Embed Size (px)
Transcript of maths project
INVENTION OF MATHSALGORITHMSLOGARITHMSARETHMETIC PROGRESSION
INVENTION OF MATHSMathematics(fromGreekmthma, knowledge, study, learning) is the abstract study of subjects encompassingquantity,structure,space,change,and more;it has no generally accepteddefinition.Mathematiciansseek outpatternsand formulate newconjectures. Mathematicians resolve the truth or falsity of conjectures bymathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work ofGiuseppe Peano(18581932),David Hilbert(18621943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishingtruthbyrigorousdeductionfrom appropriately chosenaxiomsanddefinitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions about nature.
Through the use ofabstractionandlogicalreasoning, mathematics developed fromcounting,calculation,measurement, and the systematic study of theshapesandmotionsof physical objects. Practical mathematics has been a human activity for as far back aswritten recordsexist.Rigorous argumentsfirst appeared inGreek mathematics, most notably inEuclid'sElements. Mathematics developed at a relatively slow pace until theRenaissance, when mathematical innovations interacting with newscientific discoveriesled to a rapid increase in the rate of mathematical discovery that has continued to the present day.Galileo Galilei(15641642) said, 'The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth'.Carl Friedrich Gauss(17771855) referred to mathematics as "the Queen of the Sciences".Benjamin Peirce(18091880) called mathematics "the science that draws necessary conclusions".David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."Albert Einstein(18791955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".
Mathematics is used throughout the world as an essential tool in many fields, includingnatural science,engineering,medicine, and thesocial sciences.Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such asstatisticsandgame theory. Mathematicians also engage inpure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
ALGORITHMSAlgorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously." \l Algorithmic information theory principally studies complexity measures on strings (or other data structures). Because most mathematical objects can be described in terms of strings, or as the limit of a sequence of strings, it can be used to study a wide variety of mathematical objects, including integers and real numbers.This use of the term "information" might be a bit misleading, as it depends upon the concept of compressibility. Informally, from the point of view of algorithmic information theory, the information content of a string is equivalent to the length of the shortest possible self-contained representation of that string. A self-contained representation is essentially a program in some fixed but otherwise irrelevant universal programming language that, when run, outputs the original string.Turing himself was fascinated with how the distinction between software and hardware illuminated immortality and the soul. Identifying personal identity with computer software ensured that humans were immortal, since even though hardware could be destroyed, software resided in a realm of mathematical abstraction and was thus immune to destruction.
An infinite binary sequence is said to be random if, for some constant c, for all n, the Kolmogorov complexity of the initial segment of length n of the sequence is at least n c. It can be shown that almost every sequence (from the point of view of the standard measure "fair coin" or Lebesgue measure on the space of infinite binary sequences) is random. Also, since it can be shown that the Kolmogorov complexity relative to two different universal machines differs by at most a constant, the collection of random infinite sequences does not depend on the choice of universal machine (in contrast to finite strings). This definition of randomness is usually called Martin-Lf randomness, after Per Martin-Lf, to distinguish it from other similar notions of randomness. It is also sometimes called 1-randomness to distinguish it from other stronger notions of randomness (2-randomness, 3-randomness, etc.).(Related definitions can be made for alphabets other than the set .)AlgorithmAn algorithm is any well-defined procedure for solving a given class of problems. Ideally, when applied to a particular problem in that class, the algorithm would yield a full solution. Nonetheless, it makes sense to speak of algorithms that yield only partial solutions or yield solutions only some of the time. Such algorithms are sometimes called "rules of thumb" or "heuristics."Algorithms have been around throughout recorded history. The ancient Hindus, Greeks, Babylonians, and Chinese all had algorithms for doing arithmetic computations. The actual term algorithm derives from ninth-century Arabic and incorporates the Greek word for number (arithmos ).
Algorithms are typically constructed on a case-by-case basis, being adapted to the problem at hand. Nonetheless, the possibility of a universal algorithm that could in principle resolve all problems has been a recurrent theme over the last millennium. Spanish theologian Raymond Lully (c. 12321315), in his Ars Magna, proposed to reduce all rational discussion to mechanical manipulations of symbolic notation and combinatorial diagrams. German philosopher Gottfried Wilhelm Leibniz (16461716) argued that Lully's project was overreaching but had merit when conceived more narrowly.The idea of a universal algorithm did not take hold, however, until technology had advanced sufficiently to mechanize it. The Cambridge mathematician Charles Babbage (17911871) conceived and designed the first machine that could in principle resolve all well-defined arithmetic problems. Nevertheless, he was unable to build a working prototype. Over a century later another Cambridge mathematician, Alan Turing (19121954), laid the theoretical foundations for effectively implementing a universal algorithm.Turing proposed a very simple conceptual device involving a tape with a movable reader that could mark and erase letters on the tape. Turing showed that all algorithms could be mapped onto the tape (as data) and then run by a universal algorithm already inscribed on the tape. This machine, known as a universal Turing machine, became the basis for the modern theory of computation (known as recursion theory) and inspired the modern digital computer.Turing's universal algorithm fell short of Lully's vision of an algorithm that could resolve all problems. Turing's universal algorithm is not so much a universal problem solver as an empty box capable of housing and implementing the algorithms placed into it. Thus Turing invited into the theory of computing the very Cartesian distinction between hardware and software. Hardware is the mechanical device (i.e., the empty box) that houses and implements software (i.e., the algorithms) running on it.
It is a deep and much disputed question whether the essence of what constitutes the human person is at base computational and therefore an emergent property of algorithms, or whether it fundamentally transcends the capacity of algorithms.In mathematics and computer science, an algorithm
Thelogarithmof a number is theexponentby which another fixed value, thebase, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3:1000 = 103= 101010.More generally, ifx=by, thenyis the logarithm ofxto baseb, and is writteny= logb(x), solog10(1000) = 3.LOGARITM
Logarithms were introduced byJohn Napierin the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily, usingslide rulesandlogarithm tables. These devices rely on the factimportant in its own rightthat the logarithm of aproductis thesumof the logarithms of the factors:The present-day notion of logarithms comes fromLeonhard Euler, who connected them to theexponential functionin the 18th century.The logarithm to baseb= 10is called thecommon logarithmand has many applications in science and engineering. Thenatural logarithmhas theconstante( 2.718) as its base; its use is wi