Selasa, 13 Januari 2009

MATHEMATICS HISTORY 2004

MATHEMATICS HISTORY 2004

Mathematics Figure Which Have Meritorious During The Year 2004.

1. Hans Wospakrik

Hans Jacobus Wospakrik (Serui, Papua, 10 September 1951 - Jakarta, 11 January 2005) is a Indonesia physics representing lecturer of physics teoritik in Technological Institute of Bandung.

Hans is a getting best physics appreciation by University Atma Jaya of Jakarta at the service of, consistency, and its dedication is high in research in area of theory physics. He give the contribution mean to community of world physics in the form of mathematics method utilize to comprehend the physics phenomenon in elementary particle and Common Relativity of Einstein. its Research Pickings this publicizing of in notable international journal, like Physical Review D, Journal Of Mathematical Physics, Modern of Physics Letters A, and Modern International Journal of of Physics. He die at 11 January 2005 effect leukimia.

From Atomos Till Quark.

From Atomos Till Quark is a book of result of masterpiece Hans narrating to regarding the seeking of human being as long as history of concerning smallest compiler from this natural items. Early Greek where all that moment philosopher philosophize to regarding the smallest compiler each;every items, Jazirah Arab touched by Hans as owner " knowledge torch" next after Greek, science alkemi, nuclear reaction " narrating" at us of about atom existence, proton and neutron, until finding in this time regarding set of smaller items, that is quark.

2. Christa Lorenzia

Christa Lorenzia Soesanto born in Jakarta, 21 October 1996, is a Indonesia student having many achievement in the field of mathematics. and Computer. First Child from couple of Edy Soesanto Prawirohardjo Intelbc International and this Betsy Eliane Rahardjo, succeeding well reputed of Indonesia with the champion of various competition of mathematics and Computer of either in national storey;level and also international. SINCE CHILDHOOD, Christa Lorenzia always install the goals. Either in school and also moment follow the championship, he always want to be best. Proven, he always become the public champion from play grup till pass the elementary school. Even, in various ajang competition, he also reach for the predikat champion, especially mathematics race. Christa hope can be well reputed of nation and state, so that Indonesia people do not is always stamped stupid. According to him, oftentimes Indonesian nation stamped to by lower by other nation even by nation by xself. This matter of course bother its mind as Indonesia citizen capable to show its achievement. Since October 2008, Christa have made the new champion [in] world of Mathematic Olympiad, where Christa have succeeded to get FOUR Gold Medal to Alone and TWO Gold Medal for the Beregu of alternately. This champion is first multiply made by a Indonesian nation child.

IQ

Christa start to carve the achievement of since siting in class 1 Beautiful elementary school Tirta Marta BPK Maisonette Dredger, Jakarta. He is which is on initially following as participant of Competition of Mathematics of Kumon year 2004 which is carried out in Jakarta, in the reality succeed to reach for the champion I, and also get the present of journey of a family to Malaysia. Christa is it is true pertained by a smart child, He have the IQ 150. Although often follow various race, He under developed of Iesson have never in school. Values Christa in mean school is 9.6. Even assess mathematics in raportnya always 100.

· A lot of is reached for by achievement of Christa, good in mathematics competition mount the national and also international. Achievement most glorious Christa is moment succeed to reach for the gold medal at the same time appreciation as " The Best Theory" and " the Best Overall" in Olympiad of Mathematics and National Science (OSN) Year 2007 in Surabaya. Beside that, He is noted by as young participant. Museum Rekor-Dunia Indonesia (MURI) later;then register its achievement as first woman is which reach for the gold medal in IMSO (international) and OSN (national)

· In " Po Leung Kuk Primary Mathematics World Contest" ke-12 in Hongkong [of] [at] 12-16 July 2008, Christa succeed to reach for the gold medal with the Perfect Score. In ajang followed [by] 43 team from 15 the state, Indonesia reach for 5 gold medal [pass/through] the Stefano Chiesa Suryanto (SD Theresia Jakarta), Richard Akira Heru (ex SD PL Bernadus Semarang/smp PL Domenico Savio Semarang), Peter Tirtowijoyo Young (ex SD Santa Maria Surabaya/smp Petra 1 Surabaya), Christa Lorenzia Soesanto (ex SD Tirta Martha BPK of Dredger Jakarta / smpk Tirta Martha BPK of Dredger Jakarta), and Fransisca Susan (ex SD Santa Ursula Jakarta/smp Santa Ursula Jakarta).

