A cone (or cone section) is a field curve obtained by cutting a cone with a plane through the vertex of the cone. There are three possibilities, depending on the relative position of the cone and field
If there are no lines parallel to the cone, the intersection is a closed curve, called an ellipse. If one line of the cone parallel to the plane, the intersection is an open curve with both ends asymptotic parallel, is called a parabola. Finally, there may be two parallel lines on the cone into the field, the curve in this case has two open, and is called a hyperbola.Conic section between the curves of the oldest, and is the oldest math subjects studied systematically and thoroughly. Conic sections have been discovered by Menaechmus (a, Greece c.375-325 BC), the teacher of Alexander the Great. They are nurtured in an attempt to overcome the three famous problems trisecting angle, duplication of the cube, and squaring the circle.Conic was first defined as the intersection: straight circular cone vertices from different angles, a plane perpendicular to the conical element. (An element of the cone is himpunaniap lines that form a cone) Depending on the angle is less than, equal to, or greater than 90 degrees, respectively obtained ellipse, parabola, or hyperbola.Appollonius (c. 262-190 BC) (known as The Great geometry) extended the previous results and the consolidation of a wedge cone conic section monograph, which consists of eight books with 487 propositions. Euclid's Appollonius' wedge cone Parts and Elements may represent the quintessence of Greek mathematics. Appollonius is the first to third base the theory of conic sections in the part of the circular cone, right or oblique. He also give the names ellipse, parabola, and hyperbola.In Kepler's laws of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed to high levels of conic sections. Many later mathematicians have also made contributions conic sections, particularly in the development of projective geometry where conic sections are the fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Slice the cone is a classic rich discussion that has prompted many developments in the history of mathematics.
"Pythagorean Theorem" named by the ancient Greek mathematician Pythagoras, who is considered as the person who first gave proof of this theorem. However, many people who believe that there is a special relationship between the sides of a right triangle long before Pythagoras found it. Pythagorean Theorem plays a very significant role in various fields related to mathematics. For example, to form the basis of trigonometry and arithmetic form, which form combines geometry and algebra.This theorem is a relation in Euclidean geometry among the three sides of a right triangle. It is stated that 'the number of square formed from long double-sided elbow will be equal to the number of square formed from long hipotenusanya'.Systematically, this theorem is usually written as: a ^ 2 + b ^ 2 = c ^ 2, where a and b represent the length of the two other sides of a right triangle and c represents the length of hipotenusnya (the hypotenuse).
Approximately 4000 years ago, a clay tablet Babylonian origin was found with the following text: "4 and 5 is the long diagonal. In addition, the Chinese also knew this theorem. This is due Tschou-Gun who lived in 1100 BC. He knows the characteristics of a right angle. This theorem is also known as Caldeans or "theorem Gougu '. It has long ago proved that they are aware of the fact that a triangle with a length of 3, 4, and 5 should be a right triangle siku.mereka use this concept to construct a right angle and right-angled triangle design by dividing the length of rope into 12 equal parts, as the first side of a triangle is 3, the second side is 4, and the third side is 5 units long.
In addition, the Egyptians knew that a triangle with sides 3, 4, and 5 makes 90 ° angle. As a matter of fact, they have a rope with 12 evenly spaced knots like this: That they are used to build the perfect angle in buildings and pyramids. It is believed that they only know about 3, 4, 5 triangle and not a general theorem that applies to all right-angled triangles. Although the theorem is known far in prehistory, but Pythagoras was the one who made it popular. That is why known as the Pythagorean Theorem.Pythagorean Theorem is associated with the first geometrical demonstration. There are hundreds of purely geometric demonstrations as well as limited evidence of algebra.Pythagorean theorem is one of the most important theorems in the world geometry.
Around 2500 years BC, Megalithic monuments in Egypt and northern Europe there is the composition of a right triangle with sides of a round. Bartel Leendert van der Waerden hypothesized that Triple Pythagoras identified algebraically. During the reign of Hammurabi the Great (1790-1750 BC), the Mesopotamian tablet Plimpton 32 consists of many posts related to the Pythagorean Triple. Pythagoras (569-475 BC) using algebraic methods to build a Pythagorean Triple. According to Sir Thomas L. Heath, no research because of this theorem. However, writers such as Plutarch and Cicero teoroma attribute to Pythagoras until the attribution was widely known and accepted. In 400 BC, Plato established a method to achieve a good Pythagorean Triple combined with algebra and geometry. Around 300 BC, Euclid eleman (axiomatic proof of the oldest) presents the theorem.Chinese text Chou Pei Suan Ching, written between 500 BC to 200 AD after having visual proof of Pythagoras or Teoroma called "Gougo Theorem" (as known in China) for the triangular measuring 3, 4, and 5. During the Han Dynasty (202-220 BC), Pythagorean triples appear in nine chapters on mathematical art along with the title of a right triangle. However, this has not been confirmed whether Pythagoras was the first to discover the relationship between sides of right triangles, because there is no text written by him was found. However, the name of Pythagoras was believed to be an appropriate name for this theorem.
