit numbers (Fig. 46) can be made from numbers 2, 4, 6, 8, if the numbers in the number records are not repeated?

Decision.

The first number of numbers can be any of fourdata numbers, second - any of threeothers, and the third - any of tworemaining. It turns out:

Fig. 46. \u200b\u200bTo the problem of compiling three-digit numbers

Total of these numbers can be 4 3 2 \u003d 24 three digits.

We will solve the task.

The company's board includes 5 people. From its composition, the Board should choose the president and vice president. How many ways can this be done?

Decision.

President of the company can be elected one of 5 people:

The president:

After the President is elected, the vice-president can choose any of the four remaining members of the Board (Fig. 47):

The president:

Vice President:

Fig. 47. To the problem of elections

So you can choose the president with five ways, and for each president of the president, four ways you can choose the vice-president. Hence, total number ways to choose the president and vice-president of the firm is: 5 4 \u003d 20 (see Fig. 47).

We will still task.

From the village of Anikseevo, four roads are conducted in the village of Bolsharo, and three roads in the village of Vinogradov - three roads (Fig. 48). How many ways can be reached from Anikeev in Vinogradov through the village of Bolovo?

Fig. 48. To the task of roads

Decision.

If you get on the 1st road from a in b, then there are three ways to continue the path (Fig. 49).

Fig. 49. Path options

In the same way, we get three ways to continue the way, starting to get in the 2nd, and 3rd, and on the 4th road. So, it turns out to be 4 3 \u003d 12 ways to get from Anikeev in Vinogradov.

We decide another task.

A family consisting of grandmother, dad, mom, daughters and son, presented 5 different cups. How many ways can be divided by cups between family members?

Decision. At the first family member (for example, grandmothers) there are 5 options for choosing, the following (let it be dad) remains 4 options. The next (for example, mom) will choose from 3 cups, the following - of the two, the latter receives one remaining cup. We show these methods in the diagram (Fig. 50).

Fig. 50. Scheme to solve the problem

They got that each choice of a cup of a grandmother corresponds to four possible choice of dad, i.e. Total 5 4 ways. After dad chose a cup, mom has three choices, the daughter has two, the son is one, i.e. Total 3 2 1 ways. We finally get that to solve the problem it is necessary to find a product 5 4 3 2 1.

Note that we got a product of all natural numbers from 1 to 5. Such works are written in short:

5 4 3 2 1 \u003d 5! (Read: "Five factorial").

Factorial numbers - The product of all natural numbers from 1 to this number.

So, the answer is: 5! \u003d 120, i.e. Cups between family members can be distributed in twenty ways.

In this article, we will deal with how multiplication of integers. First we introduce terms and notation, as well as find out the meaning of multiplying two integers. After that, we obtain the rules for multiplying two entire positive, whole negative and integers with different signs. In this case, we will give examples with a detailed explanation of the decision of the solution. We also raise cases of multiplication of integers when one of the multipliers is equal to one or zero. Then we will learn to check the resulting multiplication result. And finally, let's talk about multiplication of three, four and more whole numbers.

Navigating page.

Terms and notation

To describe multiplication of integers, we will use the same terms with which we described the multiplication of natural numbers. Recall them.

Multiply integer numbers are called multipliers. The result of multiplication is called work. Action multiplication is denoted by a sign multiplying the type "·". In some sources you can meet the designation of multiplication by signs "*" or "×".

The multiply integers A, B and the result of their multiplication C is conveniently recorded using equality of the form A · B \u003d c. In this record, an integer A is the first factor, an integer B - the second factor, and the number C is a work. The species A · b will also be called the work, as well as the value of this expression c.

Run ahead, we note that the product of two integers is an integer.

The meaning of multiplying integers

Multiplying whole positive numbers

Whole positive numbers are natural numbers, so multiplying whole positive numbers It is carried out according to all the rules of multiplication of natural numbers. It is clear that as a result of multiplying two integer positive numbers, a whole positive number (natural number) will be obtained. Consider a couple of examples.

Example.

What is the product of the whole positive numbers 127 and 5?

Decision.

The first factor 107 will be presenting in the form of the sum of discharge terms, that is, in the form of 100 + 20 + 7. After that, we use the rule of multiplication of the number of numbers for this number: 127 · 5 \u003d (100 + 20 + 7) · 5 \u003d 100 · 5 + 20 · 5 + 7 · 5. It remains only to finish the calculation: 100 · 5 + 20 · 5 + 7 · 5 \u003d 500 + 100 + 35 \u003d 600 + 35 \u003d 635.