· In "International Mathematic Olympiad" in Chiang Mai - Thailand of at 25-30 October 2008, Christa have succeeded to become one the other one a success Indonesian nation child grab the Gold Medal of good to Alone and also beregu.

Trace Concept

Role of its parent in educating Christa till reach this achievement is very big. Its mother always remind of concept TRACE (Goals, Reward, Appreciation, Care And Extra Miles). Old fellow Personating ' supporter' and non ' controller' so that Christa can enjoy all goals and all contest preparation. At its age is which youth, Christa Firmness have want to win the Nobel Prize of Prize in the field of computer which is also mastered by its father.

3. Frank Wilczek

Franc Wilczek born 15 May 1951 is a United States physics. H. is David Politzer and David Gross he is awarded by the Appreciation Nobel in Physics 2004.

Research.

In the year 1973, Wilczek, a laboring master student by David is Jonathan Gross in University Princeton, finding freedom asimtot, mentioning that more and moreing the near by quark one another, more and more to weaken the strong interaction (or colour payload) among both; for a while quark [of] [in] back part far, nuclear energy of among both so weaken so that berperilaku almost loo like the free particle. Teori--Yang found independently by Hugh David Politzer--Penting for the development of kromodinamika quantum.

Wilczek have assisted to express and develop the aksion, anyon, freedom asimtot, and other aspect of field theory of quantum in general, and have checked the solid goods physics, astrophysics, and particle physics.

Its research nowadays is:

· particle physics " pure": relation of among idea of teoretis and phenomenon able to be perceived.

· materials behavior: temperature ultra-high and stream structure.

· applying of particle physics to cosmology.

· applying of technique of field theory to solid goods physics.

· quantum theory black hole.

4. Yohanes Surya

Yohanes Surya born in Jakarta on 6 November 1963. He start to deepen the physics of at majors of Physics of MIPA of Indonesia University till year 1986, teaching in SMAK I of Dredger Jakarta till year 1988 and hereinafter go through the master program and its doctor in College of William and Mary, Virginia, United States. its Master Program is finished in the year 1990 and its doctor program in year 1994 by predikat is cum laude. After getting title Ph.D., Yohanes Surya become the Consultant of Theoretical Physics in TJNAF / cebaf (Continous Electron Beam Accelerator Facility) Virginia - United States (1994). Although have had the Greencard remain and work in United States, Yohanes Surya go home to Indonesia with a purpose to wish well reputed of Indonesia pass the physics olympiad (its password at that time is " Go Get Gold") and also develop the physics in Indonesia.

Come home from America, beside train and lead the Team of Olympiad of Indonesia Physics (TOFI), Yohanes Surya become the instructor and researcher of at program of pasca of master UI for the area of nuclear physics (year 1995 - 1998). From year 1993 till 2007 student binaannya succeed well reputed of nation by menyabet 54 gold medal, 33 silver medal and 42 bronze medal in so many Science competition / International physics. In the year 2006 a student binaannya, Jonathan Pradana Mailoa, succeeding to reach for the predikat " The Absolute Winner" (World Champion) in International Physics Olympiad (Ipho) XXXVII [in] Singapore.

Yohanes Surya represent the productive writer for the area of Physics/mathematics. There is 68 book have been written for the student of SD until SMA. Besides writing book, he also write hundreds of Physics article [in] erudite journal of national goodness and also international, daily of COMPASS, TEMPO, Indonesia Media and others. He also pencetus of term MESTAKUNG and three law Mestakung, and also pencetus of Whirligig study ( Easy, Besotted, Pleasing).

