What does the product of two numbers mean? Modulo difference and product of two numbers

The same terms. For example, the entry 5 * 3 means "add 5 to itself 3 times", that is, it is simply short note for 5+5+5. The result of multiplication is called work, and the multiplied numbers - multipliers or factors. There are also multiplication tables.

Recording

Multiplication is indicated by an asterisk * , a cross or a dot . Entries

mean the same thing. The multiplication sign is often omitted unless it causes confusion. For example, instead of usually write.

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

AT letter entry the product symbol is also used:

Terms and notation

To describe the multiplication of integers, we will use the same terms with which we described the multiplication natural numbers. Let's remind them.

The integers to be multiplied are called multipliers. The result of multiplication is called work. The operation of multiplication is denoted by the multiplication sign of the form "·". In some sources, you can find the designation of multiplication with the signs "*" or "×".

The multiplied integers a , b and the result of their multiplication c are conveniently written 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 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, note that the product of two integers is an integer.

Meaning of integer multiplication

Multiplication of positive integers

Positive integers are natural numbers, so multiplication of positive integers carried out according to all the rules of multiplication of natural numbers. It is clear that as a result of multiplying two positive integers, a positive integer (a natural number) will be obtained. Let's look at a couple of examples.

Example.

What is the product of the positive integers 127 and 5 ?

Solution.

We represent the first factor 107 as a sum of bit terms , that is, in the form 100+20+7 . After that, 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. It remains only to complete the calculation: 100 5+20 5+7 5= 500+100+35=600+35=635 .

So the product of the given positive integers 127 and 5 is 635.

Answer:

127 5=635 .

To multiply multivalued 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 integer positive numbers in 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.

We calculate the product of a negative integer −5 and an integer positive number 3 based on the meaning of multiplication. So (−5) 3=(−5)+(−5)+(−5)=−15. To preserve the validity of the commutative property of multiplication, the equality (−5)·3=3·(−5) must hold. That is, the product of 3·(−5) is also equal to −15 . It is easy to see that −15 is equal to the product modules of the original factors, whence it follows that the product of the original integers with different signs is equal to the product of the modules of the original factors, taken with a minus sign.

So we got multiplication rule for 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 voiced rule, we can conclude that the product of integers with different signs is always a negative integer. Indeed, as a result of multiplying the modules of factors, we get a positive integer, and if we put a minus sign in front of this number, then it will become a negative integer.

Consider examples of calculating the product of integers with different signs using the resulting rule.

Example.

Multiply a positive integer 7 by an integer a negative number −14 .

Solution.

Let's use the rule of multiplication of integers with different signs. The modules of the multipliers are 7 and 14 respectively. Let's calculate the product of modules: 7·14=98 . It remains 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 \u003d 1 044 (multiplication is best done 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 subsection, we prove the validity of the equality a·(−b)=−(a·b) , where a and −b are arbitrary integers. A special case of this equality is the voiced 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, we find the sum a (−b) + a b and verify that it is equal to zero. By virtue of the distributive property of multiplication of integers with respect to addition, the equality a·(−b)+a·b=a·((−b)+b) is true. The sum of (−b)+b is equal to zero as the sum of opposite integers, then a ((−b)+b)=a 0 . Last piece equals 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 equality a·(−b)=−(a·b) . Similarly, one 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 to obtain the rule for multiplying two negative integer numbers.

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 but a b by virtue of the definition opposite numbers. So, (−a)·(−b)=a·b .

The proved 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 voiced rule it follows that the result of multiplying two negative integers is a positive integer.

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 moduli of factors: and . It remains to calculate the product of the numbers 34 and 2, which we can do. Briefly, the whole 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 of multiplication of negative integers, the desired product is equal to the product of the modules of the factors. The multiplier modules are 1041 and 538 respectively. Let's do the multiplication by a column:

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 . By virtue of 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 result of multiplying 1 and the negative integer -989981 is -989981 .

Multiply 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 makes us accept the equality 0·a=0 . In this way, 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 some examples. The product of a positive integer 803 and zero is zero; the result of multiplying zero by a 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 zero.

Checking the result of multiplying integers

Checking the result of multiplying two integers done with 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 you get a number that is different from the other term, then somewhere a mistake was made.

Consider examples in which the result of multiplication of 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 do a check. To do this, we divide the calculated product -115 by one of the factors, for example, by -5., check the result. (−17)·(−67)=1 139 .

