In this article we will figure out how to do it multiplying integers. First, let's introduce terms and notations, and also find out the meaning of multiplying two integers. After this we will obtain the rules for multiplying two positive integers, negative integers and integers with different signs. At the same time, we will give examples with a detailed explanation of the solution process. We will also touch on cases of multiplication of integers when one of the factors is equal to one or zero. Next we will learn how to check the resulting multiplication result. And finally, let's talk about multiplying three, four and more integers.
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Terms and symbols
To describe multiplication of integers, we will use the same terms with which we described multiplication natural numbers. Let's remind them.
The integers that are multiplied are called multipliers. The result of multiplication is called work. The multiplication action is indicated by the multiply sign of the form “·”. In some sources you can find the symbol “*” or “×” for multiplication.
It is convenient to write the multiplied integers a, b and the result of their multiplication c using an equality of the form a·b=c. In this notation, integer a is the first factor, integer b is the second factor, and integer c is the product. of the form a·b will also be called a product, as well as the value of this expression c .
Looking ahead, we note that the product of two integers represents an integer.
The meaning of multiplying integers
Multiplying positive integers
Positive integers are natural numbers, so multiplying positive integers is carried out according to all the rules for multiplying natural numbers. It is clear that multiplying two positive integers results in a positive integer (natural number). Let's look at a couple of examples.
Example.
What is the product of the positive integers 127 and 5?
Solution.
Let's present the first factor 107 as a sum of bit terms, that is, in the form 100+20+7. After this, we use the rule for multiplying the sum of numbers by a given number: 127·5=(100+20+7)·5=100·5+20·5+7·5. All that remains is to complete the calculation: 100·5+20·5+7·5= 500+100+35=600+35=635.
Thus, the product of the given positive integers 127 and 5 is 635.
Answer:
127·5=635.
To multiply multi-digit positive integers, it is convenient to use the column multiplication method.
Example.
Multiply the three-digit positive integer 712 by the two-digit positive integer 92.
Solution.
Let's multiply these positive integers into a column: 
Answer:
712·92=65,504.
Rule for multiplying integers with different signs, examples
The following example will help us formulate the rule for multiplying integers with different signs.
Let's calculate the product of the negative integer −5 and the integer positive number 3 based on the meaning of multiplication. So (−5)·3=(−5)+(−5)+(−5)=−15. In order for the commutative property of multiplication to remain valid, the equality (−5)·3=3·(−5) must be satisfied. That is, the product 3·(−5) is also equal to −15. It is easy to see that −15 equal to the product moduli of the original factors, from which it follows that the product of the original integers with different signs is equal to the product of the moduli of the original factors, taken with a minus sign.
So we got rule for multiplying 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 in front of the resulting number.
From the stated rule we can conclude that the product of integers with different signs is always a negative integer. Indeed, as a result of multiplying the moduli of the factors, we get a positive integer, and if we put a minus sign in front of this number, then it becomes a negative integer.
Let's look at examples of calculating the product of integers with different signs using the resulting rule.
Example.
Multiply the positive integer 7 by an integer negative number −14 .
Solution.
Let's use the rule for multiplying integers with different signs. The moduli of the multipliers are 7 and 14, respectively. Let's calculate the product of the modules: 7·14=98. All that remains is to put a minus sign in front of the resulting number: −98. So, 7·(−14)=−98.
Answer:
7·(−14)=−98 .
Example.
Calculate the product (−36)·29.
Solution.
We need to calculate the product of integers with different signs. To do this, we calculate the product absolute values multipliers: 36·29=1,044 (it is better to multiply in a column). Now we put a minus sign in front of the number 1044, we get −1044.
Answer:
(−36)·29=−1,044 .
To conclude this section, we will prove the equality a·(−b)=−(a·b) , where a and −b are arbitrary integers. A special case of this equality is the stated rule for multiplying integers with different signs.