Thus, the product of these integer positive numbers 127 and 5 is 635.

Answer:

127 · 5 \u003d 635.

For multiplying multivalued integer positive numbers, it is convenient to use a multiplication method by a column.

Example.

Multiply a three-digit integer positive number 712 per double-digit integer positive number 92.

Decision.

Perform multiplying data of integers positive numbers in the column:

Answer:

712 · 92 \u003d 65 504.

The rule of multiplication of integers with different signs, examples

To formulate a rule of multiplication of integers with different signs will help us with the following example.

We calculate the product of a whole negative number -5 and a whole positive number 3 based on the meaning of multiplication. So (-5) · 3 \u003d (- 5) + (- 5) + (- 5) \u003d - 15. To preserve the validity of the multiplication property, the equality (-5) · 3 \u003d 3 · (-5) must be performed. That is, the product 3 · (-5) is also equal to -15. It is easy to see that -15 is equal to the product of the initial multipliers modules, from where it follows that the product of the initial integers with different signs is equal to the product of the initial multipliers modules taken with a minus sign.

So we got the rule of multiplication of integers with different signs: To multiply two integers with different signs, you need to multiply the modules of these numbers and put a minus sign before the received number.

From the voiced rule, it can be concluded that the product of integers with different signs is always a whole negative number. Indeed, as a result of multiplying multipliers modules, we will receive a whole positive number, and if you have a minus sign before this number, it will become a whole negative.

Consider examples of calculating the product of integers with different signs using the result received.

Example.

Multiplying the integer positive number 7 to a whole negative number -14.

Decision.

We use the rule of multiplication of integers with different signs. Multiplier modules are equal, respectively, 7 and 14. Calculate the product of the modules: 7 · 14 \u003d 98. It remains before the number received to put a minus sign: -98. So, 7 · (-14) \u003d - 98.

Answer:

7 · (-14) \u003d - 98.

Example.

Calculate the product (-36) · 29.

Decision.

We need to calculate the product of integers with different signs. To do this, we calculate the product of absolute magnitudes of multipliers: 36 · 29 \u003d 1 044 (multiplication is better to spend in the column). Now put the minus sign in front of the number 1 044, we get -1 044.

Answer:

(-36) · 29 \u003d -1 044.

In conclusion of this paragraph, we prove the justice of equality a · (-b) \u003d - (a · b), where a and -b are arbitrary integers. A special case of this equality is a voiced rule of multiplying integers with different signs.

In other words, we need to prove that the values \u200b\u200bof the expressions A · (-b) and A · b are opposite numbers. To prove it, we will find the amount A · (-b) + A · B and make sure that it is zero. By virtue of the distribution properties of multiplying integers relative to the addition, equality A · (-b) + A · B \u003d A · ((- b) + b). The sum (-b) + B is zero as the sum of the opposite integers, then a · ((- b) + b) \u003d A · 0. The last work is zero by the property of multiplying an integer on zero. Thus, a · (-b) + a · b \u003d 0, therefore, A · (-b) and A · b are opposite numbers, where the equality A · (-B) \u003d - (A · b) follows. Similarly, it can be shown that (-a) · b \u003d - (a · b).

Rule of multiplication of negative integers, examples

To obtain a rule of multiplying two whole negative numbers will help us equality (-a) · (-b) \u003d a · b, which we now prove.

At the end of the previous paragraph, we showed that A · (-b) \u003d - (a · b) and (-a) · b \u003d - (a · b), so we can write down the next chain of equalities (-A) · (-b) \u003d - (a · (-b)) \u003d - (- (A · b)). And the resulting expression - (- (a · b)) is nothing more than A · B due to the definition of opposite numbers. So (-a) · (-b) \u003d a · b.

Proven equality (-a) · (-b) \u003d A · B allows you to formulate rule of multiplication of whole negative numbers: The product of two negative integers is equal to the product of the modules of these numbers.

From the voiced rule, it follows that the result of the multiplication of two whole negative numbers is an integer positive number.

Consider the application of this rule when performing multiplication of entire negative numbers.

Example.

Calculate the product (-34) · (-2).

Decision.

We need to multiply two negative integers -34 and -2. We use the relevant rule. For this we find multipliers modules: and. It remains to calculate the product of numbers 34 and 2, which we can do. Briefly all the solution can be written so (-34) · (-2) \u003d 34 · 2 \u003d 68.

Answer:

(-34) · (-2) \u003d 68.

Example.

Multiplying the integer negative number -1 041 to a whole negative number -538.

Decision.