Sources:http://library-math.unm.ac.id/blog/?p=220
In about 300 BC Euclid wrote The Elements, a book that became one of the most famous books ever written. Euclid stated five postulates which he based all of his theorem. Euclid's fifth postulate, which reads:"If two lines are drawn so that they intersect a third in such a way that the sum of interior angles on one side less than two right angles, then those two lines, if extended far enough, must intersect each other on a particular side".It is clear that the fifth postulate is different from the other four.That does not satisfy Euclid and he tried to avoid its use as long as possible, in fact the first 28 propositions of The Elements proved without using it. Another comment that appears at the moment is that of Euclid and many pengikutinya, assume that the straight line is infinite.
Proclus (410-485) wrote a comment on The Elements where he comments on the evidence to try to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced evidence of 'false'. Proclus then goes on to give false evidence of his own. But he did not give the following theorem which himpunanara with the fifth postulate.Playfair axiom: "Given a line and a point not on that line, it is possible to draw exactly one line through a point parallel to the line."Although well known from the time of Proclus, this became known as the Axiom himpunanelah Playfair John Playfair wrote the famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate with these axioms.Many attempts were made to prove the fifth postulate from the other four, many of them are accepted as evidence for a period of time until the error was discovered. Always a mistake that's assuming some 'obvious' property that was himpunanara with the fifth postulate. evidence 'One such' given by Wallis in 1663 when he thought he had concluded the fifth proposition, but he really shows it is himpunanara with:"To himpunaniap triangle, there is a similar triangle of arbitrary magnitude."
One proof attempt was more important than most others. It was produced in 1697 by Girolamo Saccheri. The importance of Saccheri's job is that he is considered the fifth postulate false and attempted to get a contradiction.Here is a Saccheri quadrilateral
In figure Saccheri proved that the angle of the peak in the D and C is a testament to equal.The using the properties of congruent triangles which Euclid proved in Proposisi4 and 8 are proved before the fifth postulate digunakan.Saccheri have shown:a) The apex angle is> 90 ° (the hypothesis of an obtuse angle).b) apex angle is <90 ° (the hypothesis of acute angle).c) is the apex angle = 90 ° (the hypothesis of right angle).
Euclid's fifth postulate is c). Saccheri proved that the hypothesis of obtuse angle implied the fifth postulate, thus getting a contradiction. Saccheri then studied the acute angle hypothesis and many theorems are derived from non-Euclidean geometry without realizing what he was doing. But he finally 'prove' that the hypothesis of acute angle causes a contradiction with the assumption that there is a 'point at infinity' which is located in the area.
In 1766 Lambert followed a similar line to Saccheri. But he did not fall into the trap that Saccheri fell into and investigate hypotheses acute angle without obtaining a contradiction.Lambert noticed that, in this new geometry, the number of triangles increases as the angle decreases the triangle area.Legendre spent 40 years of his life working on the parallel postulate and the work appears in the annex to the various editions of its geometry very successful book Elements de Géométrie. Legendre proved that Euclid's fifth postulate is set by the number of angle triangle equals two right angles. Legendre showed, Saccheri has more than 100 years earlier, that the number of angle triangle can not be more than two right angles.This, once again like Saccheri, rested on the fact that an infinite straight line. In trying to show that the angle can not be less than 180 °, Legendre assumed that through a set of points in the interior angles is always possible to draw a line that meets both sides of the angle. This turned out to be another form himpunanara with the fifth postulate, but Legendre never realize his own mistakes.
Basic geometry is currently inundated with problems in the parallel postulate. D'Alembert, in 1767, called it scandalous basic geometry. The first person to really come to understand parallel problems is Gaussian. He began working on the fifth postulate in 1792 while only 15 years old, at first trying to prove the parallel postulate from the other four. In 1813 he had made little progress and writes:"In the theory of parallels we are even now not further than Euclid.This is a shameful part of mathematics ... "
But with 1817 Gauss had become convinced that the fifth postulate is independent of the other four postulates. He began working out the consequences of geometry in which more than one line can be drawn through a point parallel to a given line.Perhaps most surprising of all, Gauss never published this work, but keep it secret. At the time thought was dominated by Kant who has stated that Euclidean geometry is the inevitable requirement of thought and Gauss disliked controversy.
Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made false evidence several parallel postulate. János Bolyai Farkas Bolyai taught his son mathematics. In 1823 János Bolyai wrote to his father, saying that he knew that the Gaus has found the problem before but did not publish it. János Bolyai took two years to publish his book.
Work Bolyai reduced because Lobachevsky published a work on non-Euclidean geometry 1829. Neither Bolyai nor Gauss knew Lobachevsky work, mainly because only published in Russian in the Kazan Messenger a local university publications.Lobachevsky fared no better than Bolyai in gaining public recognition of the importance of work. He published an investigation on the geometric theory of parallel in 1840 that in 61 pages, provide the clearest record of the work of Lobachevsky.
Publishing accounts in France in Crelle's Journal in 1837 brought his work in non-Euclidean geometry mathematics community wide audience but not ready to accept ideas so revolutionary.In 1840 he Lobachevsky booklet explains clearly how the non-Euclidean geometry works."All of a straight line in the exit plane of the point can, by referring to the straight line given in the same plane, divided into two classes - into cutting and non-cut. This line of one and another class of the line will be called parallel to a given line. "
Here is a diagram of Lobachevsky
Hence Lobachevsky has replaced the fifth postulate of Euclid with Lobachevsky Parallel Postulate:"There are two parallel lines with a given line through a given point not on the phone."
Sources:
http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.htmlhttp://en.wikipedia.org/wiki/Non-Euclidean_geometry
Euclidean geometry is often called parabolic geometries, ie geometries which follow a set of propositions that are based on five postulates of Euclid has defined in his book The Elements.More specifically, Euclidean geometry is different from other types of geometry in the fifth postulate, often referred dengaan parallel postulate. Non-Euclidean geometry of the fifth postulate is replaced by one of two alternative postulates, and leads to the geometry of hyperbolic or elliptic geometry. There are two types of Euclidean geometry: the geometry of the field, which is a two-dimensional Euclidean geometry and solid geometry, which is a three-dimensional Euclidean geometry.
The five postulates of Euclid can be expressed as follows:1) It is possible to draw a straight line segment joining two points.2) It is possible to permanently extend the straight line segment himpunaniap continuously in a straight line.3) Given himpunaniap straight line segments, it is possible to have menggambarlingkaran segment as radius and one endpoint as center.4) All right angles with each other or congruent.5) If two lines are drawn so that they intersect a third in such a way that the sum of interior angles on one side less than two right angles, then those two lines, if extended far enough, must intersect each other on a particular side.
The fifth postulate is known as the parallel postulate. This parallel postulate states that given the set of straight line segments and each point is not that the line segment, there is one and only one straight line that passes through it and never cut the first row, no matter how far extended line segments. Although Euclid's fifth postulate can not be proved as a theorem, for years many claimed evidence was published. Many efforts are aimed to formulate the theorem for this postulate because it is necessary to prove an important result and it does not seem as intuitive as the other arguments. More than two thousand years of research the fifth postulate was found to be independent of the other four.This is the fifth postulate is to be continued to Euclidean geometry to be considered.
In 1826 Nikolay Lobachevsky and János Bolyai in 1832, developed independennon-Euclidean geometries are fully self-consistent in which the fifth postulate does not hold. Johann Carl Friedrich Gauss had found this but the results are not published.Euclid tried to avoid using the fifth postulate and succeeded in the first 28 propositions of The Elements, but for the proposition 29 he needed it. Part of geometry that can be derived using only the first four postulates of Euclid became known as absolute geometry. As stated above, the fifth postulate and hence describe the parallel postulate of Euclidean geometry. If part of the parallel postulate is replaced with "no line through the" geometry of the elliptical or round dijelaskan.Jika part of the parallel postulate is replaced by "at least two lines there came to pass that through the point that" the hyperbolic geometry are described.
