Arithmetic

The science of numbers and operations on sets of numbers. Arithmetic is understood to include problems on the origin and development of the concept of a number, methods and means of calculation, the study of operations on numbers of different kinds, as well as analysis of the axiomatic structure of number sets and the properties of numbers. When referring to the logical analysis of the concept of a number, the term theoretical arithmetic is sometimes employed. Arithmetic is closely connected with algebra, which includes the study of operations performed on numbers. The properties of integers form the subject of number theory (cf. Elementary number theory; Number theory).

The term "arithmetic" is also sometimes employed to denote operations performed on objects of very different kinds: "matrix arithmetic" , "arithmetic of quadratic forms" , etc.

The art of computation arose and developed long before the times of the oldest written records extant. The oldest mathematical records are the Cahoon papyri and the famous Rhind papyrus, which is believed to date back to about 2000 B.C.. An additive hieroglyphic system for the representation of numbers (cf. Numbers, representations of) enabled the ancient Egyptians to perform addition and subtraction operations on natural numbers in a relatively simple manner. Multiplication was carried out by doubling, i.e. the factors were decomposed into sums of powers of two, the individual summands were multiplied, and the components added. Operations on fractions (cf. Fraction) were reduced in Ancient Egypt to operations on aliquot fractions, i.e. on fractions of the type $ 1/n $. More complicated fractions were decomposed with the aid of tables into a sum of aliquot fractions. Division was carried out by subtracting from the number to be divided the numbers obtained by successive doubling of the divisor. The clumsy hexadecimal system of the ancient Babylonians gave rise to many difficulties in executing arithmetic operations. There are numerous tables employed by ancient Babylonians to effect multiplication and division.

In Ancient Greece arithmetic was conceived as the study of the properties of numbers; practical calculations were not included. Problems on the technique of operations on numbers, i.e. methods of calculation, were considered to be an independent science — logistics. This differentiation was inherited from the Greeks by Europeans in the Middle Ages. It was only with the advent of the Renaissance that both the beginnings of the theory of numbers and the practice of calculation began to be included in the concept of arithmetic. Greek mathematics made a sharp distinction between the concepts of a number and of a magnitude. To Greek mathematicians numbers meant only what is known as natural numbers in our own days (cf. Natural number); they also distinguished between concepts of different types, such as a number and a geometrical magnitude. There are no logistic studies by ancient Greeks extant, but it is known that their multiplication technique was close to our own. Their alphabetic number system severely complicated operations carried out on numbers. In Ancient Greece ordinary fractions were employed in calculations; however, a fraction was not considered as a number in its own right, but merely as a quotient of natural numbers.

Books 7–9 of Euclid's Elements (3rd century B.C.) deal exclusively with arithmetic in the sense in which the word was employed in ancient times. They mainly deal with the theory of numbers: the algorithm for finding the greatest common divisor (cf. Euclidean algorithm), and with theorems about prime numbers (cf. Prime number). Euclid proved that multiplication is commutative, and that it is distributive with respect to the operation of addition. He also studied the theory of proportion, i.e. the theory of fractions. Other books of his treatise include the general theory of relations between magnitudes, which may be considered as the beginning of the theory of real numbers (cf. Real number).

The manuscripts of Diophantus (probably 3rd century A.D.) that are left to us contain operations on powers (not greater than 6) of numbers, and a few examples of subtraction operations. These are implicitly operations on negative numbers. Diophantus applied his own rules to rational numbers only (cf. Rational number).

Chinese mathematicians in the 2nd century performed operations on fractions and negative numbers. At a somewhat later period they studied the extraction of square roots and cubic roots, the approximate values of which were expressed as decimal fractions (cf. Decimal fraction). The methods employed by Chinese mathematicians in solving problems in arithmetic, in particular the rule of two false assumptions, appears in several manuals on arithmetic, first in Arab, then in European manuals. Very little is known about the initial development of arithmetic in India. The simplest fractions were utilized in India long before the Christian era. Our own decimal computation system is of Indian origin. The earliest written Indian mathematical records extant were compiled in the 5th century and indicate that the knowledge of arithmetic in India of that period was of a high standard. Indian mathematicians operated on integers and fractions using methods very similar to our own. They solved many problems on proportions and on the rule-of-three, and could compute percentages. Studies on negative numbers began in India in the 7th century. The works of Bhaskara II The wreath of science (12th century) contain rules for the multiplication and division of negative numbers.

Indian mathematics exerted a decisive influence on the development of the knowledge of the arithmetic of the Arabs. The treaty on arithmetic written by Muhammed Al-Khwarizmi in the 9th century greatly contributed to the dissemination of the Indian decimal notation and of the methods of addition, subtraction, multiplication, division and extraction of square roots.

Many ancient nations performed their calculations on an abacus, which had replaced the primitive counting on fingers. The external form of the abacus varied, but its principle remained the same: ruling a column, or some other cascade-type marking of numbers. An abacus was used by the Greeks long before the Christian era. The Chinese suan-pan and the Russian schety, which externally resemble each other, are also variants of the abacus.