In other words, we need to prove that the values of the expressions a·(−b) and a·b are opposite numbers. To prove this, let's find the sum a·(−b)+a·b and make sure that it is equal to zero. Due to the distributive property of multiplication of integers relative to addition, the equality a·(−b)+a·b=a·((−b)+b) is true. The sum (−b)+b is equal to zero as the sum of opposite integers, then a·((−b)+b)=a·0. Last piece is equal to zero by the property of multiplying an integer by zero. Thus, a·(−b)+a·b=0, therefore, a·(−b) and a·b are opposite numbers, which implies the validity of the equality a·(−b)=−(a·b) . Similarly, we can show that (−a) b=−(a b) .
Rule for multiplying negative integers, examples
The equality (−a)·(−b)=a·b, which we will now prove, will help us obtain the rule for multiplying two negative integers.
At the end of the previous paragraph, we showed that a·(−b)=−(a·b) and (−a)·b=−(a·b) , so we can write the following chain of equalities (−a)·(−b)=−(a·(−b))=−(−(a·b)). And the resulting expression −(−(a·b)) is nothing more than a·b due to the definition of opposite numbers. So, (−a)·(−b)=a·b.
The proven equality (−a)·(−b)=a·b allows us to formulate rule for multiplying negative integers: The product of two negative integers is equal to the product of the moduli of these numbers.
From the stated rule it follows that the result of multiplying two negative integers is a positive integer.
Let's consider the application of this rule when performing multiplication of negative integers.
Example.
Calculate the product (−34)·(−2) .
Solution.
We need to multiply two negative integers −34 and −2. Let's use the corresponding rule. To do this, we find the modules of the multipliers: and . It remains to calculate the product of the numbers 34 and 2, which we know how to do. Briefly, the entire solution can be written as (−34)·(−2)=34·2=68.
Answer:
(−34)·(−2)=68 .
Example.
Multiply the negative integer −1041 by the negative integer −538.
Solution.
According to the rule for multiplying negative integers, the desired product is equal to the product of the moduli of the factors. The moduli of the multipliers are 1,041 and 538, respectively. Let's do column multiplication: 
Answer:
(−1,041)·(−538)=560,058 .
Multiplying an integer by one
Multiplying any integer a by one results in the number a. We already mentioned this when we discussed the meaning of multiplying two integers. So a·1=a . Due to the commutative property of multiplication, the equality a·1=1·a must be true. Therefore, 1·a=a.
The above reasoning leads us to the rule for multiplying two integers, one of which is equal to one. The product of two integers in which one of the factors is one is equal to the other factor.
For example, 56·1=56, 1·0=0 and 1·(−601)=−601. Let's give a couple more examples. The product of the integers −53 and 1 is −53, and the product of one and the negative integer −989,981 is −989,981.
Multiplying an integer by zero
We agreed that the product of any integer a and zero is equal to zero, that is, a·0=0. The commutative property of multiplication forces us to accept the equality 0·a=0. Thus, the product of two integers in which at least one of the factors is zero is equal to zero. In particular, the result of multiplying zero by zero is zero: 0·0=0.
Let's give a few examples. The product of the positive integer 803 and zero is equal to zero; the result of multiplying zero by the negative integer −51 is zero; also (−90 733)·0=0 .
Note also that the product of two integers is equal to zero if and only if at least one of the factors is equal to zero.
Checking the result of multiplying integers
Checking the result of multiplying two integers carried out using division. It is necessary to divide the resulting product by one of the factors; if this results in a number equal to the other factor, then the multiplication was performed correctly. If the result is a number different from the other term, then a mistake was made somewhere.
Let's look at 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. Is the product calculated correctly?
Solution.
Let's check. To do this, divide the calculated product −115 by one of the factors, for example, −5., check the result. (−17)·(−67)=1 139 .
Multiplying three or more integers
The combinatory property of multiplication of integers allows us to uniquely determine the product of three, four, or more integers. At the same time, the remaining properties of multiplication of integers allow us to assert that the product of three or more integers does not depend on the method of placing parentheses and on the order of the factors in the product. We substantiated similar statements when we talked about multiplying three or more natural numbers. In the case of integer factors, the rationale is completely the same.
Let's look at the example solution.
Example.
Calculate the product of five integers 5, −12, 1, −2 and 15.
Solution.
