arithmetic(redirected from arithmetical)
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the science of numbers, primarily of the natural (positive integers) numbers and the (rational) fractions and of the operations performed on them. Possession of a sufficiently developed concept of the natural number and the skill to perform operations with numbers are necessary for man’s practical and cultural activities. Therefore, arithmetic is part of a child’s preschool education and a compulsory subject in the school curriculum.
Many mathematical concepts can be constructed using natural numbers; for example, the basic concept of mathematical analysis is the real number. In connection with this, arithmetic is one of the fundamental mathematical sciences. When the logical analysis of the concept of number is emphasized, the term “theoretical arithmetic” is sometimes used. Arithmetic is closely related to algebra, which, in particular, studies operations on numbers without considering their individual properties. The individual properties of integers constitute the subject of number theory.
History. Arithmetic developed in remote antiquity as a result of the practical necessities of counting and of the simplest measurements. It developed as complications arose in economic activity and social relations; monetary calculations; problems of distance, time, and area measurements; and the demands of other sciences.
The origins of counting and the initial stages of the formation of arithmetic concepts are usually conjectured by observing counting among primitive peoples and, indirectly, by studying traces of analogous stages, which have been preserved in the languages of civilized peoples and observed in the learning process of children. Data indicate that the development of those elements of intellectual activity that form the basis of the process of counting proceeds through a number of intermediate phases. These include the ability to recognize the one and the same object and to distinguish objects in a totality of objects which are to be counted; the ability to establish an exhaustive resolution of this totality into elements that differ from each other and at the same time that are equivalent during the counting process (the use of a concrete “unit” of counting); and the ability to establish a correspondence between the elements of two sets, at first directly and then by comparing them with elements of a permanently ordered set of objects, that is, a set of objects arranged in a definite order. The elements of such a standard ordered set become the words (numerals) that are used in the counting of items of any qualitative nature and that correspond to the formation of the abstract concept of a number. Under the most varied conditions it is possible to observe similar features of the gradual origin and refinement of the skills enumerated above and the arithmetic concepts that correspond to them.
At first, counting proves to be possible only for sets of a comparatively small number of objects, beyond which quantitative differences are perceived vaguely and are characterized by words synonymous with the word “many.” In this case, counting tools include notches in wood (tally counting); stones, beads, fingers, and the like; and also sets containing a constant number of elements—for example, the word “eyes” as a synonym of the number 2, the wrist (metacarpus) as a synonym and practical base of the number 5 and so forth.
Verbal ordinal counting (one, two, three, and so forth), whose direct dependence on finger counting (the sequential utterance of the names of fingers and parts of hands), can be sometimes traced directly, is later connected with the counting of groups containing a definite number of objects. This number forms the basis of the corresponding number system, usually, as a result of counting the fingers of both hands, which is equal to 10. However, groupings by five, by 20 (French 80 quatre-vingt= 4 x 20), by 40, by 12 (dozen), by 60, and even by 11 (New Zealand) are encountered. In the era of well-developed commercial relations, numeration methods, both oral and written, naturally exhibited a tendency toward uniformity among tribes and peoples who associated with each other. This circumstance played a decisive role in the establishment and dissemination of the present numeration system (numbering system), the principle of the placed (ordered) significance of numbers, and the methods of performing arithmetic operations. Apparently, analogous reasons can explain the well-known resemblance in the names of numbers in different languages: for example, the number 2 is dva in Russian, dva in Sanskrit, duo in Greek, and duo in Latin.
The source of the first reliable information on the state of arithmetical knowledge in the era of ancient civilizations is the written documents of ancient Egypt (mathematical papyruses), written in approximately 2000 B.C. These are collections of problems with instructions for their solution and rules for operations on whole numbers and fractions with auxiliary tables, with no theoretical explanations whatsoever. Some problems in these collections are essentially solved by means of the setting up and solution of equations; arithmetic and geometric progressions are also encountered.
Cuneiform mathematical texts are evidence of the fairly high level of Babylonian arithmetical knowledge between 2000 and 3000 B.C. The written numeration in the cuneiform texts is a curious combination of the decimal system (for numbers less than 60) with the sexagesimal with classifying units 60, 602, and so forth. The most significant indicator of the high level of arithmetic is the use of sexagesimal fractions using the same numeration system as modern decimal fractions. The Babylonian techniques of performing arithmetic operations, which was theoretically analogous to ordinary methods in the decimal system, were complicated by the necessity of resorting to extensive multiplication tables (for the numbers 1 to 59). The extant cuneiform materials, which were apparently textbooks, also contain the corresponding tables of reciprocal numbers (two-digit and three-digit, that is, to within 1/602 and 1/603), used in division.