As stated above, two types of Euclidean geometry, solid geometry and geometry of the field, which is clearly different.Geometry of the field is part of the geometry in two-dimensional space associated with an overview of the field, such as lines, circles and polygons. Solid Geometry is a part of the geometry of three-dimensional space associated with solid foods, such as polyhedra, spheres, and lines and fields. In both types of the fifth postulate of Euclidean geometry according to Euclid, but each figure illustrates the various types of space. Euclidean space is the space of all n-tuples of real numbers and is denoted as R ^ n.This space is a vector space and has a topological dimension (Lebesgue covering dimension) n (see topology). Contravariant and the number of covariance that himpunanara in Euclidean space. R ^ 1 is the real line, namely line with the scale fixed in accordance with the number of unique points on the line.Generalization real line in two-dimensional space is called Euclidean field and is denoted R ^ 2.
Sources:http://www.bookrags.com/research/euclidean-geometries-wom/
Here are explained the history and development of the number (number theory) from the ancient time until being used now.
a. History of Ancient MathematicalAt first, in ancient times, many nations who reside along the major rivers. The Egyptians along the Nile in Africa, the people of Babylonia along the river Tigris and Eufrat, Hindu race along the river Indus and the Ganges, the Chinese people along the Huang Ho and the Yang Tze. Nations is in need of skills to deal with floods, drying the marshes, making irrigation to cultivate the land along the river into the agricultural area for the required practical knowledge, that knowledge and mathematical techniques together.History shows that the initial Math from people who reside along the river flow. They require calculations, removal of which can be used in accordance with the changing seasons. Necessary measuring instruments to measure Persil Persil-owned land. The increase of civilization requires evaluating the trade, finance and tax collection. For practical needs it is needed the numbers.The number was originally used only to remember the number, but in the long himpunanelah treasury specialists add mathematical symbols and the right words to define the number then becomes the subject of mathematics is essential for life and we can not pungkiri that in everyday life we will always meet with the name of, because the number is always required both in technology, science, economy or the world of music, philosophy and entertainment, and many other aspects of life.Number of previously used as a symbol to replace an object such as pebbles, twigs, each tribe or nation has its own way to describe the number in the form of symbols.In further development, the X ditemukanlah century Spanish manuscript that contains the number of symbols written by ancient Hindu-Arabic nations and style of writing that has been a symbol of the embryo of writing we used so far.
b. Development of Number Theory1) Number Theory Babylonia quarterBabylonian mathematics refers to the mathematics developed by the people of Mesopotamia (now Iraq) since the beginning of the Sumerian to the beginning of Hellenistic civilization. Called the "Babylonian Mathematics" because the main role of the area of Babylonia as the place to learn. At the time of Hellenistic civilization, Mathematics Babylonian Mathematics united with Greece and Egypt to raise the Greek Math. Then under Islamic Kekhalifahan, Mesopotamia, specialized Baghdad, once again became an important center of Islamic study Mathematics.Contrary to langkanya resources on Egyptian Mathematics, Babylonian Mathematical knowledge passed down from more than 400 plates of clay excavated since the 1850s. Written in nail plates while still wet clay, and baked in the oven or dried in the sun. Some of them were home-based work.Terdini mathematical proof is the work of the people writing the Sumerians, who developed an ancient civilization in Mesopotamia. They developed complex system of metrology since 3000 BC. From about 2500 BC to the face, the Sumerian people write multiplication tables on clay plates and dealing with geometrical exercises and sharing issues. Terdini Trail system also refers to the number of Babylonia during this period.Most of the clay plates are known to originate from the years 1800 until 1600 BC, and covers the topics of fractions, algebra, quadratic and cubic equations, and calculation of the number of regular, inverse multiplication, and the number of prime twins.Plates also include multiplication tables and linear equation solving methods and quadratic equations. 7289 BC Babylonia plates give the approximate to √ 2 accurate to five decimal places.Babylonian mathematics was written using a system of sexagesimal (base-60). From here down the use of 60 seconds to a minute, 60 minutes to an hour, and 360 (60 x 6) degrees to a circle rotation, the use of seconds and minutes on the arc of a circle represents the fraction of degrees. Also, unlike the Egyptians, Greeks, and Roman, Babylonian people have a system where the true value, where the numbers are written in the left column specifies the value that is larger, as in the decimal system
2) Theoretical Number In Ancient Egypt EthnicsEgyptian mathematics refers to mathematics written in the language of Egypt. Since civilization Hellenistic Egyptians melt with math math Greek and Hellenistic Babylonia who raised Math. Continues the study of mathematics in Egypt under the Islamic Caliphate as a part of Islamic mathematics, when Arabic became the written language of Egyptian intellectuals.Egypt's mathematical writings is how long is the Sheet Rhind (sometimes also called "Ahmes Sheet" by the author), is estimated to originate from 1650 BC, but probably the sheet is a copy of older documents from the Central Government, namely the years 2000-1800 BC . User instruction sheet is for students Arithmetic and geometry. In addition to providing extensive formula and methods of multiplication, sharing, and processing breakdown, is also a proof sheet for other mathematical knowledge, including composite and prime numbers; Arithmetic average, geometric, and harmonic and simple understanding of the Sieve of Eratosthenes and theory of perfect (ie, number 6).The sheet also includes how to solve linear equations of the line order Arithmetic and geometry.Other important Egyptian mathematical manuscripts are sheets Moscow, also from the Middle Kingdom period, dated about 1890 BC. This script contains the word or question about the story, which perhaps is intended as entertainment.
