Riddle of elementary arithmetic
Mathematics is the queen of all sciences,
arithmetic is the queen of mathematics.
K.F.Gauss
How are the four arithmetic operations related? You will laugh, but the lack of a comprehensive answer to this question significantly slows down the development of physics, chemistry and related sciences. Scientists, unfortunately, can only guess about this inhibition. If this question had been investigated in a timely manner, we would have no problems with the development of the ideas of D.I. Mendeleev, and according to the results of the hadron collider, most likely, computer models of elementary particles and atomic nuclei would be created.
The article on arithmetic in the English Wikipedia article briefly describes arithmetic operations, gives their properties, but there is practically no information about their interrelation. In the Russian version it is indicated that all actions of arithmetic have inverse: addition has subtraction, multiplication has division, and in addition only Naper's ideas are given, which will be discussed later. In the English description of arithmetic and for such scant information there was no place, while arithmetic is indulgently characterized as the oldest and most elementary part of mathematics. Even in the literature on the history of arithmetic, it is difficult to find information about research on the relationship between arithmetic operations. The history of arithmetic itself is mainly devoted to the theory of numbers, which is sometimes called the highest arithmetic.
The search for the relationship of arithmetic actions went to the Renaissance [1]. In 1515, the first German textbook of arithmetic, compiled by Jacob Köbel, emphasizes the equivalence of all four actions. In 1518, G. Grammatheus, in his essay “A New Easy and Accurate Book on the Account, on Solving Different Questions by the Triple Rule, and so on,” notes the interdependence of addition with multiplication, and subtraction with division. In the "Book of the Number" by E. Mizrahi, published in Istanbul in 1533, multiplication is considered as a special case of addition and is not included in the number of arithmetic operations. In the book “Logistic Art”, published in 1839 according to Napier’s notes of the 70s of the 16th century, arithmetic operations differ in degrees: multiplication and division are described as higher-order actions regarding addition and subtraction.
')
These ideas were the beginning of the study of the problem of the relationship of arithmetic operations. But, probably due to insufficient attention to elementary arithmetic, a solution was not found. We will continue the search for a bygone era and count the number of soldiers in the convoy. The solution of the problem by addition is the sequential increase by one unit of the number of warriors counted to obtain the resulting amount. Counting can be accelerated by counting the number of soldiers in the line and multiplying it by the number of rows in the column. Now let's divide the column in half. With the help of subtraction, one soldier can be successively separated into each of the two new columns. To speed up the counting, you can divide the number of ranks in half and recalculate the detachable ranks.
The solution of these two problems suggests that multiplication and division under certain conditions (in our case it is the organization of the objects of the account in the matrix) are a generalization of addition and subtraction, respectively. Usually, generalizations (let's call them generalizations of A) are associated with the extension of a mathematical operation to a previously unused area. Should we speak of generalizations if the only result of their use is to speed up calculations (generalizations B)? Let's say yes and analyze the result.
Generalizations B is easier to find and quite a lot of them are known. It is assumed that the study of their properties will allow to predict the existence and properties of the still unknown generalizations of A. Here you can use the analogy with geochemical exploration of deposits. Ore bodies of small size (bodies A) are included in rather extensive bodies of increased contents of chemical elements relative to the background. Initially, the task of finding body A is replaced by the easier task of finding body B, within which further searches are carried out.
As a generalization of multiplication, the factorial of a natural number n is known, which is interpreted as the number of permutations of a set of n elements. Factorial is defined in combinatorics, but is widely used in mathematics. There is also a generalization of factorial for positive real numbers - the gamma function, known in mathematical statistics. Gamma function is defined as factorial extensions for all complex numbers, excluding negative integers. The generalization of factorial to the set of real (and complex) numbers is also the pi-function. The list can be continued with double factorial, superfactorial, etc.
But what about the division? On the basis of it, generalizations are unknown, but one case simply cannot be ignored. This division of one number by another, when they have the same dimension or dimensionless. The case of different dimensions is well interpreted. For example, if the 6 km path is covered in 1 hour, then, by dividing the path by time, it can be argued that the speed is 6 km / h. We proceed further to proportional values.