Multiplication of three or more integers

The associative 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 the multiplication of integers allow us to assert that the product of three or more integers does not depend on the way the brackets are arranged and on the order of the factors in the product. We substantiated similar statements when we talked about the multiplication of three or more natural numbers. In the case of integer factors, the justification is completely the same.

Let's consider an example solution.

Example.

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

Solution.

We can successively replace two adjacent factors from left to right by their product: 5 (−12) 1 (−2) 15= (−60) 1 (−2) 15= (−60) (−2 ) 15= 120 15=1 800 . This version of the calculation of the product corresponds to the following way of placing brackets: (((5 (−12)) 1) (−2)) 15.

We could also rearrange some of the factors and arrange the brackets differently, if this allows us to calculate the product of these five integers more rationally. For example, it was possible to rearrange the factors in the following order 1 5 (−12) (−2) 15 , 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 variants parentheses and different order the succession of factors led us to the same result.

Answer:

5 (−12) 1 (−2) 15=1 800.

Separately, we note that if in the product of three, four, etc. 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 , −90 321 , 0 and 111 is zero; the result of multiplying three integers 0 , 0 and −1 983 is also zero. The converse statement is also true: if the product is equal to zero, then at least one of the factors is equal to zero.

Let's analyze the concept of multiplication with an example:

The tourists were on the road for three days. Every day they walked the same path of 4200 m. How far did they walk in three days? Solve the problem in two ways.

Solution:
Let's consider the problem in detail.

On the first day the hikers covered 4200m. On the second day, the same path was covered by tourists 4200m and on the third day - 4200m. Let's write in mathematical language:
4200+4200+4200=12600m.
We see the pattern of the number 4200 repeating three times, therefore, we can replace the sum by multiplication:
4200⋅3=12600m.
Answer: tourists covered 12,600 meters in three days.

Consider an example:

In order not to write a long record, we can write it as a multiplication. The number 2 is repeated 11 times, so the multiplication example would look like this:
2⋅11=22

Summarize. What is multiplication?

Multiplication 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 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 laws of multiplication in mathematics. Consider them:

Commutative law of multiplication.

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, then the result will not change.

Property displacement law multiplications:
The product does not change from changing the places of factors.
m⋅ n=n⋅m

Associative 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 law of multiplication:
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.

Swapping multiple factors and putting them in parentheses does not change the result or product.

These laws are true for any natural numbers.

Multiplication of any natural number by one.

Consider an example:
7⋅1=7 or 1⋅7=7
a⋅1=a or 1⋅a= a
When multiplying any natural number by one, the product will always be the same number.

Multiplication of any natural number by zero.

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

Questions to the topic “Multiplication”:

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

What is multiplication for?
Answer: in order not to write a 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 the 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?
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 in the form of 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 sweets 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 per 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, and so on.
7+7+7+7+7+7+7+7=56
The record turned out to be inconvenient and long, we will replace the sum with the product.
7⋅8=56
The answer is 56 pencils.

- (product) The result of the multiplication. product of numbers, algebraic expressions, vectors or matrices; can be shown with a dot, a slash, or simply by writing them one after the other, i.e. f(x).g(y), f(x) x g(y), f(x)g(y)… … Economic dictionary

The science of whole numbers. The concept of an integer (See number), as well as arithmetic operations over numbers has been known since ancient times and is one of the first mathematical abstractions. Special place among integers, i.e. numbers ..., 3 ... Great Soviet Encyclopedia

Ex., s., use. often Morphology: (no) what? works for what? work, (see) what? work of what? work about what? about the work; pl. what? works, (no) what? works, why? works, (see) what? works, ... ... Dictionary Dmitrieva

Matrix mathematical object, written as a rectangular table of numbers (or ring elements) and allowing algebraic operations (addition, subtraction, multiplication, etc.) between it and other similar objects. Execution rules ... ... Wikipedia

In arithmetic, multiplication is understood as a short record of the sum of identical terms. For example, the notation 5*3 means "add 5 to itself 3 times," which is just a shorthand notation for 5+5+5. The result of multiplication is called a product, and ... ... Wikipedia

Section of number theory, the main task of which is to study the properties of integers of fields algebraic numbers finite degree over the field rational numbers. All integers of the extension field K of a field of degree n can be obtained using ... ... Mathematical Encyclopedia

Number theory, or higher arithmetic, is a branch of mathematics that studies integers and similar objects. In number theory in broad sense both algebraic and transcendental numbers are considered, as well as functions various origins which ... ... Wikipedia

