Arithmetic

The arithmetic (Greek αριθμητική [ τέχνη ], arithmitiké [ téchne ], literally " the Numerical [ art ]") is a branch of mathematics. It includes the calculation with the numbers, especially the natural numbers. It deals with the basic arithmetic operations, ie with the addition ( adding together ), subtraction ( subtracting ), multiplication ( multiplication ), Division ( parts ) as well as the associated computational laws. For arithmetic includes the divisibility with the laws of divisibility of integers and the division with remainder, the arithmetic can be understood as part of algebra, such as the " doctrine of the algebraic properties of numbers. " Arithmetic leads to number theory, which employed in the broadest sense the characteristics of the numbers. Arithmetic is a calculus.

History

As a science, arithmetic was founded by the Greeks. From the pre-Hellenic times are us, for example, of the Egyptians and the Babylonians merely handed empirical rules for solving problems from practical life. For the Pythagoreans make the natural numbers from the nature of things. In the books VII -X of Euclid's Elements, the then-known arithmetic / algebraic / number-theoretic results are first summarized. Especially after the fall of Toledo ( 1085 ) reaches the collected by the Arabs Greek mathematics, enriched by the introduced by the Indians number 0 and the fully developed with this supplement decimal, returned to Western Europe. During the Renaissance, a revival of Greek mathematics takes place.

On this basis, the arithmetic is developed primarily through the introduction of an appropriate sign language for numbers and operations continued in the 16th and 17th centuries. This makes it possible to survey relationships that seem very opaque with verbal play at a glance. François Viète ( Vieta, 1540 to 1603 ) divides the time " logistics " called arithmetic in a " Logistica numerosa ", in our sense, arithmetic, and a " Logistica speciosa ", from which the algebra evolves. He uses for number sizes as letters and operation signs for addition, - for subtraction, and the fraction line for the division. William Oughtred (1574 - 1660) used "x" as a sign of multiplication, but also that he can get away. The now usual multiplication point goes back to Leibniz. Johnson used since 1663 the current normal colon ( :) for the division. Thomas Harriot (1560 - 1621) used the usual signs today for " greater than" (>) and "less than" (< ) and small letters as variables for numbers. Robert Recorde (1510 - 1558) introduces the equal sign (=). By René Descartes (1596 - 1650) derived the notation for squares. Gottfried Wilhelm Leibniz (1646 - 1716) increases with the attempt of an axiomatic justification of arithmetic with natural numbers thoughts of modern mathematical basic research beforehand.

Carl Friedrich Gauss (1777 - 1855) is often quoted as saying: - This neologism can the love of number theory in CF Gauss recognize and shows how " Mathematics is the queen of sciences and arithmetic is the queen of mathematics. " very mathematicians can prescribe this sub-discipline. As Gauss himself in the preface of his famous " studies on higher arithmetic " (see references ) notes, include the theory of cyclotomic or the regular polygons which is treated in the seventh section, though in itself not in the arithmetic; but must be drawn solely from the higher arithmetic their principles. Since the current number theory has developed far beyond the elementary number theory is only referred to as arithmetic number theory ( = higher arithmetic Gaussian ). The term " arithmetic " ( elementary arithmetic Gaussian ) in the true sense is reserved for the main computation.

Leopold Kronecker (1839 - 1914) the statement is attributed to: " The integers has made the good God, all else is the work of man. "

Content

Tags: cardinal, ordinal, 0 or 1 natural as the smallest number, natural number, Peano axioms, decimal place value system, number fonts, number sign. The question of the foundation of the natural numbers into the foundations of mathematics, including set theory.

Tags: seclusion relation to the particular Grundrechenart, commutative, associative law, neutral element, inverse element, inverse operation, distributive power of the set of natural numbers. Generalization and abstraction lead to Algebra.

Tags: The number zero (0 ) (if not already introduced as the smallest natural number), integers, counter number value of a number, sign of a number, fraction, inverse, rational number, cardinality of the sets of numbers. Generalization and abstraction lead to Algebra. Sets of numbers, such as the real numbers, the complex numbers or the quaternions do not belong to arithmetic.

Tags: divider, divisibility, Teilbarkeitssätze, greatest common divisor (gcd ), the least common multiple ( LCM ), Euclidean algorithm, prime number, sieve of Eratosthenes, Primzahlsieb Sundaram, prime factorization, Fundamental Theorem of Arithmetic, cardinality of the set of primes. Generalization and abstraction lead to number theory.