Outside aktifitasnya of above, Yohanes Surya act in so many international organization as Board member of the International Physics Olympiad, Vice of President of The First step to Nobel Prize (1997); conceptor And President Asian Physics Olympiad (2000); Chairman Of The first Asian Physics Olympiad, in Karawaci, Tangerang (2000); officer of Member of the World Physics Federation Competition; Chairman Of The International Econophysics Conference 2002; Chairman The World Conggress Physics Federation 2002; Board Of Experts in magazine of National Geographic Indonesia and also become the Chairman of Asian Science Camp 2008 in Denpasar, Bali. During have career to in area of physics development, Yohanes Surya have got various award/fellowship for example CEBAF/sura award ACE ' 92-93 (one of the best student in the field of nuclear physics at American south-east region), creativity appreciation 2005 from Institution of Creativity Development, Loyal Badge award of Masterpiece Hero (2006) from President of RI Susilo Bambang Yudhoyono. Is same in the year, he chosen as Indonesia proxy in the field of education to come in contact with the United States President, George W. Bush. In the year 2007, he write the book " Mestakung: Successful Secret of World Champion" getting appreciation as quickest writer Best Seller in Indonesia.

mathematics history which still used hitherto (Number of Zero)

mathematics history which still used hitherto

(Number of Zero)



On a day-to-day basis, in fact we do not require the number of zero, really do not butuh. When you asked, ' Have how much/many your orange ?', hence you'd tend to to tell ' I have no the orange' compared to telling ' I have zero orange'. When we haveing a asked. How many year old age your sister/ brother is ?. Hence more opting us to answer Its age newly 1 month;moon' than have to answer with Its age newly 0 year. This is its problem, because in practice we is not at all need the number of zero.

Hence during very old at history of human being journey, number of zero do not emerge. And in the reality number of zero by xself relative not yet too is old found, because it is true 'insignificant'. Guide regarding early human being recognize the calculation found by arkeolog Karl Absolom year 1930 in a cutting of wolf bone - in the reality they more bernyali, because more opting us to use the media of paper of dibading of wolf bone - what [is] estimated old age 30.000 year.

In the reality along the nun of time, they start to string up the number there is. Tribe of Bacairi and Baroro have the system calculate ' one', ' two', ' two and one', ' two and two', ' two and two and one', etc. They have the system number base on two and we now mention it by system is binary - in this time we often study it if we study the system calculation used by a computer. In this time even also we write down eleven as ten and one, etc.

Egypt start to make the sign to show ' one', other sign to show ' five', dsb. Before a period of pyramid, ancient Egypt people have used the picture for the system of number denary - bases ten, my two hand finger - they. Egypt Nation will draw six symbol to note the angaka one hundred twenty three compared to drawing 123 line. Egypt Nation recognized by very mastering of mathematics. Expert Meraka of astral and reliable time marker and even have created the calendar. invention of System of sun calendar represent the big breakthrough and added with the artistic invention of geometry . Though they have reached the high level mathematics, but number of zero in the reality not yet emerged also in Egypt. This because of they use the mathematics to be practical and do not use it for the something that of do not relate to fact.

Later;Then we make a move to Greek. Before year 500 SM, they have comprehended the mathematics is eminently compared to by a Egypt. They also use the bases 10. Greece , for example, writing down number 87 by 2 is symbol, compared to by a Egypt which must write down it by 15 is symbol, what exactly lost ground at number Romawi needing 7 symbol - LXXXVII. If Egypt nation assume the mathematics of only appliance to know the day commutation - with the calendar system - and arrange the division of farm - with the geometry , hence greece look into the serious philosophy and number considerably.

