- (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 on numbers, has been known since ancient times and is one of the first mathematical abstractions. A 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 of Dmitriev
A matrix is a 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
A branch of number theory, the main task of which is to study the properties of integers of fields of algebraic numbers of 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 a broad sense, both algebraic and transcendental numbers are considered, as well as functions of 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 the Tula Scientific School of Biophysics of Fields and…
The same 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 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
In literal notation, the product symbol is also used: 
see also
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See what "Product (mathematics)" is in other dictionaries:
- (mathematics) the result of multiplication. Piece of art. Musical composition. Audiovisual work. Service work ... Wikipedia
The product of two or more objects is a generalization in category theory of such concepts as the Cartesian product of sets, the direct product of groups, and the product of topological spaces. The product of a family of objects is in ... ... Wikipedia
The Kronecker product is a binary operation on matrices of arbitrary size, denoted. The result is a block matrix. The Kronecker product should not be confused with ordinary matrix multiplication. The operation is named after the German ... ... Wikipedia
History of science By subject Mathematics Natural sciences ... Wikipedia
I. Definition of the subject of mathematics, connection with other sciences and technology. Mathematics (Greek mathematike, from máthema knowledge, science), the science of quantitative relations and spatial forms of the real world. "Pure... Great Soviet Encyclopedia
Category theory is a branch of mathematics that studies the properties of relations between mathematical objects that do not depend on the internal structure of objects. Some mathematicians [who?] consider category theory too abstract and unsuitable for ... ... Wikipedia
Vector This term has other meanings, see Vector ... Wikipedia
This term has other meanings, see function. "Display" request is redirected here; see also other meanings ... Wikipedia
This term has other meanings, see Operation. A mapping operation that associates one or more set elements (arguments) with another element (value). The term "operation" is usually applied to ... ... Wikipedia
This term has other meanings, see Rotor. Rotor, or vortex is a vector differential operator over a vector field. It is designated (in Russian-language literature) or (in English-language literature), as well as vector multiplication ... Wikipedia
Books
- A set of tables. Maths. 4th grade. 8 tables + methodology, . Educational album of 8 sheets (format 68 x 98 cm): - Doli. - Multiplication and division of a number by a product. - Addition and subtraction of values. - Multiplication and division of values. - Written multiplication by...
- Kirik Novgorodets - a Russian scientist of the 12th century in Russian book culture, Simonov R.A. …
To solve many problems "to the maximum and minimum", i.e. to find the largest and smallest values of a variable, you can successfully use some of the algebraic statements that we will now get acquainted with.
x y
Consider the following problem:
Into what two parts must a given number be divided so that their product is the largest?
Let the given numbera. Then the parts into which the number is divideda, can be denoted by
a / 2 + x and a / 2 - x;
number X shows how much these parts differ from half the number a. The product of both parts is
(a / 2 + x) · ( a / 2 - x) = a 2 / 4 - x 2.
It is clear that the product of the taken parts will increase with decreasing X, i.e. while decreasing the difference between these parts. The greatest product will be x= 0 , i.e. when both sides are equal a/2.
So,
The product of two numbers, the sum of which is unchanged, will be the largest when these numbers are equal to each other.
x y z
Consider the same question for three numbers.
What three parts must be divided into a given number so that their product is the largest?
In solving this problem, we will rely on the previous one.
Let the number a broken into three parts. Assume first that none of the parts are equal a/3.Then among them there is a part, a large a/3(all three cannot be less than a/3); let's denote it by
a / 3 + x.
In the same way, among them there is a part, a smaller a/3; let's denote it by
a / 3 - y.
Numbers X and at are positive. The third part will obviously be equal to
a / 3 + y - x.
Numbers a/3 and a / 3 + x - y have the same sum as the first two parts of the number a, and the difference between them, i.e. x - y, less than the difference between the first two parts, which was equal to x + y. As we know from the solution of the previous problem, it follows that the product
a/3 · ( a / 3 + x - y)
greater than the product of the first two parts of the number a.
So if the first two parts of a number a replace with numbers
a/3 and a / 3 + x - y,
and leave the third unchanged, then the product will increase.
Let now one of the parts is already equal a/3. Then the other two look like
a / 3 + z and a / 3 - z.
If we make these two last parts equal a/3 (why their sum does not change), then the product will increase again and become equal to
a / 3 a / 3 a / 3 = a 3 / 27 .
So,
if the number a is divided into 3 parts that are not equal to each other, then the product of these parts is less than a 3 / 27, i.e. than the product of three equal factors that add up to a.
In a similar way, this theorem can be proved for four factors, for five, and so on.
xp yq
Let us now consider a more general case.
For what values of x and y is the expression x p y q the largest if x + y = a?
We need to find at what value of x the expression
x r(a - x) q
reaches its maximum value.
Multiply this expression by the number 1 / р p q q. Get a new expression
x p / p p · (a-x ) q / q q,
which, obviously, reaches its maximum value at the same time as the original one.