It is considered that two interdependent quantities are proportional, if the ratio of their values ​​does not change. The result of the division is called the proportionality coefficient. The ratio of the contents of gold and silver in gold-silver deposits can serve as an example of the proportionality coefficient for the values ​​of one dimension. This is important geological information. The ores of the Mexican Pachuka deposit for 1 ton of gold contain approximately 200 tons of silver, that is, the ratio of gold to silver in this deposit is 1: 200. Why should the gold content be divided by the silver content, and not vice versa? Because it is so accepted and satisfied with many, but not the author of the article. The reason for disagreement with the accepted division rule lies in the need for both a further transition to the study of proportionality of dimensionless quantities, and an increase in the number of quantities for which the proportionality coefficient is calculated.
In these cases, the result of dividing numbers will not please everyone without exception. The proportionality ratio of the atomic mass of hydrogen and helium can be calculated differently: in the form of fractions 1.008 / 4.0026 and 4.0026 / 1.008. The result in terms of the degree of uncertainty exceeds the old joke about multiplication, in which the colonel states that twice two is approximately five and more precisely for these calculations it is not necessary. Dream with the coefficients of our example! Imagination is clearly not enough to calculate one proportionality factor for three or more numbers, for example, the atomic masses of hydrogen, helium and lithium.
Moreover, it seems unreasonable to require a constant value for the proportionality coefficient. The interdependence of quantities is also a separate problem. The distribution of calculations of the coefficients of proportionality to sets of one or more numbers was obtained using the information coefficient of proportionality [2], [3], [4]. It has in common with the usual coefficient of proportionality the most important property to maintain its value when multiplying the original numbers by the same number. For calculations, the equations of information theory and a 3 Ă— 3 square matrix are used, similar to the tic-tac-toe field.
If three initial numbers a , b and c are taken three times and arranged in such a matrix so that they are all present in each row and column, then you can eliminate the need to “assign” numerators and denominators for calculations. For example, three lines can be represented as a, b, c; b, c, a and c, a, b . When changing places of any two triads of the same numbers, the information proportionality coefficient will not change. The coefficient calculation can be performed using the proportionality calculator available for free use on the Internet.
By analogy, for four numbers a 4 × 4 matrix is ​​needed, for five numbers - 5 × 5, etc. In this case, a problem arises: when using different matrices it is impossible to jointly process the results of calculations. Universal calculations can be done with a 3 × 3 matrix, calculating a large set of informational proportionality coefficients. The numbers in the matrix are randomly located, and the sum of eight others is used as the ninth element. Source numbers instead of a single coefficient are characterized by the probability distribution of a set of information proportionality coefficients. This is customary for mathematical statistics, in which such distributions are considered.
Two articles on practical applications of informational coefficients of proportionality were published in Russian in 2008 in the scientific journal of the Siberian Federal University and from the moment of publication they are available on the Internet for free review. Scientific articles in English can be found on the website of Cornell University (www.arxiv.org).
How important are proportionality factors? Without them, our communication would be impossible, since Newton's gravitational constant is a coefficient of proportionality, ensuring our presence on Earth. The proportionality coefficients can be considered as an important characteristic of chemical compounds. The number of chemical elements found in nature, the maximum possible numbers of minerals, inorganic and organic chemical compounds are probably determined respectively as combinations of 1, 2, 3 and 4 of 95 natural chemical elements. A hypothesis is proposed that the distribution of informational proportionality coefficients for atomic masses of chemical elements of minerals and other chemical compounds corresponds to the distribution of these sets for combinations of 2, 3, and 4 atomic masses of 95 natural chemical elements [2], [3].
Such a strange ratio of chemical elements and chemical compounds could not be explained otherwise. Minerals, for example, contain up to 12 chemical elements without taking into account impurities. Why can we speak of their number as a combination of 2 out of 95? The explanation of this pattern is due to the fact that the probability distribution of information coefficients of proportionality of atomic masses of more than two chemical elements coincides with that distribution for any two chemical elements.