We can sequentially from left to right replace two adjacent factors with their product: 5·(−12)·1·(−2)·15= (−60)·1·(−2)·15= (−60)·(−2 )·15= 120·15=1,800. This option for calculating the product corresponds to the following method of arranging brackets: (((5·(−12))·1)·(−2))·15.
We could also rearrange some factors and arrange the parentheses differently if this allows us to calculate the product of the given five integers more efficiently. For example, it was possible to rearrange the factors in the following order 1·5·(−12)·(−2)·15, and then arrange the brackets like this ((1·5)·(−12))·((−2)·15). In this case, the calculations will be as follows: ((1·5)·(−12))·((−2)·15)=(5·(−12))·((−2)·15)= (−60)·(−30)=1 800 .
As you can see, different options placement of brackets and different order succession of factors led us to the same result.
Answer:
5·(−12)·1·(−2)·15=1 800.
Separately, we note that if in a product there are three, four, etc. of integers, at least one of the factors is equal to zero, then the product is equal to zero. For example, the product of four integers 5, −90321, 0 and 111 is equal to zero; The result of multiplying three integers 0, 0 and −1983 is also zero. Fair and converse statement: if the product is equal to zero, then at least one of the factors is equal to zero.
Let's look at the concept of multiplication using an example:
The tourists were on the road for three days. Every day they walked the same path of 4200 m. How much distance did they cover in three days? Solve the problem in two ways.
Solution:
Let's consider the problem in detail.
On the first day, tourists walked 4200m. On the second day, tourists walked the same path 4200m and on the third day – 4200m. Let's write it in mathematical language:
4200+4200+4200=12600m.
We see a pattern in which the number 4200 is repeated three times, therefore, the sum can be replaced by multiplication:
4200⋅3=12600m.
Answer: tourists walked 12,600 meters in three days.
Let's look at an example:
To avoid writing a long entry, we can write it in the form of multiplication. The number 2 is repeated 11 times, so an example with multiplication would look like this:
2⋅11=22
Let's summarize. What is multiplication?
Multiplication– this is an action that replaces the repetition of the term m n times.
The notation m⋅n and the result of this expression are called product of numbers, and the numbers m and n are called multipliers.
Let's look at this with an example:
7⋅12=84
The expression 7⋅12 and the result 84 are called product of numbers.
The numbers 7 and 12 are called multipliers.
There are several multiplication laws in mathematics. Let's look at them:
Commutative law of multiplication.
Let's consider the problem:
We gave two apples to 5 of our friends. Mathematically, the entry will look like this: 2⋅5.
Or we gave 5 apples to two of our friends. Mathematically, the entry will look like this: 5⋅2.
In the first and second cases, we will distribute the same number of apples equal to 10 pieces.
If we multiply 2⋅5=10 and 5⋅2=10, the result will not change.
Property of the commutative multiplication law:
Changing the places of the factors does not change the product.
m⋅
n=n⋅m
Combinative law of multiplication.
Let's look at an example:
(2⋅3)⋅4=6⋅4=24 or 2⋅(3⋅4)=2⋅12=24 we get,
(2⋅3)⋅4=2⋅(3⋅4)
(a⋅
b) ⋅
c=
a⋅(b⋅
c)
Property of the associative multiplication law:
To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second.
By swapping multiple factors and putting them in parentheses, the result or product will not change.
These laws are true for any natural numbers.
Multiplying any natural number by one.
Let's look at an example:
7⋅1=7 or 1⋅7=7
a⋅1=a or 1⋅a=
a
When any natural number is multiplied by one, the product will always be the same number.
Multiplying any natural number by zero.
6⋅0=0 or 0⋅6=0
a⋅0=0 or 0⋅a=0
When any natural number is multiplied by zero, the product will equal zero.
Questions for the topic “Multiplication”:
What is a product of numbers?
Answer: the product of numbers or the multiplication of numbers is the expression m⋅n, where m is a term, and n is the number of repetitions of this term.
What is multiplication used for?
Answer: in order not to write long addition of numbers, but to write abbreviated. For example, 3+3+3+3+3+3=3⋅6=18
What is the result of multiplication?