Let us represent the expression obtained now in the form
(a-x) / q (a-x) /q · ... · (a-x) / q ,
where factors of the first kind are repeated p once, and the second q once.
The sum of all factors of this expression is equal to
x/p+x/p+...+x/p+ (a-x) /q+ (a-x) /q + ... + (a-x) / q =
= px / p + q (a-x) / q = x + a - x = a ,
those. the value of a constant.
On the basis of what was proved earlier, we conclude that the product
x / p x / p ... x / p (a-x) / q (a-x) /q · ... · (a-x) / q
reaches a maximum when all its individual factors are equal, i.e. when
x/p= (a-x) / q.
Knowing that a - x = y, we obtain, by rearranging the terms, the proportion
x / y = p / q.
So,
the product x p y q at a constant sum x + y reaches its maximum value when
x: y = p: q .
In the same way, one can prove that
works
x p y q z r , x p y q z r t u etc.
with constant sums x+y+z, x + y + z + t etc. reach their maximum when
x:y:z=p:q:r,x: y: z: t = p: q: r: u, etc.
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 the same signs, then assign X the value of the difference modulo the original numbers, and T 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 is less than zero, then 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 sign. 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. This will reduce the total number of lines of source code a little. How to do it - think for yourself.
If the 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, i.e. 75.
The sum in which all the terms are equal to each other is written shorter: instead of 25 + 25 + 25 they write 25 3. Hence, 25 3 \u003d 75 (Fig. 43). The number 75 is called work numbers 25 and 3, and the numbers 25 and 3 are called multipliers.
Rice. 43. The product of the numbers 25 and 3
To multiply a number m by a natural number n means to find 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. Numbers that multiply are called multipliers. Those. m and n are factors.
The products of 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.
displaceable
a × b = b × a .
The products (5 3) 2 \u003d 15 2 and 5 (3 2) \u003d 5 6 have the same value 30. Hence, 5 (3 2) \u003d (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 multiplication property is called associative. It is written in letters like this:
a (bc) = (abWith).
The sum of n terms, each of which is 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.
For the commutative property of multiplication to be true for n = 1 and n = 0, we agreed that m 1 = m and m 0 = 0.
Before alphabetic factors, they usually do not write the multiplication sign: instead of 8 X write 8 X, instead of ab write ab.
Omit the multiplication sign before the brackets. 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 brackets in the product notation, the multiplication is performed in order from left to right.
The works are read, calling each factor in genitive case. For example:
1) 175 60 - the product of one hundred seventy-five and sixty;
2) 80 (X+ 1 7) is the 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 entry do not repeat?
Solution.
The first digit of the number can be any of four given digits, the second - any of three others, and the third - any of two the rest. It turns out:
Rice. 46. On the problem of compiling three-digit numbers
In total, from these numbers, you can make 4 3 2 = 24 three-digit numbers.
Let's solve the problem.
The board of the company consists of 5 people. The board must elect a president and a vice president from among its members. In how many ways can this be done?
Solution.
One of 5 people can be elected as the President of the company:
The 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 problem of elections
So there are five ways to choose a president, and for each elected president, there are four ways to choose a vice president. Consequently, total number ways to choose the president and vice president of the company is: 5 4 \u003d 20 (see Fig. 47).
Let's solve another problem.
Four roads lead from the village of Anikeevo to the village of Bolshovo, and three roads lead from the village of Bolshovo to the village of Vinogradovo (Fig. 48). In how many ways can you get from Anikeevo to Vinogradovo through the village of Bolshovo?
Rice. 48. On the problem of roads
Solution.
If you get from A to B along the 1st road, then there are three ways to continue the path (Fig. 49).
Rice. 49. Way options
Arguing in the same way, we get three ways to continue the path, starting to get along the 2nd, and the 3rd, and the 4th road. 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 grandmother, father, mother, daughter and son was presented with 5 different cups. In how many ways can the cups be divided among family members?
Solution. The first member of the family (for example, grandmother) has 5 choices, the next one (let it be dad) has 4 choices. The next one (for example, mom) will choose from 3 cups, the next one from two, the last one gets one remaining cup. We will show these methods in the diagram (Fig. 50).
Rice. 50. Scheme for solving the problem
We got that each choice of a cup by a grandmother corresponds to four possible choices dads, i.e. total 5 4 ways. After dad has chosen a cup, mom has three choices, daughter has two, son has one, i.e. total 3 2 1 ways. Finally, we get that to solve the problem, we need to find the product 5 4 3 2 1.
Note that we got the product of all natural numbers from 1 to 5. Such products are written shorter:
5 4 3 2 1 = 5! (read: "five factorial").
Factorial of a number is the product of all natural numbers from 1 to this number.
So, the answer to the problem is: 5! = 120, i.e. cups among family members can be distributed in one hundred and twenty ways.
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