The arithmetic (from Greek ἀριθμός Arithmos , " number ", derived from the adjective ἀριθμητικός arithmētikós , "for counting or calculating belonging" and τέχνη téchnē , " Art ", literally "the associated counting or calculating Art"), is a sub-region of mathematics . It includes calculating with numbers, especially natural numbers . It deals with the basic arithmetic operations , i.e. with addition (counting up), subtraction (deduction), multiplication(Multiply), division (divide) and the associated laws of calculation ( mathematical operators or calculi ). Arithmetic also includes the theory of divisibility with the laws of divisibility of whole numbers and division with the remainder. Arithmetic can be understood as part of algebra , for example as the "theory of the algebraic properties of numbers". Arithmetic leads to number theory , which deals in the broadest sense with the properties of numbers. The arithmetic is a calculus.
As a science, arithmetic was founded by the Greeks. From the pre-Greek times, for example, only empirical rules for solving problems from practical life have been handed down to us by the Egyptians and the Babylonians. For the Pythagoreans , natural numbers are the essence of things. In books VII-X of Euclid's Elements , the then known arithmetic / algebraic / number theoretic results are summarized for the first time. Especially after the fall of Toledo (1085), the Greek mathematics collected by the Arabs, enriched by the number 0 introduced by the Indians and the decimal system fully developed with this addition , returned to the West. During the Renaissance there is a revival of Greek mathematics.
On this basis, arithmetic was further developed in the 16th and 17th centuries, primarily through the introduction of a useful sign language for numbers and operations. This makes it possible to see connections that appear very opaque when verbally reproduced at a glance. François Viète (Vieta, 1540–1603) divides the art of arithmetic, then known as “logistics”, into a logistica numerosa , in our sense arithmetic, and a logistica speciosa , from which algebra develops. He uses letters for the size of numbers and + as an operation sign for addition, - for subtraction and the fraction bar for division. William Oughtred (1574–1660) uses x as a symbol of multiplication, which he sometimes leaves out. The multiplication point common today goes back to Gottfried Wilhelm Leibniz (1646–1716). John Johnson has been using the colon (:) for the division since 1663. Thomas Harriot (1560–1621) uses the symbols commonly used today for “greater than” (>) and “less than” (<) as well as small letters as variables for numbers. Robert Recorde (1510–1558) introduced the equal sign (=). The spelling for squares comes from René Descartes (1596–1650). Leibniz anticipates the ideas of modern basic research in mathematics with the attempt to provide an axiomatic justification for calculating with natural numbers.
Carl Friedrich Gauß (1777–1855) is often quoted with the statement: "Mathematics is the queen of the sciences, and arithmetic is the queen of mathematics." - This word creation shows CF Gauß 'love for number theory and shows how very mathematicians can devote themselves to this sub-discipline. As Gauss himself notes in the preface to his famous “Investigations into Higher Arithmetic” (see literature), the theory of the division of circles or regular polygons , which is treated in the seventh section, does not belong in and of itself to arithmetic; but its principles must be drawn solely from higher arithmetic. Since today's number theory has developed far beyond this, only elementary number theory is also referred to as arithmetic number theory (= higher arithmetic according to Gauss). The term "arithmetic" (elementary arithmetic according to Gauss) in the actual sense is mainly reserved for arithmetic.
Leopold Kronecker (1823-1891) is ascribed the saying: "God made the whole numbers, everything else is human work."
- 1. Natural numbers and their notation.
Keywords: cardinal number , ordinal number , 0 or 1 as the smallest natural number, natural number , Peano axioms , decimal system , place value system , number fonts , numerals . The question of the foundation of natural numbers leads to the foundations of mathematics , especially set theory .
- 2. The four basic arithmetic operations and comparisons of numbers.
Keywords: isolation with regard to the respective basic arithmetic operation , commutative law , associative law , neutral element , inverse element , inverse operation , distributive law , comparison . Generalization and abstraction lead to algebra .
- 3. Payment range extensions.
Keywords: The number zero (0) (if not already introduced as the smallest natural number), whole numbers , opposite number , amount of a number , sign of a number , fraction , reciprocal value , rational number , power of the number sets. Generalization and abstraction lead to algebra. Sets of numbers such as real numbers , complex numbers or quaternions no longer belong to arithmetic.
- 4. Divisors and divisibility.
Keywords: divisors, divisibility , divisibility theorems, greatest common divisor (GCF), least common multiple (lcm), Euclidean algorithm , prime number , sieve of Eratosthenes , prime number sieve of Sundaram, prime factorization , fundamental theorem of arithmetic , power of the set of prime numbers. Generalization and abstraction lead to number theory.
- Klaus Denecke & Kalčo Todorov: Algebraic Basics of Arithmetic. Heldermann, Berlin 1994, ISBN 3-88538-104-4 .
- Friedrich Ernst Feller and Carl Gustav Odermann , edited by Abraham Adler and BR. Fights: The Whole of Commercial Arithmetic , Part 1 and 2, BG Teubner Verlag, Leipzig and Berlin 1924
- Carl Friedrich Gauß: Studies on higher arithmetic. Edited by Hermann Maser. Springer, Berlin 1889; Kessel, Remagen-Oberwinter 2009, ISBN 978-3-941300-09-5 .
- Gottlob Frege: The basics of arithmetic: A logical-mathematical investigation into the concept of number . Wilhelm Koebner, Breslau 1884.
- Donald E. Knuth : Arithmetic. Springer, Berlin [a. a.] 2001, ISBN 3-540-66745-8 .
- Gerhard Kropp: History of Mathematics. Problems and shapes. Quelle and Meyer, Heidelberg 1969; Aula-Verlag, Wiesbaden 1994, ISBN 3-89104-546-8 .
- Reinhold Remmert & Peter Ullrich: Elementary number theory. 2nd Edition. Birkhäuser, Basel / Boston / Berlin 1995, ISBN 3-7643-5197-7 .
- Student Duden: The mathematics. 1, p. 30.
- Oliver Deiser: Real Numbers: The Classical Continuum and the Natural Sequences. P. 79 ( books.google.de ).
- Kropp, p. 19.
- Kropp, p. 23.
- Kropp, pp. 35/6.
- Kropp, p. 75.
- H. Weber: Leopold Kronecker . In: German Mathematicians Association (Hrsg.): Annual report of the German Mathematicians Association . tape 2 . Reimer, 1893, ISSN 0012-0456 , p. 5–31 ( uni-goettingen.de - quote on p. 19).