Answer: the meaning of the work.
What does multiplication 3⋅5 mean?
Answer: 3⋅5=5+5+5=3+3+3+3+3=15
If you multiply a million by zero, what is the product equal to?
Answer: 0
Example #1:
Replace the sum with the product: a) 12+12+12+12+12 b)3+3+3+3+3+3+3+3+3
Answer: a) 12⋅5=60 b) 3⋅9=27
Example #2:
Write it down as a product: a) a+a+a+a b) c+c+c+c+c+c+c
Solution:
a)a+a+a+a=4⋅a
b) s+s+s+s+s+s+s=7⋅s
Task #1:
Mom bought 3 boxes of chocolates. Each box contains 8 candies. How many candies did mom buy?
Solution:
There are 8 candies in one box, and we have 3 such boxes.
8+8+8=8⋅3=24 candies
Answer: 24 candies.
Task #2:
The art teacher told her eight students to prepare seven pencils for each lesson. How many pencils did the children have in total?
Solution:
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 to be inconvenient and long, let’s replace the sum with the product.
7⋅8=56
The answer is 56 pencils.
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A sum is the result of addition, and the word can refer not only to numbers.
The difference is what is obtained after subtracting numbers.
Product is what is obtained after multiplication; the word has another meaning.
The quotient is what is obtained after division.
I. Mathematical concepts SUM, DIFFERENCE, PRODUCT, QUARTER are interrelated with mathematical terms ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION.
All definitions are given here on the set of natural numbers.
Each pair of numbers is assigned a number called their AMOUNT.
The sum consists of as many units as there are in numbers (commands) from a given pair.
SUM is the result of adding the number terms.
Subtraction is the inverse operation of addition. It consists of finding one of the terms from the sum and the other term. This amount is called the minuend, this term is called the subtrahend, and the required term is called BY DIFFERENCE.
DIFFERENCE- this is the number that is the result of subtraction, the remainder of the subtraction.
Each pair of numbers can be associated with a number that consists of as many units as are contained in the first number from the pair, taken as many times as there are units contained in the second number from the pair. This number corresponding in this way to a pair of numbers (they are called factors) is called WORK.
WORK is the result of multiplication.
Division is the inverse operation of multiplication.
Division is finding one of the factors from the product and the other factor. This product is called divisible, this factor is called a divisor, and the required factor is PRIVATE, that is, the number obtained by dividing one number by another.
II. OTHER MEANINGS OF THE WORDS SUM, DIFFERENCE, PRODUCT, QUARTER.
All words used as mathematical concepts can have other lexical meanings.
SUM figuratively means a collection, total quantity anything.
For example. The professionalism of a teacher lies in the amount of knowledge, skills and abilities that he transfers to his students. The lack of the required amount of money forced me to abandon the purchase.
DIFFERENCE has the meaning of difference, dissimilarity, difference in something.
For example. Differences in interests are much worse than differences in age. Friendship can begin with the idea of a commonality of views, and enmity can begin with a difference of views.
WORK means something produced in the process of labor, the creation of something, a product of labor, creativity, art, etc.
For example. High work of art makes a person think about his life. At a competition for young pianists, the boy played a piece by P.I. Tchaikovsky. This box is a real work of art.
PRIVATE- this is something personal, personal, belonging only to one person, it is his property, his and only his property. And be it personal thoughts, be it property or something else, but it belongs only to him, a private person.
For example. A friend gave me a notebook with the inscription Private. Is it good to contrast the private with the public?
In fact, all four words in the question, namely sum, difference, product and quotient, reflect four basic mathematical operations, which are the basics. It is with learning these actions that the fascinating journey into the world of mathematics begins. Thus,
Sum, difference, product, quotient - this is the result of mathematical operations with which we all began our acquaintance with mathematics. In life we also use these words, but we put more mathematical meaning into them, although we can’t add numbers. A work can also be artistic. This is a completely different meaning of the word that we use in life.
All four of these terms are used primarily in mathematics.
A sum is when two numbers are added together;
Difference is the subtraction of one number from another;
A quotient is the division of one number by another;
A product is the multiplication of one number by another.