Zeno bearing paradox of very undifinite and Pytagoras is we recognize with the its right triangle theorem - known by that this formula in fact have been known by since 1000 year previously, borne here. We also recognize the Aristoteles and Ptolomeus

Return to east world, Babilonia - Iraq now - in the reality have the system calculate much more ancient go forward. They use the system base on 60, seksagesimal , so that they have 59 sign. differentiating this system with the Egypt and Greek is, that a sign can mean 1, 60, 3600 or other bigger number yg. Them defining appliance assist to count/calculate the abax - soroban in Japan, suan-pan in China, s'choty in Russia, coulbadi in Turki, etc which here we mention with the abaci. System calculate the them of like our system in this time where 222 showing value ' two', ' twenty' and ' two hundreds'. So also symbol i show ' one' or ' six puluh' in two different position. People Babilonia do not have the method to show the correct column to symbol written, whereas by abakus is this matter is shown by easier of which number is which are such. A stone which is located in distinguishable second column easily from stone which is there are in third column and so on. Thereby i can mean 1, 60 or 3600 or assess the larger ones. So that ii earn more intrude again, because bsa mean 61, 3601, dsb. Hence needed by penanda and they use ii as vacant lot, a empty column at abakus. So that now ii mean 61 and iiii mean 3601. Although they have found the empty penanda column by ii, but in fact number of zero just remain to be not yet emerged at culture ini.ii remain to haven't assessed the separate numerik.

Return to east world, Babilonia - Iraq now - in the reality have the system calculate much more ancient go forward. They use the system base on 60, seksagesimal , so that they have 59 sign. differentiating this system with the Egypt and Greek is, that a sign can mean 1, 60, 3600 or other bigger number yg. Them defining appliance assist to calculate the abax – soroban in Japan, suan-pan in China, s'choty in Russia, coulbadi in Turki, etc which here we mention with the abaci. System calculate the them of like mathematics Provenance in India still impersonate. A text written in the year 476 M show the influence of Greek mathematics, Egypt And Babilonia brought by Alexander of conquest moment it. Expert at one time India Mathematics alter the system calculate the them from Greek system to Babilonia of but being based on ten. But from first reference of Hindu number coming from a Bishop Suriah in the year 662 mentioning that they use 9 sign of rather than ten.

With the fall of empire Romawi of at century VII, West even also lost ground and East experience of the evocation. During West star sink at the opposite of firmament, published other star, Islam.

After Rasulullah Muhammad saw pass away hence started [by] a period of/to Khulafur Rasyidin which dipimpim by Dusty Khalifah Burn the Ash Shiddiq ra, Amirul Mukminin Umar Bin Khattab Al Faruq ra, Amirul Mukminin Usman Bin Affan Dzunnurrain ra and Amirul Mukminin Ali Bin Abi Thalib kw. And in this time Islam have been spread over to reach the Egypt, Suriah, Mesopotamia And Persian as well as Yerusalem. In the year 700 M, Islam have reached the river Hindus in the East and Algiers in West. Year 711 M, Islam have mastered the Spanyol to region Prancis and in year 751 M have defeated the Chinese. And in Spanyol which more knowledgeable by Andalusia, experiencing of top kejayaanya of at century VIII.

At century IX, Khalifah Al Ma'Mun found the luxury library, Bayt Al Hikmah - Wisdom House. And one of the man of science of terkemukannya is Muhammad Ibnu of Mozes of Al Khawarizmi. Its his Important Article for example Al-Jabr Wa Al-Muqabala and from here emerge the algebra term - solution. As well as propagating Algorithm from word Al-Khawarizmi.

And from here nation in other world cleft will follow the new arabic numerals system. Number consisted of by ten sign. And finally number of zero even also emerge and finish our journey. And we remain not know surely whether number of zero first appear in the India of or in Andalusia of or in Arab.

IDEA HISTORY

IDEA HISTORY

Epistemologi is philosophy branch concerning in with the knowledge theory. Epistemologi is science studying various elementary recognition form of knowledge, reality and its value. traditionally, becoming fundamental of[is problem of epistemologi is the source of, provenance and nature of knowledge base: area, boundary and knowledge reach.