3) Theory of Numbers In India EthnicsSulba sutras (about 800-500 BC) is the geometry of the writings of using irrational numbers, prime numbers, the order of three cubic root; calculate the square root of 2 to a portion of one hundred thousand; provide a wide circle construction method approaches the square given, solving linear and quadratic equations; develop Pythagorean triples algebraically, and provide numerical evidence for the statement and the Pythagorean theorem.About the 5th century BC to formulate rules of Sanskrit grammar using the same notation with modern mathematical notation, and using meta rules, transformations, and recursion. Pingala (about the 3rd century until the first century BC) in the pamphlet prosodynya use in accordance with the number of binary systems. Pembahasannya about kombinatorika compatible with a basic version of the binomial theorem. Pingala paper also contains a basic idea of the number of Fibonacci.At about the 6th century BC, Pythagoras developed a group of properties is complete (perfect number), the number bersekawan (amicable number), the number of prime (prime number), the number of triangles (triangular number), the number of squares (square number), the number of hexagons (pentagonal number) and the numbers of polygon (figurate numbers) to another. One of the features of the famous triangle until now called Pythagorean triple, ie: aa + bb = cc of discovery through the calculation of the broad area of square sides are the sides of the triangular square with sloping sides (hypotenosa) is c, and the other side is a and b. Other findings were very popular until now is the classification of prime and composite numbers.The number of prime is a positive integer greater than one that does not have positive factors except 1 and the number itself.Positive number other than one and the other prime number called a composite number. Historical records show that the problem of prime numbers has attracted the attention of mathematicians for thousands of years, especially with regard to how many prime numbers and how the formula can be used to find and make a list of prime numbers.With the expansion of literacy and numeracy systems, methods and procedures developed are aritmetis for track work, particularly to address the general problem, through specific measures, which clearly referred to the algorithm. First the algorithm worked by Euclid.At around 4 century BC, Euclid developed the concepts of geometry and the theory of policy. Book VII of Euclid to take an algorithm to find the Greatest Federation factor of two positive integers using a technique or procedure that efficiently, through a finite number of steps. Word comes from the algorism algorithm.At the time of Euclid, the term is not known. Algorism said came from the name of a famous Muslim and the author of 825 in M., which is Abu Ja'far Mohammed ibn Musa al-Khowarizmi. The end of his name (Al-Khowarizmi), inspired the birth of the term Algorism. Algorithm in terms of vocabulary at the beginning of most of the computer revolution, that is the end of 1950.In the 3rd century BC, marked by a number of theoretical development work Erathosthenes, now known as Screening Erastosthenes (The Sieve of Erastosthenes). In the next six centuries, Diopanthus published a book called Arithmetika, which discusses solving the equations in whole numbers and rational number, in the form of a symbol (not the form / up geometrically, as developed by Euclid). With the work of this symbol, referred to as one of Diopanthus founder of algebra.
4) Theory of Numbers The History of Time (AD)Early rise of modern number theory pioneered by Pierre de Fermat (1601-1665), Leonhard Euler (1707-1783), JL Lagrange (1736-1813), AM Legendre (1752-1833), Dirichlet (1805-1859), Dedekind (1831-1916), Riemann (1826-1866), Giussepe Peano (1858-1932), Poisson (1866-1962), and Hadamard (1865-1963). As a prince of mathematics, Gauss was so entranced to the theory of beauty and charm, and to melukiskannya, it mentions the theory of numbers as the queen of mathematics.At this time, the theory is not only expanding the extent of the concept, but also much applied in many fields of science and technology. This can be seen on the utilization of the concept of the method of lines code, cryptography, computer, and so forth.