The quotient is the result of dividing numbers, the product is the result of multiplying numbers, the sum is the result of adding numbers, the difference is the result of subtraction. These are basic mathematical operations that can be done with numbers.
These are mathematical concepts.
The sum is the result of addition. Numbers that are added are called the first addend and the second addend. It is indicated by the following sign: +.
The difference is the result of the subtraction. The numbers that are subtracted are called the minuend (the one that is greater) and the subtrahend (the one that is less). Indicated by the following sign: -.
A product is the result of multiplication. The numbers that are multiplied are called the first factor and the second factor. Indicated by the following sign: *.
The quotient is the result of division. The numbers that divide are called the dividend (the one that is greater), the divisor (the one that is less). Indicated by this sign: :.
All these concepts are taught in elementary school.
There are four in mathematics simple operations, which can be applied to two numbers and get the following results:
the sum is the result of adding numbers,
the difference is the result of subtracting one number from another,
product is the result of multiplying numbers,
the quotient is already the result of dividing numbers.
In mathematics, a sum is a number that is obtained by adding one number to another. Difference is the opposite number of addition, it is when it is subtracted from more less. A product is a number that results from multiplying one number by another. The difference is the opposite number to the product. We get the difference like this: divide one number by another.
I am a mathematician by training, specialty: mathematics teacher. She worked all her life as a mathematics teacher at a pedagogical university.
It is necessary to make a reservation. In the future we will talk about the sum, difference, product, quotient numbers.
The answers to these questions, although simple, cause difficulties for students. To be able to consider this general topic in more detail, I bring to your attention useful material on her. The note is called Mathematics for Blondes.
I liked the study method.
A provocative question is asked:
Is difference divided or multiplied?
They are trying to interest (not a single proposed version is correct!)))
Then they answer:
The difference is to take away. The result of the subtraction is called the difference.
Similarly you get:
The sum is to add up. The result of addition is called the sum.
Product is multiplication. The result of multiplication is called a product.
The quotient is a division. The result of division is called the quotient.
So in simple language The correct concepts of sum, difference, product and quotient in mathematics are explained. Only phrases are written a little simplified: difference is to subtract, sum is to add, product is to multiply, quotient is to divide. To be precise, they don't say so.
So, the result of adding numbers(terms) - these are theirs sum, result of subtracting numbers(minuend and subtrahend) - this is difference, the result of multiplying numbers(factors) is work, A the result of dividing numbers(divisible by the divisor), and the divisor must not be equal to zero, otherwise the division cannot be performed, there is private these numbers.
I don’t think about other meanings of these words; mathematics overshadows everything.)))
The words Sum, Difference, Product and Partial are very familiar to students of schools and other educational institutions teach them these definitions in every math lesson.
1) Sum
A sum is the result obtained after adding (+) two or more numbers.
The amount is also the final cost of the product (the amount to be paid), the total body of knowledge, impressions and much more.
2) Difference
In mathematics, it means the result of subtracting a number (-).
The word difference can also be used as a word for the difference between something. For example, difference of opinions, difference of views, difference of indicators, etc.
3) Work
The product is the result obtained after multiplying numbers (*).
In addition to mathematics, this word is also used to denote the result creative process(work of art), as a verb of produce.
4) Honest
This word denotes the result of dividing two numbers (:).
We can also hear the word private when denoting the ownership of something by one owner (private person, private property, private business).
If a concert hall is illuminated by 3 chandeliers with 25 bulbs each, then the total number of bulbs in these chandeliers will be 25 + 25 + 25, that is, 75.
The sum in which all terms are equal to each other is written shorter: instead of 25 + 25 + 25, write 25 3. This means 25 3 = 75 (Fig. 43). The number 75 is called work numbers 25 and 3, and the numbers 25 and 3 are called multipliers.
Rice. 43. Product of numbers 25 and 3
Multiplying the number m by the natural number n means finding the sum of n terms, each of which is equal to m.
The expression m n and the value of this expression are called work numbersmAndn. The numbers that are multiplied are called multipliers. Those. m and n are factors.
The products 7 4 and 4 7 are equal to the same number 28 (Fig. 44).