The ancient Greeks did not develop arithmetic’s practical aspect further. Their system of written numeration using the letters of the alphabet was considerably less suited for complex calculations than the Babylonian system. (It is significant, in particular, that ancient Greek astronomers preferred to use the sexagesimal system.) On the other hand, ancient Greek mathematicians began the theoretical development of arithmetic in the part which concerned the teaching of the natural numbers, the theory of proportions, measurements of magnitudes, and the theory of irrational numbers (in implicit form). Euclid’s Elements, written in the third century B.C., contains a proof of the infinite number of prime numbers which has retained its value to the present, as well as basic theorems of divisibility, algorithms for finding the common measure of two segments and the greatest common divisor of two numbers, a proof of the nonexistence of a rational number whose square is 2 (the irrationality of the number ), and the theory of proportions stated in geometric form. The theoretical and numerical problems under consideration also included the related problems of perfect numbers (Euclid), Pythagorean numbers, and also—in an even later era—an algorithm for the isolation of prime numbers (sieve of Eratosthenes) and the solution of a number of indeterminate equations of the second and higher degrees (Diophantine equations).
A significant role in the formation of the concept of the infinite series of natural numbers was played by Archimedes’ Sandreckoner, written in the third century B.C., in which the possibility of naming and designating any large numbers in an arbitrary manner is demonstrated. The treatises of Archimedes testify to considerable skill in obtaining approximate values of unknown quantities: root extraction of multi-digital numbers and finding rational approximations for irrational numbers; for example,
The Romans did not advance the techniques of calculation, but they did leave a numeration system—the Roman numerals—which has survived to this day. However, this system is poorly suited for performing operations and is presently used almost exclusively for designating ordinal numbers.
It is difficult to trace the continuity in the development of mathematics in relation to former, more ancient, cultures. However, extremely important phases in arithmetic’s development are connected with the culture of India, which influenced the countries of Southwest Asia and Europe as well as the countries of East Asia (China and Japan). Besides the application of algebra to the solution of arithmetic problems, the most significant service of the Hindus was the introduction of the positional number system, using ten digits, including zero to indicate the absence of units in any of the digits. This system made it possible to develop comparatively simple rules for performing basic arithmetic operations.
The scholars of the medieval East not only preserved in translations the legacy of the ancient Greek mathematicians, but also promoted the dissemination and further development of Hindu achievements. Beginning in the tenth century A.D., the methods of performing arithmetic operations, which differed considerably from modern ones but already used the advantages of the positional number system, gradually began to penetrate into Europe, starting with Italy and Spain.
By the beginning of the 17th century, arithmetic, which barely progressed during the Middle Ages, underwent a rapid improvement of computational methods in connection with increased practical interest in the techniques of computation—problems of nautical astronomy, mechanics, complicated commercial calculations, and so forth. Fractions with decimal denominators, which were already used by the Hindus for extracting square roots and which repeatedly attracted the attention of European scholars, were first applied in implicit form in trigonometric tables in the form of whole numbers which expressed the lengths of the lines of the sine, tangent, and the like for a radius of 105. In 1427, al-Kashi first described in detail a system of decimal fractions and the rules for operations with them. The decimal fraction notation, essentially the same as the modern notation, is encountered in the 1585 work of S. Stevin, and from this time received universal dissemination. J. Napier’s invention of logarithms in the beginning of the 17th century belongs to the same era. In the beginning of the 18th century, methods of performing and writing calculations acquired modern form.
A numeration similar to the Greek was used in Russia until the beginning of the 17th century. The system of verbal numeration which extended to the 50th order was unique and well developed. L. F. Magnitskii’s Arithmetic (1703), which was highly rated by M. V. Lomonosov, was the most significant Russian arithmetic textbook of the early 18th century. It contains the following definition of arithmetic: “Arithmetic, or computation, is an art that is honest, non-envious, and easily understood by all and the most useful and most praised art invented and exposited by the finest mathematicians of ancient and recent times.” In addition to problems of numeration, the exposition of the techniques of calculation with integers and fractions, including decimals, and the corresponding problems, this textbook includes elements of algebra, geometry, and trigonometry and also practical information pertaining to commercial calculations and problems of navigation. The exposition of arithmetic acquired a more or less modern form from L. Euler and his students.
Theoretical problems. The theoretical development of problems pertaining to the study of the number and the measurement of quantity cannot be separated from the development of mathematics as a whole. Its decisive stages are related to the moments that determined to an equal extent the development of algebra, geometry, and analysis. Of great importance is the formulation of the general study of quantities, the corresponding abstract study of the number (integral, rational, and irrational), and the literal apparatus of algebra.