c. History of Zero FiguresIntroduced as the number of zeros, and as a symbol to fill the empty space the first time by al-Khwarizmi. Zero (0) is in the English language that could mean zero is empty or blank.Around the year 300 BC the Babylonian had started with two slashes (/ /) to indicate an empty place, an empty column in Abacus. This symbol provides an easy way to determine the place of a symbol. Zero is very useful and is a symbol that describes an empty spot in Abacus, a column with stones placed at the bottom. Its purpose is to ensure that these items are in the right places, the number zero does not have a numeric value of its own.At zero computer can harm the system, because there is no zero mean. Whatever the number multiplied by zero the result is not there. Well this is confusing the calculation operations. Note that this example:0 = 0 (zero equal to zero, true)0 x 3 = 0 x 89 (both zero multiplied by a number, because it will be worth zero)(0 x 3) / 0 = (0 x 89) / 0 (a number divided by the number of the same, will be worth it)3 = 89 (???, these results are confusing)Zero conflict with one of the key principles of western philosophy, a dictum which terhujam roots in the philosophy of numbers and the importance grows Phythagoras of Zeno's paradox. the Cosmos Greek erected on pillars: there is no vacancy.December = Greek cosmos created by Phytagoras, Aristotle and are still enduring Ptolemeus himpunanelah Greek civilization collapse. In this cosmos there is no unavailable. Therefore, most of the two Millennium western people not willing to accept zero.Frightening consequences. The absence of zero inhibits the development of mathematics, science and hinder innovation even more dangerous, demoralize removal system.
Resources:http://eduklinik.info/2010/11/20/sejarah-teori-bilangan/http://translate.google.co.id/translate?hl=id&langpair=en|id&u=http://en.wikipedia.org/wiki/Number
Mathematicians have been using the set since the beginning of the subject. For example, the Greek mathematician defines a circle as the set of fixed points at a distance r from a fixed point P. However, the concept of 'infinite set' & limited 'to avoid the set of mathematicians and philosophers for centuries. For example, the Hindu mind is understood in their infinite Ishavasy-opanishad scripture text as follows: "Overall there. The whole being here. From the hole imanates whole. Get rid of the whole of the whole, what remains is still a Whole". Phythagoras (~ 585 -500 BC), a Greek mathematician, good and evil associated with the finite and infinite, respectively. Aristotle (384-322 BC) said, "infinity is not perfect, not yet finished and therefore, unthinkable, it's out of shape and confused. "Roman Emperor and philosopher Marcus Aqarchus (121-180 AD) said infinity is a fathomless gulf, where everything is gone" philosopher. British Thomas Hobbes (1588-1679) said, "When we say something is infinite, we signify only that we can not conceive ends and boundaries thing named".
Mathematicians work, as well as roads, rarely associated with unusal question: what numbers? However, attempts to answer this question it has driven a lot of work by the mathematician and philosopher at the foundations of mathematics during the last hundred years. Characterization of integers, rational and real numbers has become a classic problem of the center for the study of Weierstrass, Dedekind, Kronecker, Frege, Peano, Russell, Whitehead, Brouwer, and others. Researchers from the Georg Cantor around 1870 in a theory with infinite series and related topics of analysis provides a new direction for the development of set theory. Cantor, who is usually regarded as the founder of set theory as a mathematical discipline, led by his work into consideration the set of infinite or arbitrary character classes.
However, Cantor's results are not immediately accepted by his contemporaries. Also, it was found that the definition of the set leads to logical contradictions and paradoxes. The most famous among these is given in 1918 by Bertrand Russell (1872-1970), now known as Russell's paradox.
In an effort to resolve this paradox, mathematicians first reaction is to 'axiomatize' Cantor's intuitive set theory. Axiomatization is as follows: starting with a clear set of statements called axioms, the truth is assumed, one can deduce all the remaining propositions of the theory of logical axioms using the inference axioms. Russell and Alfred North Whitehead (1861-1974) in 1903 proposed the axiomatic set theory in a three-volume work called Principia Mathematicians find it awkward to digunakan.Sebuah axiomatic set theory can be done and logistics is fully given in 1908 by Ernst Zermello ( 1871 to 1953).wa was increased in 1921 by A. Fraenkel Ibrahim (1891-1965) and T. Skolem (1887-1963) and now known as' Zermello-Frankel (ZF)-axiomatic set theory.
Sources:
http://www-groups.dcs.st-and.ac.uk/ ~ history / HistTopics / Beginnings_of_set_theory.htmlhttp://en.wikipedia.org/wiki/Set_theoryhttp://www.mathresource.iitb.ac.in/project/history.htmhttp://stanford.library.usyd.edu.au/entries/set-theory/