The answer to the question of the importance of using informational proportionality coefficients also provides an idea of ​​the structure of the atomic nucleus of any chemical element in the form of a cube consisting of 27 elementary cubes [5], in essence - a Rubik's cube. Such a cube contains integers from 1 to 8, a total of 9 components. This structure leads us to an explanation of the number of isotopes of each chemical element and the different occurrence of isotopes in nature. It also explains the reasons for the joint finding of chemical elements in nature, the appearance of nuggets of some chemical elements, for example, gold, and the practical absence of others, for example, tin. A program designed for such calculations is also available to any researcher.
The potential application of information proportional factors is enormous. With their help, data were obtained on the possible existence in the quartz vein of the Vasilyevsky gold deposit on the Yenisei Ridge of a double spiral structure - a possible analogue of DNA [2]. The open program Agemarker can be used to classify rocks and ores according to the results of their chemical analysis while determining their relative age [6] (the most important task in geology).
Using this program, minerals can also be combined into packages of the periodic system of chemical compounds [7]. Such a combination of minerals is the key to a similar periodic systematization of all inorganic and organic compounds in packages of 95. The classification indicator is calculated only on the basis of the atomic masses of chemical elements, and the atomic masses at one time allowed the creation of the Periodic System of Chemical Elements.
I conclude the article with the statement that in mathematics both practical and theoretical studies of informational coefficients of proportionality and the development of the theory of mathematical generalizations are necessary. The starting point of this theory can be arithmetic with the problem of generalizing the actions of addition and subtraction. In mathematics, generalized functions are known - perhaps it is time to define generalized and generalizing mathematical actions. If we ignore these questions in silence, then the mystery of arithmetic operations can turn into increasing problems, since the proposed calculations of proportionality indicators are probably irreplaceable in medicine and ecology, as, indeed, in other natural sciences.
arithmetic is the queen of mathematics.
K.F.Gauss
How are the four arithmetic operations related? You will laugh, but the lack of a comprehensive answer to this question significantly slows down the development of physics, chemistry and related sciences. Scientists, unfortunately, can only guess about this inhibition. If this question had been investigated in a timely manner, we would have no problems with the development of the ideas of D.I. Mendeleev, and according to the results of the hadron collider, most likely, computer models of elementary particles and atomic nuclei would be created.
The article on arithmetic in the English Wikipedia article briefly describes arithmetic operations, gives their properties, but there is practically no information about their interrelation. In the Russian version it is indicated that all actions of arithmetic have inverse: addition has subtraction, multiplication has division, and in addition only Naper's ideas are given, which will be discussed later. In the English description of arithmetic and for such scant information there was no place, while arithmetic is indulgently characterized as the oldest and most elementary part of mathematics. Even in the literature on the history of arithmetic, it is difficult to find information about research on the relationship between arithmetic operations. The history of arithmetic itself is mainly devoted to the theory of numbers, which is sometimes called the highest arithmetic.
The search for the relationship of arithmetic actions went to the Renaissance [1]. In 1515, the first German textbook of arithmetic, compiled by Jacob Köbel, emphasizes the equivalence of all four actions. In 1518, G. Grammatheus, in his essay “A New Easy and Accurate Book on the Account, on Solving Different Questions by the Triple Rule, and so on,” notes the interdependence of addition with multiplication, and subtraction with division. In the "Book of the Number" by E. Mizrahi, published in Istanbul in 1533, multiplication is considered as a special case of addition and is not included in the number of arithmetic operations. In the book “Logistic Art”, published in 1839 according to Napier’s notes of the 70s of the 16th century, arithmetic operations differ in degrees: multiplication and division are described as higher-order actions regarding addition and subtraction.
')
These ideas were the beginning of the study of the problem of the relationship of arithmetic operations. But, probably due to insufficient attention to elementary arithmetic, a solution was not found. We will continue the search for a bygone era and count the number of soldiers in the convoy. The solution of the problem by addition is the sequential increase by one unit of the number of warriors counted to obtain the resulting amount. Counting can be accelerated by counting the number of soldiers in the line and multiplying it by the number of rows in the column. Now let's divide the column in half. With the help of subtraction, one soldier can be successively separated into each of the two new columns. To speed up the counting, you can divide the number of ranks in half and recalculate the detachable ranks.
