ARITHMETIC

branch of mathematics in which numbers, relations among numbers, and observations on numbers are studied and used to solve problems. Arithmetic (a term derived from the Greek word arithmos, number) refers generally to the elementary aspects of the theory of numbers, arts of mensuration (measurement), and numerical computation (that is, the processes of addition, subtraction, multiplication, division, raising to powers, and extraction of roots). Its meaning, however, has not been uniform in mathematical usage. An eminent German mathematician, Carl Friedrich Gauss, in Disquisitiones Arithmeticae (1801), and certain contemporary mathematicians have used the term to include some aspects of number theory that are considerably more advanced. The reader interested in the latter is referred to number theory. branch of mathematics in which numbers, relations among numbers, and operations on numbers are studied and used to solve problems. The numbers under consideration may be the natural numbers (1, 2, 3, . . . ), the whole numbers or integers ( . . . -3, -2, -1, 0, 1, 2, 3, . . . ), the rational numbers (positive and negative fractional or decimal numbers, such as 1/3, 0.68, 2 2/3, 3.7, -7/9, along with the integers), the real numbers (irrational numbers, such as , p, sin 32, along with the rational numbers), or various more exotic number systems. The choice of number system depends on the physical situation that is to be dealt with by the arithmetic. Virtually every known society has had some method of dealing with numbers. Very primitive systems may be as simple as having a name for one and a name for more than one. A system of arithmetic developed among the Sumerians of Mesopotamia more than 5,000 years ago, and other societies in various parts of the world independently developed sophisticated systems of arithmetic. The numeration system (or system for writing number symbols) widely used throughout the world today is a place-value system based on the number 10 and usually called the Arabic, or Hindu-Arabic, numeration system. In this system the position a symbol occupies helps determine the value of the symbol. For example, in 333, the 3 on the right means three, but the 3 in the middle means three tens and the 3 on the left means three hundreds. In modern Roman numerals (where I, V, X, L, C, D, and M are 1, 5, 10, 50, 100, 500, and 1,000, respectively), on the other hand, CCC means 300each C stands for one hundred, and the relative position of the C's is of no importance. There are some rules regarding the order of symbols in the Roman numeral system, however (for example, IX means 9, while XI means 11), though generally position is not as important as in place-value systems. A place-value system, such as the Arabic numeral system, has clear advantages in economy of symbolism and in efficiency of computation. Only the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 and the decimal point are needed to write numbers of any size. Arithmetic includes such esoteric subjects as the arithmetic of complex numbers and of quaternions, and number theory; but, for most practical purposes, the arithmetic of whole numbers and of rational numbers and the relation of these to the real world is sufficient. Additional reading Isaac Asimov, Realm of Numbers (1959, reissued 1981), is an introductory presentation at an elementary level. Constance Reid, From Zero to Infinity, 4th ed. (1992), is an up-to-date introduction to elementary number theory accessible to nonmathematicians. Other introductory works include Robert L. Hershey, How to Think with Numbers (1982), an analysis of consumer applications of arithmetic; Peter Hilton and Jean Pedersen, Fear No More: An Adult Approach to Mathematics (1983), showing practical applications of elementary material; Sandra Preis and George Cocks, Arithmetic: An Individualized Approach, 3rd ed. (1990), a programmed text that can be successfully used for self-instruction; and Stephen P. Richards, A Number for Your Thoughts: Facts and Speculations About Numbers from Euclid to the Latest Computers (1982), a lucid explanation of number theory for a wide range of readers.Harold D. Larsen and H. Glenn Ludlow, Arithmetic for Colleges, 3rd ed. (1963); Sidney G. Hacker, Wilfred E. Barnes, and Calvin T. Long, Fundamental Concepts of Arithmetic (1963); Carl B. Allendoerfer, Mathematics for Parents (1965); and Gary L. Musser and William F. Burger, Mathematics for Elementary Teachers: A Contemporary Approach, 3rd ed. (1994), are written from the point of view of education. A very good introductory college text is Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An Introduction to the Theory of Numbers, 5th ed. (1991). H. Davenport, The Higher Arithmetic, 6th ed. (1992); William J. LeVeque, Elementary Theory of Numbers (1962, reprinted 1990); and Kenneth H. Rosen, Elementary Number Theory and Its Applications, 3rd ed. (1992), discuss fundamental concepts associated with the theory of numbers.Graham Flegg (ed.), Numbers Through the Ages (1989), explores the history of counting systems, including basic techniques of calculation. Frank J. Swetz, Capitalism and Arithmetic: The New Math of the 15th Century (1987), translates the Treviso Arithmetic (or Arte dell'abbaco) of 1478, an early work demonstrating methods and applications of arithmetic, and analyzes its content and impact. J.L. Berggren, Episodes in the Mathematics of Medieval Islam (1986), chronicles the history of Islamic mathematics. Works emphasizing the history of arithmetic as a part of number theory include ystein Ore, Number Theory and Its History (1948, reprinted with supplement, 1988); Leonard Eugene Dickson, History of the Theory of Numbers, 3 vol. (191923, reprinted 1992); Tobias Dantzig, Number: The Language of Science, 4th ed. rev. and augmented (1954, reissued 1967); and David Eugene Smith, Rara Arithmetica: A Catalogue of the Arithmetics Written Before the Year MDCI . . ., 4th ed. (1970).The following are classic and advanced studies: Carl Frber, Arithmetik (1911); Carl Friedrich Gauss, Disquisitiones Arithmeticae (1966, reprinted 1986; originally published in Latin, 2nd ed. 1870); Gottlob Frege, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, 2nd rev. ed. (1953, reprinted 1980; originally published in German, 1884); Felix Klein, Elementary Mathematics from an Advanced Standpoint, vol. 1, Arithmetic, Algebra, Analysis (1924, reissued 1953; originally published in German, 3rd ed. 1924); G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th ed. (1979); and J.-P. Serre, A Course in Arithmetic (1973; originally published in French, 1970). C.C. MacDuffee David Eugene Smith William Judson LeVeque The Editors of the Encyclopdia Britannica