According to the rule of multiplication of whole negative numbers, the desired work is equal to the product of multipliers modules. Multiplier modules are equal, respectively, 1 041 and 538. Perform a multiplication by the Stage:

Answer:

(-1 041) · (-538) \u003d 560 058.

Multiplying an integer per unit

Multiplication of any integer A per unit results in the number a. We have already mentioned this when we discussed the meaning of multiplying two integers. So A · 1 \u003d a. Due to the transmitting properties of multiplication, equality A · 1 \u003d 1 · a should be fair. Consequently, 1 · a \u003d a.

The above arguments lead us to the rule of multiplication of two integers, one of which is equal to one. The product of two integers in which one of the multipliers is the unit equal to another multiplier.

For example, 56 · 1 \u003d 56, 1 · 0 \u003d 0 and 1 · (-601) \u003d - 601. We give a couple more examples. The product of integers -53 and 1 is -53, and the result of multiplication of a unit and a negative integer -989 981 is the number -989 981.

Multiplying a whole number to zero

We agreed that the product of any whole number A to zero is zero, that is, A · 0 \u003d 0. The variety of multiplication makes us accept the equality 0 · a \u003d 0. In this way, the product of two integers in which at least one of the multipliers is zero, equal to zero. In particular, the result of multiplication of zero to zero is zero: 0 · 0 \u003d 0.

We give a few examples. The product of the integer positive number 803 and zero is zero; The result of the multiplication of zero to a whole negative number -51 is zero; Also (-90 733) · 0 \u003d 0.

We also note that the product of two integers then and only then is zero, when at least one of the multipliers is zero.

Checking the result of multiplication of integers

Check the result of multiplying two integers carried out by division. It is necessary to divide the resulting work on one of the factors if the number equal to another multiplier is obtained, then the multiplication was fulfilled. If a number is different from another complimentary, then an error was made somewhere.

Consider the examples in which the result of multiplying integers is checked.

Example.

As a result of multiplying two integers -5 and 21, the number -115 was obtained, the work is calculated correctly?

Decision.

Perform a check. To do this, we divide the calculated product -115 per factor, for example, on -5., Check the result. (-17) · (-67) \u003d 1 139.

Multiplication of three and more integers

The combination property of multiplication of integers allows us to definitely determine the product of three, four and more integers. At the same time, the remaining properties of multiplying integers make it possible to assert that the product of three and more integers does not depend on the method of arrangement of brackets and on the procedure for following the multipliers in the work. Similar statements we justified when they talked about the multiplication of the three and more natural numbers. In case of entire factors, the rationale is completely coincided.

Consider the solution of the example.

Example.

Calculate the product of five integers 5, -12, 1, -2 and 15.

Decision.

We can consistently from left to right to replace two adjacent factors by their work: 5 · (-12) · 1 · (-2) · 15 \u003d (-60) · 1 · (-2) · 15 \u003d (-60) · (-2 ) · 15 \u003d 120 · 15 \u003d 1 800. This option for calculating the work corresponds to the following method of laying brackets: (((5 · (-12)) · 1) · (-2)) · 15.

We could also rearrange some factor places and arrange brackets otherwise, if it makes it possible to calculate the product of these five integers more rationally. For example, it was possible to rearrange multipliers in the following order 1 · 5 · (-12) · (-2) · 15, after which the brackets arrange so ((1 · 5) · (-12)) · ((- 2) · 15). In this case, the calculations will be such: ((1 · 5) · (-12)) · ((- 2) · 15) \u003d (5 · (-12)) · ((- 2) · 15) \u003d (-60) · (-30) \u003d 1 800.

As you can see different variants The arrangements of the brackets and the different order of the factors led us to the same result.

Answer:

5 · (-12) · 1 · (-2) · 15 \u003d 1 800.

Separately, we note that if in the work of three, four, etc. In the integers at least one of the factors is zero, then the work is zero. For example, the product of four integers 5, -90 321, 0 and 111 is zero; The result of multiplying three integers 0, 0 and -1 983 is also zero. Reverse statement is also true: if the work is zero, then at least one of the multipliers is zero.

We will analyze the concept of multiplication by the example:

Tourists were on the way for three days. Every day they passed the same path of 4200 m. What distance did they go over for three days? Decide the task in two ways.

Decision:
Consider the task in detail.

On the first day, tourists passed 4200m. On-day day, the same path was the tourists of 4200m and on the third day - 4200m. We write the mathematical language:
4200 + 4200 + 4200 \u003d 12600m.
We see the pattern number 4200 repeats three times, therefore, you can replace the amount by multiplication:
4200⋅3 \u003d 12600m.
Answer: Tourists passed 12,600 meters for three days.