The solution of these two problems suggests that multiplication and division under certain conditions (in our case it is the organization of the objects of the account in the matrix) are a generalization of addition and subtraction, respectively. Usually, generalizations (let's call them generalizations of A) are associated with the extension of a mathematical operation to a previously unused area. Should we speak of generalizations if the only result of their use is to speed up calculations (generalizations B)? Let's say yes and analyze the result.
Generalizations B is easier to find and quite a lot of them are known. It is assumed that the study of their properties will allow to predict the existence and properties of the still unknown generalizations of A. Here you can use the analogy with geochemical exploration of deposits. Ore bodies of small size (bodies A) are included in rather extensive bodies of increased contents of chemical elements relative to the background. Initially, the task of finding body A is replaced by the easier task of finding body B, within which further searches are carried out.
As a generalization of multiplication, the factorial of a natural number n is known, which is interpreted as the number of permutations of a set of n elements. Factorial is defined in combinatorics, but is widely used in mathematics. There is also a generalization of factorial for positive real numbers - the gamma function, known in mathematical statistics. Gamma function is defined as factorial extensions for all complex numbers, excluding negative integers. The generalization of factorial to the set of real (and complex) numbers is also the pi-function. The list can be continued with double factorial, superfactorial, etc.
But what about the division? On the basis of it, generalizations are unknown, but one case simply cannot be ignored. This division of one number by another, when they have the same dimension or dimensionless. The case of different dimensions is well interpreted. For example, if the 6 km path is covered in 1 hour, then, by dividing the path by time, it can be argued that the speed is 6 km / h. We proceed further to proportional values.
It is considered that two interdependent quantities are proportional, if the ratio of their values ​​does not change. The result of the division is called the proportionality coefficient. The ratio of the contents of gold and silver in gold-silver deposits can serve as an example of the proportionality coefficient for the values ​​of one dimension. This is important geological information. The ores of the Mexican Pachuka deposit for 1 ton of gold contain approximately 200 tons of silver, that is, the ratio of gold to silver in this deposit is 1: 200. Why should the gold content be divided by the silver content, and not vice versa? Because it is so accepted and satisfied with many, but not the author of the article. The reason for disagreement with the accepted division rule lies in the need for both a further transition to the study of proportionality of dimensionless quantities, and an increase in the number of quantities for which the proportionality coefficient is calculated.
In these cases, the result of dividing numbers will not please everyone without exception. The proportionality ratio of the atomic mass of hydrogen and helium can be calculated differently: in the form of fractions 1.008 / 4.0026 and 4.0026 / 1.008. The result in terms of the degree of uncertainty exceeds the old joke about multiplication, in which the colonel states that twice two is approximately five and more precisely for these calculations it is not necessary. Dream with the coefficients of our example! Imagination is clearly not enough to calculate one proportionality factor for three or more numbers, for example, the atomic masses of hydrogen, helium and lithium.
Moreover, it seems unreasonable to require a constant value for the proportionality coefficient. The interdependence of quantities is also a separate problem. The distribution of calculations of the coefficients of proportionality to sets of one or more numbers was obtained using the information coefficient of proportionality [2], [3], [4]. It has in common with the usual coefficient of proportionality the most important property to maintain its value when multiplying the original numbers by the same number. For calculations, the equations of information theory and a 3 Ă— 3 square matrix are used, similar to the tic-tac-toe field.
If three initial numbers a , b and c are taken three times and arranged in such a matrix so that they are all present in each row and column, then you can eliminate the need to “assign” numerators and denominators for calculations. For example, three lines can be represented as a, b, c; b, c, a and c, a, b . When changing places of any two triads of the same numbers, the information proportionality coefficient will not change. The coefficient calculation can be performed using the proportionality calculator available for free use on the Internet.
By analogy, for four numbers a 4 × 4 matrix is ​​needed, for five numbers - 5 × 5, etc. In this case, a problem arises: when using different matrices it is impossible to jointly process the results of calculations. Universal calculations can be done with a 3 × 3 matrix, calculating a large set of informational proportionality coefficients. The numbers in the matrix are randomly located, and the sum of eight others is used as the ninth element. Source numbers instead of a single coefficient are characterized by the probability distribution of a set of information proportionality coefficients. This is customary for mathematical statistics, in which such distributions are considered.