While the European studies in the domain of number theory are based on Greek mathematics, in particular on the work of Euclid and Diophantus, this is not true of computational techniques. The development of arithmetic in Europe is connected with the arrival of the Indian positional decimal system and of Arabic numerals in Europe. The technique of arithmetic operations was not taken over from India directly, but rather by way of the work of Al-Khwarizmi and other Arab mathematicians.

The abacus was extensively employed in the Middle Ages. It also became a synonym of the word arithmetic, so that Leonardo da Pisa (13th century) called his arithmetical treatise The book of abacus. The book quotes calculation methods taken from Arab sources, while introducing many original improvements. Thus, the addition of fractions is performed by way of the least common multiple of the denominators, and the operations are checked not only, like the Indians, in the base nine, but also in certain other bases. The problems dealt with in the book include the rule-of-three, the rule of alligation, problems involving recurrent sequences, arithmetic progressions and geometric progressions (cf. Arithmetic progression; Geometric progression). In Europe decimal fractions began to be employed in the 15th century, but became widely known only in the 16th century following the publication of the studies of S. Stevin.

Various methods for the multiplication and division of multiplace numbers were proposed in the 15th century, 16th century and in later centuries. They differed from each other only by the system of notation of the intermediate computations. A. Riese, through numerous textbooks, was most influential in the move to replace the old computation (in terms of counters and Roman numerals) by the newer methods (using pen and Hindu–Arab numerals).

In Europe negative numbers were first used by Leonardo da Pisa, who treated them as one would treat debts. A system of operations on negative numbers was presented in the 16th century by M. Stifel. He referred to these numbers as "fictitious" . Proofs for the rules of operation on negative numbers were still studied as late as the 18th century, and it is only owing to the critical reasoning in the second half of the 19th century that such studies have ceased to be taken seriously.

In Europe, up to the 15th or 16th centuries, arithmetic operations on irrational numbers were limited to the extraction of square roots. Nevertheless, Leonardo da Pisa considered the problem of the approximate calculation of cubic roots as well as square roots. S. dal Ferro (around 1500) and N. Tartaglia (16th century) used cubic roots in solving equations of the third degree. There was no general treatment of operations performed on real numbers. The concept of a real number was assimilated in mathematics only gradually, in connection with the development of analytic geometry and mathematical analysis. Up to the 18th century, proofs of operations on irrational numbers were limited to magnitudes expressible by radicals.

Complex numbers were encountered at various times, beginning with Indian mathematics, in solving quadratic equations. However, imaginary solutions were discarded as non-existent. The arithmetic of complex numbers (cf. Complex number) begins with R. Bombelli (16th century), who gave formal rules for performing arithmetic operations on such numbers. However, even as late as the 17th century such operations were performed in a manner similar to those on real numbers, which frequently led to errors. It is only in the 18th century that a precise definition of the arithmetic of complex numbers could be given, owing to the formulas of A. de Moivre and L. Euler.

The idea of logarithms dates back to Archimedes (3rd century B.C.), who compared the terms of arithmetic and geometric progressions. Stifel (16th century) extended these progressions to the left by adding negative powers. He showed that there is a connection between operations carried out on these series, thus introducing the fundamental idea of a logarithm. Taking logarithms as a routine stage in calculations was introduced by J. Napier and J. Burgi in the first half of the 17th century.

The first calculating machines were built in the 17th century by W. Schickard and B. Pascal, who worked independently of each other; they were the prototypes of modern calculators. However, it was only in the 19th century that such machines began to be extensively employed in practical work. The rise of fast electronic computers in middle of the 20th century enhanced the importance of performing an algorithmic operation in a minimum number of elementary operations.

It has been considered ever since the times of Euclid that in order to built a theory, it is sufficient to decompose it into a small number of clear, simple, primary assumptions and to ensure that all basic postulates of the theory can be deduced from them in a purely logical manner. It was tacitly assumed that the connection of these fundamental principles with the real world should be immediately perceptible.

The method of models, used to give foundations of mathematical theories, was discovered in the 19th century. The use of such a method was inevitable, since certain objects and theories studied in mathematics had not found a real-life interpretation. These include, mainly, complex numbers, ideals, and non-Euclidean and $ n $- dimensional geometries. The method of models permitted one to reduce the consistency of one mathematical theory to the consistency of another. Thus, on the assumption that Euclidean geometry is consistent, is was proved that Lobachevskii's geometry was consistent, while the consistency of Euclidean geometry was reduced to the consistency of the arithmetic of real numbers.

At the end of the 19th century, the foundation of arithmetic seemed complete. R. Dedekind and G. Peano, independently of each other, gave an axiom system for the natural numbers from which all known propositions of this science could be deduced. K. Weierstrass proposed to take pairs of natural numbers as a model for the integers and positive rational numbers. The geometrical representation of complex numbers, discovered by J. Argand, C. Wessel and C.F. Gauss, is in essence a model for the theory of complex numbers in the framework of the theory of real numbers. Finally, the set-theoretic approach allowed Dedekind, G. Cantor and Weierstrass to built the theory of real numbers.