Consider an example:

To do not write a long entry, you can write it in the form of multiplication. The number 2 is repeated 11 times. Therefore, an example with multiplication will look like this:
2⋅11=22

Summarize. What is multiplication?

Multiplication- This is an action replacing the repetition of N times the term M.

Recording M⋅n and the result of this expression is called production of numbers, and the numbers M and N are called multipliers.

Consider what has been said on the example:
7⋅12=84
Expression 7⋅12 and the result 84 are called production of numbers.
Numbers 7 and 12 are called multipliers.

In mathematics there are several laws of multiplication. Consider them:

Movement law multiplication.

Consider the task:

We gave two apples to our friends. Mathematically recording will look like this: 2⋅5.
Or we gave 5 apples to our two friends. Mathematically recording will look like this: 5⋅2.
In the first and second case, we will distribute the same amount of apples equal to 10 pieces.

If we multiply 2⋅5 \u003d 10 and 5⋅2 \u003d 10, the result will not be changed.

Property of the Multiplication Movement:
From change places of multipliers, the work does not change.
m.⋅ n.\u003d N⋅.m.

The combination law of multiplication.

Consider on the example:

(2⋅3) ⋅4 \u003d 6⋅4 \u003d 24 or 2⋅ (3⋅4) \u003d 2⋅12 \u003d 24 get,
(2⋅3)⋅4=2⋅(3⋅4)
(a.⋅ b.) ⋅ c.= a.⋅(b.⋅ c.)

Property of a combination law of multiplication:
To multiply the number of two numbers, you can first multiply it to the first factor, and then the resulting product is multiplied to the second.

By changing several multipliers in places and entering them into brackets, the result or work will not change.

These laws are true for any natural numbers.

Multiplying any natural number per unit.

Consider an example:
7⋅1 \u003d 7 or 1⋅7 \u003d 7
a.⋅1 \u003d a or 1⋅a.= a.
When multiplying any natural number per unit, the work will always be the same.

Multiplying any natural number to zero.

6⋅0 \u003d 0 or 0⋅6 \u003d 0
a.⋅0 \u003d 0 or 0⋅a.=0
When multiplying any natural number to zero, the product will be zero.

Questions to the topic "Multiplication":

What is the number of numbers?
Answer: the number of numbers or multiplication of numbers is called the expression M⋅n, where M is the term, and n is the number of repetitions of this term.

Why do you need multiplication?
Answer: In order not to write a long addition of numbers, but to write abbreviated. For example, 3 + 3 + 3 + 3 + 3 + 3 \u003d 3⋅6 \u003d 18

What is the result of multiplication?
Answer: The value of the work.

What does the multiplication record 3⋅5 mean?
Answer: 3⋅5 \u003d 5 + 5 + 5 \u003d 3 + 3 + 3 + 3 + 3 \u003d 15

If you multiply a million to zero, what will be the work equal?
Answer: 0.

Example number 1:
Replace the amount of the work: a) 12 + 12 + 12 + 12 + 12 b) 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3
Answer: a) 12⋅5 \u003d 60 b) 3⋅9 \u003d 27

Example number 2:
Write down in the form of a work: a) a + a + a + a b) C + C + C + C + C + C + C
Decision:
a) a + a + a + a \u003d 4⋅a
b) C + C + C + C + C + C + C \u003d 7⋅С

Task number 1:
Mom bought 3 boxes of candy. In each box of 8 candies. How many candies bought mom?
Decision:
In one box of 8 candies, and we have such boxes of 3 pieces.
8 + 8 + 8 \u003d 8⋅3 \u003d 24 candy
Answer: 24 candies.

Task number 2:
Drawing teacher said to prepare her eight students for seven pencils on the lesson. How many pencils together were children?
Decision:
You can calculate the sum of the task. The first student had 7 pencils, the second student had 7 pencils, etc.
7+7+7+7+7+7+7+7=56
The recording turned out uncomfortable and long, replace the amount on the work.
7⋅8=56
Answer 56 pencils.

To solve many tasks "at maximum and minimum", i.e. On the location of the greatest and smallest variable values, you can successfully use some algebraic statements with which we will now meet.

x · y.

Consider the following task:

What two parts should be broken down by this number so that their work is the greatest?

Let this numberbut. Then parts on which the number is brokenbut, you can designate through

a / 2 + x and A / 2 - X;

number h. shows for which magnitude these parts differ from half of the number but. The work of both parts is equal

( a / 2 + x) · ( A / 2 - X) \u003d a 2/4 - x 2.