The fundamental importance of arithmetic as a science, adequate for the study of different types of continuous quantities, was realized only at the end of the 17th century in connection with the incorporation into arithmetic of the concept of the irrational number, which is defined by a sequence of rational approximations. In addition, an important role has been played by the apparatus of decimal fractions and the use of logarithms, which has broadened the range of operations that could be performed with the required accuracy on real numbers, both irrational and rational.
I. Newton was the first to state the general definition of a number as the ratio of two values of some quantities, but he avoided writing down the laws which he discovered in formula form expressing the value of one of the quantities in terms of the values of the other, nonhomogeneous with it, and preferred to give such relations the form of proportions—for example, y1/y2 = x12\x22 instead of the corresponding formula y = kx2. Elementary teaching still at times does not realize to a sufficient extent the modern point of view, according to which all the letters in formulas denote simply numbers and operations are performed on numbers that are equivalent among themselves and are independent of their concrete origin. (This is evidenced in the names used in the writing down of operations, in the excessive caution in determining derivatives of physical quantities, and so forth.)
Axiomatic construction of arithmetic. The next stage, the axiomatization of arithmetic, began in the 19th century and is connected with the general process of critical revision of the logical foundations of mathematics, in which the work of N. I. Lobachevskii on geometry, in particular, played the most significant role. The very simplicity and obvious indisputability of the initial propositions of arithmetic hampered the identification of the basic propositions—axioms and definitions—which could serve as the starting point for the construction of a theory. The first allusions to the possibility of such a construction were already seen in G. Leibniz’ proof of the relation 2•2 = 4.
Only in the middle of the 19th century, H. Grassmann succeeded in selecting a system of basic axioms defining the operations of addition and multiplication in such a way that the remaining propositions of arithmetic grew out of it as a logical consequence. If there is a natural series of numbers, beginning with 1, and 2 is defined as 1 + 1, 3 as 2 + 1, 4 as 3 + 1, and so forth, then one general proposition a + (b+ 1) = (a + b) + 1, which is used as an axiom or definition of addition, is sufficient not only to derive formulas of a particular type, as, for example, 3 + 2=5, but, using the method of mathematical induction, is also sufficient to prove the general properties of addition, which are correct for any natural numbers—the commutative and associative laws. A similar role in multiplication is played by the formulas a · 1 = a and a(b + 1) = ab + a. Thus, the aforementioned proof of the relation 2•2 = 4 can be represented in the form of a chain of equalities that follow from the formulas presented here and the definitions of the numbers 2, 3, and 4, namely: 2•2 = 2(1 + 1) = 2•1 + 2• 1 = 2 + 2 = 2 + (1 + 1) = (2 + 1) + 1 = 3+1=4.
After proving the commutative, associative, and distributive (with respect to addition) laws of the operation of multiplication, the further construction of the theory of arithmetic operations on the natural numbers no longer presents major difficulties. To remain at the same level of abstraction, it is necessary to introduce fractional numbers as pairs of whole numbers (numerator and denominator) that are subject to specific laws of comparison and operations.
Grassmann’s construction was subsequently completed in the work of G. Peano, in which a system of basic concepts (not definable in terms of other concepts) was clearly isolated, namely: the concept of the natural number, the concept of the succession of one number directly after another in a natural sequence, and the concept of the initial term of a natural series (which can be either 0 or 1). These concepts are connected by five axioms which can be considered as the axiomatic definition of the indicated basic concepts.
Peano’s axioms are (1) 1 is a natural number; (2) a natural number is followed by a natural number; (3) 1 is not the successor of any natural number; (4) if the natural number a follows the natural numbers b and c, then b and c are identical; and (5) if any proposition is proven for 1 and if from the assumption that it is correct for the natural number η it follows that it is correct for the natural number that follows n, then this proposition is true for all natural numbers. This last axiom—the axiom of complete induction—makes it possible to subsequently employ Grassmann’s definitions of operations and to prove the general properties of natural numbers.
These constructions, which yield the solution for the problem of substantiating the formal propositions of arithmetic, avoid the question of the logical structure of the arithmetic of natural numbers in the broader sense of the word, including those operations that themselves determine the applications of arithmetic both within mathematics itself and in practical life. The analysis of this aspect of the question, having elucidated the substance of the concept of a quantitative number, at the same time showed that the question of substantiating arithmetic is closely connected with the more general fundamental problems of the methodological analysis of mathematical disciplines. If the simplest propositions of arithmetic that pertain to the elementary counting of objects and are a generalization of mankind’s centuries-old experience naturally fall into the simplest logical schemes, then arithmetic, as a mathematical discipline investigating the infinite set of natural numbers, requires the study of the self-consistency of the corresponding system of axioms and a more detailed analysis of the meaning of the general propositions that follow from it.
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I. V. ARNOL’D