Section of number theory, in which distribution patterns are studied prime numbers(p.h.) among natural numbers. Central is the problem of the best asymptotic. expressions for the function p(x), denoting the number of p.h., not exceeding x, but ... ... Mathematical Encyclopedia

- (in foreign literature scalar product, dot product, inner product) an operation on two vectors, the result of which is a number (scalar) that does not depend on the coordinate system and characterizes the lengths of the factor vectors and the angle between ... ... Wikipedia

A symmetric Hermitian form defined on a vector space L over a field K, usually considered as an integral part of the definition of this space, making a space (depending on the type of space and the properties of the internal ... Wikipedia

Books

Collection of problems in mathematics, V. Bachurin. The questions on mathematics considered in the book fully correspond to the content of any of the three programs: school, preparatory departments, entrance exams. Even though this book is called...
Living matter. Physics of Living and Evolutionary Processes, Yashin A.A. This monograph summarizes the author's research over the past few years. The experimental results presented in the book were obtained by Tulskaya scientific school field biophysics and…

Task 1.2
Two integers X and T are given. If they have different signs, then assign X the value of the product of these numbers, and T the value of their modulo difference. If the numbers have identical signs, then assign X the value of the difference modulo initial numbers, and T is the value of the product of these numbers. Display the new X and T values on the screen.

The task is also easy. “Misunderstandings” can arise only if you forgot what the modulo difference is (I hope that this is the product of two integers, you still remember))).

Difference modulo two numbers

The modulo difference of two integers (although not necessarily integers - it doesn't matter, it's just that the numbers are integers in our problem) - this, speaking in a simple way, when the result of the calculation is the modulus of the difference of two numbers.

That is, the operation of subtracting one number from another is first performed. And then the modulus of the result of this operation is calculated.

Mathematically, this can be written as:

If anyone has forgotten what a modulus is or how to calculate it in Pascal, then see.

Algorithm for determining the signs of two numbers

The solution to the problem is generally quite simple. Difficulty for beginners can only cause the definition of the signs of two numbers. That is, it is necessary to answer the question: how to find out whether the numbers have the same signs or different ones.

First, it begs the alternate comparison of numbers with zero. This is acceptable. But the source code will be quite large. Therefore, it is more correct to use the following algorithm:

Multiply numbers with each other
If the result less than zero, so the numbers have different signs
If the result is zero or greater than zero, then the numbers have the same signs

I performed this algorithm in the form of a separate . And the program itself turned out to be the same as shown in the Pascal and C++ examples below.

Solution of problem 1.2 in Pascal program checknums; var A, X, T: integer; //************************************************** **************** // Checks if the numbers N1 and N2 have the same signs. If yes, then // returns TRUE, otherwise - FALSE //**************************************** ************************** function ZnakNumbers(N1, N2: integer) : boolean; begin := (N1 * N2) >= 0; end; //************************************************** **************** // MAIN PROGRAM //**************************** ************************************ begin Write("X = "); ReadLn(X); Write("T = "); ReadLn(T); if ZnakNumbers(X, T) then //If the numbers have the same signs begin A:= (X - T); //Get the difference modulo the original numbers T:= X * T; end else //If numbers have different signs begin A:= X * T; T:= Abs(X - T); end; X:=A; //Write value A to X WriteLn("X = ", X); //Output X WriteLn("T = ", T); //Output T WriteLn("The end. Press ENTER..."); ReadLn; end.

Solution of problem 1.2 in C++#include #include using namespace std; int A, X, T; //************************************************** **************** // Checks if the numbers N1 and N2 have the same signs. If yes, then // returns TRUE, otherwise - FALSE //**************************************** ****************************** bool ZnakNumbers(int N1, int N2) ( return ((N1 * N2) >= 0); ) //**************************************************** ***************** // MAIN PROGRAM //****************************** ************************************* int main(int argc, char *argv) ( cout > X; cout > T; if (ZnakNumbers(X, T)) //If the numbers have the same signs ( A = abs(X - T); //Get the difference modulo the original numbers T = X * T; ) else // If the numbers have different signs ( A = X * T; T = abs(X - T); ) X = A; // Write the value A cout to X

Optimization

This a simple program can be further simplified if you do not use the function and slightly alter the source code of the program. Wherein total lines source code will shrink a little. How to do it - think for yourself.

Portal for the student. Self-training

What does the product of two numbers mean? Modulo difference and product of two numbers

Recording

see also

See what "Product (mathematics)" is in other dictionaries:

Books

Terms and notation