It is clear that the piece of parts taken will increase by decreasing h.. With a decrease in the difference between these parts. The greatest work will be with x \u003d.0, i.e. In the case when both parts are equal A / 2..

So,

the product of two numbers whose amount is unchanged will be the highest when these numbers are equal to each other.

x · y · z

Consider the same question for the three numbers.

What three parts need to break this number so that their work is the greatest?

When solving this task, we will rely on the previous one.

Let the number but broken into three parts. Suppose first that none of the parts are equal A / 3.. Then there is some among them, large A / 3. (all three can not be less A / 3.); Denote it through

a / 3 + x.

In the same way among them there is a part, less A / 3.; Denote it through

A / 3 - Y.

Numbers h. and w. positive. The third part will be obviously equal to

A / 3 + Y - X.

Numbers A / 3. and A / 3 + X - Y have the same amount as the first two parts of the number but, and the difference between them, i.e. x - Y., less than the difference between the first two parts, which was equal x + y.. As we know from the decision of the previous task, it follows that the work

A / 3. · ( A / 3 + X - Y)

more than the work of the first two parts of the number but.

So, if the first two parts of the number but replace numbers

A / 3. and A / 3 + X - Y,

and the third is not a change, the work will increase.

Let now one of the parts are already equal A / 3.. Then the other two are

A / 3 + Z and A / 3 - Z.

If we make these two parts with equal A / 3. (why the amount will not change), then the work will increase again and will become equal

A / 3 · A / 3 · A / 3 \u003d A 3/27 .

So,

if the number A is divided into 3 parts, not equal to each other, then the product of these parts is less than a 3/2 27, i.e. than a product of three equal in factors, in the amount of components a.

Similarly, you can prove this theorem for four multipliers, for five, etc.

x p · y q

Consider now a more general case.

Under what values \u200b\u200bx and y expression x P in q is the greatest, if x + y \u003d e?

It is necessary to find, with what value x expression

x r ·(a - H.) Q.

reaches the greatest value.

Multiply this expression on the number 1 / P q q q. We get a new expression

x p / p p · (a - X. ) Q / Q Q,

which obviously reaches the greatest value at the same time when initial.

Imagine the expression obtained now in the form

(a - X.) / Q · (a - X.) / Q · ... · (a - X.) / Q. ,

where the multipliers of the first type are repeated p. Once, and the second - q. time.

The sum of all factors of this expression is equal

X / P + X / P + ... + X / P + (a - X.) / Q +. (a - X.) / Q + ... + (a - X.) / Q. =

\u003d px / p + q ( A - X.) / q \u003d x + a - x \u003d a ,

those. The magnitude is constant.

Based on previously proven, we conclude that the work

x / p · x / p · ... · x / p · (a - X.) / Q · (a - X.) / Q · ... · (a - X.) / Q.

maxima reaches the equality of all of its individual factors, i.e. when

x / p \u003d (a - X.) / Q..

Knowing what a - x \u003d y, we get, rearring the members, proportion

X / Y \u003d P / Q.

So,

the product X P y Q is constantly the amount of X + y reaches the greatest value when

x: y \u003d p: q.

In the same way, you can prove that

work

x p y q z r, x p y q z r t u, etc.

with the constancy of sums x + y + z, x + Y + Z + T etc. achieve the greatest value when

x: y: z \u003d p: Q: R, x: y: z: t \u003d p: Q: R: U, etc.

Identical terms. For example, a 5 * 3 entry indicates "5 fold with you 3 times, that is, it is simply a brief record for 5 + 5 + 5. The result of multiplication is called work, and multiplying numbers - multipliers or in fact. There are also multiplication tables.

Record

Multiplication is indicated by an asterisk *, crossed or point. Entries

denote the same thing. Multiplication sign often misses if it does not lead to confusion. For example, instead they usually write.

If there are many factors, then some of them can be replaced with a lot. For example, the product of integers from 1 to 100 can be written as

The letter of the work is also applied in the letter record:

Multiplying natural numbers and its properties. Work (mathematics)

Terms and notation

The meaning of multiplying integers

Multiplying whole positive numbers

The rule of multiplication of integers with different signs, examples

Rule of multiplication of negative integers, examples

Multiplying an integer per unit

Multiplying a whole number to zero

Checking the result of multiplication of integers

Multiplication of three and more integers

Movement law multiplication.

The combination law of multiplication.

Multiplying any natural number per unit.

Multiplying any natural number to zero.

x · y.

x · y · z

x p · y q

Record

see also

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