Calculation Mathematics
In mathematics , a calculation is an operation or set of operations on quantities Etymology The word "calculus" comes from the original Latin calculus means "pebble." According to Georges Ifrah , shepherds with their sheep comptabilisaient pebbles in a jar at the entrance and exit of the fold . These pebbles are the source of one of the oldest accounting systems discovered today . The use of rocks to symbolize people, animals or measures of grain and to carry out additions and subtractions is fundamental in the development of mathematical calculation. First calculator silent and symbolic, it is the precursor of a whole family assistance calculation are the charts. The first calculations have focused on whole numbers (number of animals in a herd, number of soldiers in an army, number of days in a calendar, price in a transaction or a tax). The development of number systems can then perform calculations on fractional numbers (representing the length or duration) as in Sumer at the end of the fourth millennium or more later in Egypt . In antiquity, the Greeks seem to have less immediate concerns. The calculation example is oriented in order to "measure the earth," Working geometry , but with the philosophical meaning attached to it : "Let none enter here is not a geometer" proclaimed, according to legend, , the epigraph of the pediment of the Academy of Plato. On one count tool for use by shepherds and accounting, computing has progressively moved towards the abstract. Greek mathematicians are working on length and study the notion of commensurability (is there a unit that can measure two lengths?) which is closer to the current concept of rational number. In seeking to calculate the diagonal of the square of side 1, that is to say the square root of two , they discover the existence of incommensurable numbers , (it seems nowadays irrational numbers ) and invented the concept of building length. For several centuries, the calculations are performed on these types of numbers. Finding solutions of quadratic equations leads to calculations negative numbers or complex, that d'Alembert in his encyclopedia , describes respectively false roots and imaginary roots and do not accept them as final result of a calculation . As for all real numbers , it was not until the late nineteenth century to make it clear . Along with calculations on numbers (numeric), can develop in the Arabic mathematicians ( Ibn al-Banna , Al Khwarizmi ), precursors of algebra computations on polynomials . Symbolic notations developed by Francois Vieta and Descartes introduce this type of computing in Europe. Symbolic notations release calculations and field numbers are performed in Europe calculations on objects as diverse as the functions (XVII century), or vectors (XIX century). In the late nineteenth century, the German school creates the sets ( commutative fields , rings which are defined operations that have little to do with classical addition and multiplication, although the same notation their is assigned (+ and ). It is the birth of algebraic structures. In the nineteenth and twentieth centuries, the development of mathematical logic offers a new scope: the logical propositions. This is the domain of propositional calculus. Found in the area of operations followed a similar trend. The first four operations are, in order of complexity, the addition , the subtraction , the multiplication and division. Calculation rules are established for these four operations ranging from addition tables or multiplication to multiplication algorithms or division. The extraction of root ( square root , cube root, etc..) is a higher level of complexity. The Chinese book The Nine Chapters , with commentaries by Liu Hui (263), presents algorithms Extraction of square roots that are related to the division algorithm, the operation also called it the most often divide by square root. " The exponentiation (calculation of a b) b classical world, is later for b rational or real. Gradually, as the object of calculation are diversifying operations do the same. Besides the conventional operations of addition, subtraction and multiplication by a real one then finds the matrix product , the cross product or dot on vectors. It can also be the product of polynomials, into a Euclidean division but also the drift. We can also calculate the derivative of a differentiable function, integrate an integrable function, make the product function digital or dial applications. The mathematical calculation then includes all branches of mathematics, statistical calculation ( mean , variance , estimator ) to the integral calculus , the calculus or computer algebra. On logical propositions, operations are the logical operators (and, or, negation, etc.).. A calculation is accurate when the results provided no different from the desired result. Calculating a sum, difference or product can be performed accurately if the starting values are accurate and if the size of the number does not exceed the computing capacity. However, it is common for the calculation of a quotient or root can only lead to approximate value. This is called approximate calculation. It often seeks to provide, with the result approached, an increase of the error. For example, 7 / 3 is approximately equal to 2.33 with an error less than 0.01 by default, or else is approximately equal to 256/81. This approximate calculation of was known to Egyptians as the seventeenth century BC. AD . Some calculations of area and volume can be carried out only approximate value. The approximate calculation appears very early in the history of computing. He is responsible for creating tables of numerical values approximate: table of sines in India and in Arabic mathematicians , tables of logarithms in Europe in the seventeenth century . It is an object of study in Europe from the seventeenth century with the development of functions in power series , and research values approached zero of a function. It is still very present and linked to the capabilities of computers. Long, the calculation called human operators, although they were assisted by mechanical aids such as the abacus or the abacus. Complex calculation methods are described very early by using algorithms that free the user from the search process for him to leave the steps of the calculation to make. This applies, for example the algorithms contained in the Babylonian mathematics or the The Nine Chapters on the Mathematical Art in China (263). Things have changed with the advent of automatic calculation. The changing rules of calculation in mathematics has led to the discovery of new algorithms , which break the simple instructions. These new methods are fundamental in computer science and robotics , and are widely used by other sciences such as physics or chemistry. The objects of the calculations
Transactions
Calculating exact and approximate calculation
Tools to calculate
References
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