Two articles on practical applications of informational coefficients of proportionality were published in Russian in 2008 in the scientific journal of the Siberian Federal University and from the moment of publication they are available on the Internet for free review. Scientific articles in English can be found on the website of Cornell University (www.arxiv.org).
How important are proportionality factors? Without them, our communication would be impossible, since Newton's gravitational constant is a coefficient of proportionality, ensuring our presence on Earth. The proportionality coefficients can be considered as an important characteristic of chemical compounds. The number of chemical elements found in nature, the maximum possible numbers of minerals, inorganic and organic chemical compounds are probably determined respectively as combinations of 1, 2, 3 and 4 of 95 natural chemical elements. A hypothesis is proposed that the distribution of informational proportionality coefficients for atomic masses of chemical elements of minerals and other chemical compounds corresponds to the distribution of these sets for combinations of 2, 3, and 4 atomic masses of 95 natural chemical elements [2], [3].
Such a strange ratio of chemical elements and chemical compounds could not be explained otherwise. Minerals, for example, contain up to 12 chemical elements without taking into account impurities. Why can we speak of their number as a combination of 2 out of 95? The explanation of this pattern is due to the fact that the probability distribution of information coefficients of proportionality of atomic masses of more than two chemical elements coincides with that distribution for any two chemical elements.
The answer to the question of the importance of using informational proportionality coefficients also provides an idea of ​​the structure of the atomic nucleus of any chemical element in the form of a cube consisting of 27 elementary cubes [5], in essence - a Rubik's cube. Such a cube contains integers from 1 to 8, a total of 9 components. This structure leads us to an explanation of the number of isotopes of each chemical element and the different occurrence of isotopes in nature. It also explains the reasons for the joint finding of chemical elements in nature, the appearance of nuggets of some chemical elements, for example, gold, and the practical absence of others, for example, tin. A program designed for such calculations is also available to any researcher.
The potential application of information proportional factors is enormous. With their help, data were obtained on the possible existence in the quartz vein of the Vasilyevsky gold deposit on the Yenisei Ridge of a double spiral structure - a possible analogue of DNA [2]. The open program Agemarker can be used to classify rocks and ores according to the results of their chemical analysis while determining their relative age [6] (the most important task in geology).
Using this program, minerals can also be combined into packages of the periodic system of chemical compounds [7]. Such a combination of minerals is the key to a similar periodic systematization of all inorganic and organic compounds in packages of 95. The classification indicator is calculated only on the basis of the atomic masses of chemical elements, and the atomic masses at one time allowed the creation of the Periodic System of Chemical Elements.
I conclude the article with the statement that in mathematics both practical and theoretical studies of informational coefficients of proportionality and the development of the theory of mathematical generalizations are necessary. The starting point of this theory can be arithmetic with the problem of generalizing the actions of addition and subtraction. In mathematics, generalized functions are known - perhaps it is time to define generalized and generalizing mathematical actions. If we ignore these questions in silence, then the mystery of arithmetic operations can turn into increasing problems, since the proposed calculations of proportionality indicators are probably irreplaceable in medicine and ecology, as, indeed, in other natural sciences.
Literature
- Depman I.Ya. The history of arithmetic. 2nd ed., Corr. - M .: Enlightenment, 1965. 416s.
- Labushev, MM Borzykh O.S. The use of information coefficients of proportionality for the analysis of the distribution of gold in the ore body of the Vasilyevsky deposit. Journal of Siberian Federal University. Engineering & Technologies 1 (2008) 40-46
- Labushev, MM About the maximum possible number of minerals, inorganic and organic chemical compounds. Journal of Siberian Federal University. Engineering & Technologies 3 (2008) 221–233
- Labushev, MM (2011). Table 5. Comprieved from arxiv.org/abs/1103.3972
- Labushev, MM (2012). Computer Simulation of Atoms Nuclei Structured Accuracy, Information Retrieval, 14. Retrieved from arxiv.org/abs/1207.4671
- Labushev, MM, Khokhlov AN (2012). For their Potential Energy Index, 19. Retrieved from arxiv.org/abs/1212.2628
- Labushev, MM (2013). Three Packets of Minerals and Chemicals Compounds, 15. Retrieved from arxiv.org/abs/1304.1280
Source: https://habr.com/ru/post/209580/
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