Veština računanja, nauka o brojevima; nauka o računanju određenim brojevima koji se pišu ciframa; politička aritmetika, primena aritmetike na društvene i državne ustanove (npr. osiguranje života, lutrije i dr.).
arithmetic / erɪθmetɪk /
arithmetic je nebrojiva imenica
ETYM Old Eng. arsmetike, Old Fren. arismetique, Latin arithmetica, from Greek arithmein to number, from arithmos number, prob. from same root as Eng. arm, the idea of counting coming from that of fitting, attaching.
The branch of mathematics dealing with the addition, subtraction, multiplication, and division of real numbers.
The branch of pure mathematics dealing with the theory of numerical calculations.
Branch of mathematics concerned with the study of numbers and their properties. The fundamental operations of arithmetic are addition, subtraction, multiplication, and division. Raising to powers (for example, squaring or cubing a number), the extraction of roots (for example, square roots), percentages, fractions, and ratios are developed from these operations.
Forms of simple arithmetic existed in prehistoric times. In China, Egypt, Babylon, and early civilizations generally, arithmetic was used for commercial purposes, records of taxation, and astronomy. During the Dark Ages in Europe, knowledge of arithmetic was preserved in India and later among the Arabs. European mathematics revived with the development of trade and overseas exploration. Hindu-Arabic numerals replaced Roman numerals, allowing calculations to be made on paper, instead of by the abacus.
The essential feature of this number system was the introduction of zero, which allows us to have a place–value system. The decimal numeral system employs ten numerals (0,1,2,3,4,5,6,7,8,9) and is said to operate in “base ten”. In a base-ten number, each position has a value ten times that of the position to its immediate right; for example, in the number 23 the numeral 3 represents three units (ones), and the number 2 represents two tens. The Babylonians, however, used a complex base-sixty system, residues of which are found today in the number of minutes in each hour and in angular measurement (6 x 60 degrees). The Mayas used a base-twenty system.
There have been many inventions and developments to make the manipulation of the arithmetic processes easier, such as the invention of logarithms by Scottish mathematician John Napier 1614 and of the slide rule in the period 1620–30. Since then, many forms of ready reckoners, mechanical and electronic calculators, and computers have been invented.
Modern computers fundamentally operate in base two, using only two numerals (0,1), known as a binary system. In binary, each position has a value twice as great as the position to its immediate right, so that for example binary 111 (or 1112) is equal to 7 in the decimal system, and binary 1111 (or 11112) is equal to 15. Because the main operations of subtraction, multiplication, and division can be reduced mathematically to addition, digital computers carry out calculations by adding, usually in binary numbers in which the numerals 0 and 1 can be represented by off and on pulses of electric current.
Modular or modulo arithmetic, sometimes known as residue arithmetic or clock arithmetic, can take only a specific number of digits, whatever the value. For example, in modulo 4 (mod 4) the only values any number can take are 0, 1, 2, or 3. In this system, 7 is written as 3 mod 4, and 35 is also 3 mod 4. Notice 3 is the residue, or remainder, when 7 or 35 is divided by 4. This form of arithmetic is often illustrated on a circle. It deals with events recurring in regular cycles, and is used in describing the functioning of gasoline engines, electrical generators, and so on. For example, in the mod 12, the answer to a question as to what time it will be in five hours if it is now ten o’clock can be expressed 10 + 5 = 3.
Properties of numbers.
All the properties of numbers may be deduced from this law, which states that the sum of a set of numbers is the same whatever the order of addition, and that the product of a set of numbers is the same whatever the order of multiplication.
A special case of the associative law produces commutativity where there are only two numbers in the set. For example.
A + b = b + a.
Ab = ba.
The distributive law for multiplication over addition states that, given a set of numbers a, b, c, . and a multiplier m.
M(a + b + c + .) = ma + mb + mc + .
9 × 132 = (9 × 100) + (9 × 30) + (9 × 2).
The distributive law does not apply for addition over multiplication; for example.
7 + (3 × 5) ą (7 + 3) × (7 + 5).
Zero is described as the identity for addition because adding zero to any number has no effect on that number.
N + 0 = 0 + n = n.
One is the identity for multiplication because multiplying any number by one leaves that number unchanged.
N × 1 = 1 × n = n.
Every number has a negative -n such that.
N + (-n) = 0.
Every number (except 0) has an inverse 1/n such that.
N × 1/n = 1.
calculus / kælkjələs /
Množina reči calculus je calculuses.
Branch of mathematics which uses the concept of a derivative (see differentiation) to analyze the way in which the values of a function vary. Calculus is probably the most widely used part of mathematics. Many real-life problems are analyzed by expressing one quantity as a function of another—position of a moving object as a function of time, temperature of an object as a function of distance from a heat source, force on an object as a function of distance from the source of the force, etc.—and calculus is used to deal with such functions. There are several branches of calculus. Differential and integral calculus, each of which deals with small quantities which during manipulation are made smaller and smaller, compose the infinitesimal calculus. Differential equations relate to the derivatives of a set of variables and may include the variables. Many give the mathematical models for physical phenomena such as simple harmonic motion. Differential equations are solved generally by integration, depending on their degree. If no analytical processes are available, integration can be performed numerically. Other branches of calculus include calculus of variations and calculus of errors. Calculus methods have been developed slowly since the ancient Greek mathematicians. In the 17th century Isaac Newton and Gottfried Leibniz were the first to give (independently) general rules for calculus but it was very difficult to put the subject on a secure logical basis, mainly because of the difficult concepts of limit and continuity involved. Instead of using the idea of limit, 18th- and 19th-century mathematicians sought to base calculus on the ideas of “infinitesimals” (roughly, very small quantities) and “differentials” and the subject has in the past been known as “infinitesimal calculus” or “differential calculus”. The first complete presentation of calculus using limits was given by Augustin Cauchy in 1821, but his ideas were not generally adopted (particularly in Britain) for many years.