However, following the discovery of paradoxes in set theory, there arose the question of how to give a foundation for the arithmetic of natural and real numbers. How can one be sure there are no paradoxes in those branches of mathematics as well? Direct perception does not tell one that the universe is infinite or that matter is infinitely divisible. For this reason the ideas of the infinite set of natural numbers and the continuous number axis may be considered as not directly connected with the physical world. On the other hand, there is no simpler model of the arithmetic of natural numbers than the theory of natural numbers itself, while models of the theory of real numbers are based to a considerable extent on the apparatus of set theory, the reliability of which seems to be open to doubt.

What should be the methods and means of reasoning, not involving models, which would directly indicate the absence of inconsistencies in a given theory — or, in other words, which would show that a logical reasoning based on the axioms of the theory will never yield results which contradict each other?

In the view of D. Hilbert, the reason for the appearance of paradoxes in set theory is that modes of reasoning that are undoubtedly applicable to finite systems of objects, are illegitimately applied to infinite families. This may be avoided if the symbols employed are considered as objects of some new theory, and if the logical conclusions are expressed with the aid of a formal process. In such a case a statement in a theory is expressed as a formula constructed out of a finite set of symbols, while a proof is represented by a finite chain of formulas, obtained according to certain rules from formulas known as axioms (cf. Axiom). In this way, according to Hilbert, it will be possible to operate on finite rather than on infinite objects and to obtain a reliable procedure for establishing the consistency of any theory. Hilbert hoped that this approach would yield, in the first place, a positive solution to the problem of the consistency of the arithmetic of natural numbers and that it would show that the addition of any unprovable formula in number theory to the formulas of arithmetic converts this system of axioms into an inconsistent system.

However, these hopes were smashed. K. Gödel in 1931 demonstrated that formal arithmetic (cf. Arithmetic, formal) is incomplete. He found, moreover, that for any consistent formal system containing the axioms of arithmetic, it is possible to find an explicit description of some closed formula $ u $ such that neither the formula $ u $ itself nor its negation can be deduced in the formal system.

Using this result it is possible to prove that non-isomorphic models of formal arithmetic exist. At the same time, the system of Peano axioms is categorical. How is this to be explained? The system of Peano axioms contains the induction axiom: Any natural number has a certain property $ P $ if the number 1 has this property and if for any natural number $ n $ that has the property $ P $ the natural number $ n + 1 $ also has this property. In this axiom $ P $ may stand for any conceivable property of natural numbers. In the corresponding axiom of formal arithmetic, $ P $ may denote only such properties of natural numbers as can be expressed by the methods of the given formalism. The difference between these two axioms is immaterial if one discusses theorems of elementary number theory, but is very material in the explanation of the properties of the formal theory.

Gödel also showed that a consistent formal system which includes formal arithmetic contains a formula expressing its consistency, and that this formula cannot be proved in this system. Thus, the consistency of such a formal system can only be substantiated by means that are stronger than those formalized in the system itself. G. Gentzen in 1936 gave a proof of the consistency of formal arithmetic, using transfinite induction up to the transfinite number $ E _ {0} $. One must naturally inquire into the consistency of the means used in the proof. Other approaches to the problem of consistency of the arithmetic of natural numbers were also investigated in this context.

Attempts to overcome the difficulties involved in the foundation of the theory of real numbers played a certain role in the development of the constructive approach in mathematics.

References

[1]	, The history of mathematics , Moscow (1970–1973) (In Russian)
[2]	B.L. van der Waerden, "Ontwakende wetenschap" , Noordhoff (1957)
[3]	S.C. Kleene, "Mathematical logic" , Wiley (1967) MR0216930 Zbl 0149.24309
[4]	D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) pp. Appendix VIII MR1859422 MR1807508 MR1807507 MR1732507 MR1676305 MR1109913 MR0981143 MR0874532 MR0851072 MR0799771 MR0474006 MR0309913 MR0262046 MR0229120 MR0177322 MR0098003 MR0080308 MR1511181 Zbl 44.0543.02
[5]	, Encyclopaedia of elementary mathematics , 1 , Moscow-Leningrad (1951–1963) (In Russian)
[6]	V.N. Molodshii, "Fundamentals of the study of numbers in the 18-th and the beginning of the 19-th century" , Moscow (1963) (In Russian)
[7]	V.I. Nechaev, "Number systems" , Moscow (1975) (In Russian)
[8]	D.J. Struik, "A concise history of mathematics" , 1–2 , Dover, reprint (1967) (Translated from Dutch) MR0207502 Zbl 0161.00104
[9]	H.A. Thurston, "The number-system" , Blackie (1956) MR0079558 Zbl 0071.04901

References

[a1]	B.L. van der Waerden, "Geometry and algebra in ancient civilisations" , Springer (1983)