Rice. 44. Product 7 4 = 4 7
1. The product of two numbers does not change when the factors are rearranged.
commutative
a × b = b × a .
The products (5 3) 2 = 15 2 and 5 (3 2) = 5 6 have the same value 30. This means 5 (3 2) = (5 3) 2 (Fig. 45).
Rice. 45. Product (5 3) 2 = 5 (3 2)
2. To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.
This property of multiplication is called associative. Using letters it is written like this:
A (bc) = (abWith).
The sum of n terms, each equal to 1, is equal to n. Therefore the equality 1 n = n is true.
The sum of n terms, each of which is equal to zero, is equal to zero. Therefore, the equality 0 n = 0 is true.
In order for the commutative property of multiplication to be true for n = 1 and n = 0, it is agreed that m 1 = m and m 0 = 0.
The multiplication sign is usually not written before alphabetic factors: instead of 8 X write 8 X, instead of Ab write Ab.
The multiplication sign is also omitted before the parentheses. For example, instead of 2 ( a +b) write 2 (a+b) , and instead of ( X+ 2) (y + 3) write (x + 2) (y + 3).
Instead of ( ab) with write abc.
When there are no parentheses in the product notation, the multiplication is performed in order from left to right.
The works are read, naming each factor in genitive case. For example:
1) 175 60 is the product of one hundred seventy-five and sixty;
2) 80 (X+ 1 7) – product of r.p. r.p.
eighty and the sum of x and seventeen
Let's solve the problem.
How many three-digit numbers (Fig. 46) can be made from the numbers 2, 4, 6, 8, if the numbers in the number are not repeated?
Solution.
The first digit of a number can be any of four given numbers, the second – any of three others, and the third – any of two the remaining ones. It turns out:
Rice. 46. To the problem of composing three-digit numbers
In total, from these numbers you can make 4 3 2 = 24 three-digit numbers.
Let's solve the problem.
The company's board consists of 5 people. From among its members, the board must elect a president and vice president. In how many ways can this be done?
Solution.
One of 5 people can be elected president of the company:
President:
After the president is elected, any of the four remaining board members can be chosen as vice president (Fig. 47):
|
Rice. 47. On the election problem
This means that there are five ways to select a president, and for each elected president, there are four ways to select a vice president. Hence, total number The number of ways to choose the president and vice president of the company is: 5 4 = 20 (see Fig. 47).
Let's solve another problem.
There are four roads leading from the village of Anikeevo to the village of Bolshovo, and three roads from the village of Bolshovo to the village of Vinogradovo (Fig. 48). In how many ways can you get from Anikeev to Vinogradovo through the village of Bolshevo?
Rice. 48. To the problem of roads
Solution.
If you get from A to B along the 1st road, then there are three ways to continue the journey (Fig. 49).
Rice. 49. Path options
Reasoning in the same way, we get three ways to continue the journey, starting to get along the 2nd, 3rd, and 4th roads. This means that in total there are 4 3 = 12 ways to get from Anikeev to Vinogradov.
Let's solve one more problem.
A family consisting of a grandmother, father, mother, daughter and son was given 5 different cups. In how many ways can cups be divided among family members?
Solution. The first family member (for example, grandmother) has 5 choices, the next (let it be dad) has 4 choices. The next one (for example, mom) will choose from 3 cups, the next one from two, and the last one gets one remaining cup. Let's show these methods in the diagram (Fig. 50).
Rice. 50. Scheme for solving the problem
We found that each choice of cup by the grandmother corresponds to four possible choices dads, i.e. only 5 4 ways. After dad has chosen a cup, mom has three choices, daughter has two, son has one, i.e. only 3 2 1 ways. Finally, we find that to solve the problem we need to find the product 5 4 3 2 1.
Note that we have obtained the product of all natural numbers from 1 to 5. Such products are written more briefly:
5 4 3 2 1 = 5! (read: “five factorial”).
Factorial of a number– the product of all natural numbers from 1 to this number.
So, the answer to the problem is: 5! = 120, i.e. Cups can be distributed among family members in one hundred and twenty ways.