Knowledge is a word used to subject to what known by someone of about something. Knowledge always have the subjek: knowing. Without there is knowing hence may not be knowledge. Knowledge also suppose the object. Without object or matter knew also have to be told may not be knowledge. knowledge Berelasi by the problem of the truth of. Truth is according to knowledge with the knowledge object. Its problem is truth of a knowledge object cannot is at a time obtained in a[n certain knowledge time. a knowledge object present the truism. Truth of searched in knowledge step compiled methodically, is rational and systematic. There is three knowledge type, that is ordinary knowledge or erudite pre knowledge, erudite knowledge and philosophic knowledge.

A knowledge surely have the source. Whether in fact becoming the source of knowledge? Some philosopher mention that source of knowledge is mind of kindness or ratio. Kindness mind have the important function in course of knowledge. Some other philosopher have a notion that source of knowledge is experience inderawi. Knowledge basically lean and base on the the five senses and also at experience of empirik inderawi. Polar oposition of idea of this rasionalitas and empirisme is pacified by Immanuel Kant expressing that even entire/all idea and concept have the character of apriori, the concept and idea of application can only if there is experience. Without experience, all idea and concept can never application.

Other problems in epistemologi is certainty and truth a knowledge. Criterion of what weared to measure a knowledge can be referred as by a correctness and surely? How a knowledge can be told by as knowledge which sahih? In epistemologi of there are some theory of kesahihan knowledge, that is theory koherensi, correspondence theory, pragmatic theory, semantic theory and abundant theory logikal.

Khant tell, that human being have been provided with a set willingness, so that we can give the form to raw data is which we perceive. So that thereby we possible have the knowledge apriori, and needn't experience of by xself to get the basal knowledge, knowledge which aposteriori. We is not ordinary know the realita which in fact, but a[n realita reflected by conscious experience of us. God is as an realita is outside experience, and represent the knowledge object.

(1724–1804). The philosopher of the 1700s who ranks with Aristotle and Plato of ancient times is Immanuel Kant. He set forth a chain of explosive ideas that humanity has continued to ponder since his time. He created a link between the idealists—those who thought that all reality was in the mind—and the materialists—those who thought that the only reality lay in the things of the material world. Kant's ideas on the relationship of mind and matter provide the key to understanding the writings of many 20th-century philosophers.

Kant was born on April 22, 1724, in Königsberg, Germany (now Kaliningrad, Russia). His father was a saddle and harness maker. He attended school at the Collegium Fredericianum where he studied religion and the Latin classics. When he was 16 years old Kant entered the University of Königsberg. He enrolled as a student of theology but soon became more interested in physics and mathematics.

After leaving college he worked for nine years as a tutor in the homes of wealthy families. In 1755 he earned his doctorate at the university and became a lecturer to university students, living on the small fees his students paid him. He turned down offers from schools that would have taken him elsewhere, and finally the University of Königsberg offered him the position of professor of logic and metaphysics.

Kant never married and he never traveled farther than 50 miles (80 kilometers) from Königsberg. He divided his time among lectures, writing, and daily walks. He was small, thin, and weak, but his ideas were powerful.

Kant's most famous work was the ‘Critique of Pure Reason' (published in German in 1781). In it he tried to set up the difference between things of the outside world and actions of the mind. He said that things that exist in the world are real, but the human mind is needed to give them order and form and to see the relationships between them. Only the mind can surround them with space and time. The principles of mathematics are part of the space-time thoughts supplied by the mind to real things.

For example, we see only one or two walls of a house at any one time. The mind gathers up these sense impressions of individual walls and mentally builds a complete house. Thus the whole house is being created in the mind while our eyes see only a part of the whole.

Kant said that thoughts must be based on real things. Pure reason without reference to the outside world is impossible. We know only what we first gather up with our senses. Yet living in the real world does not mean that ideals should be abandoned. In his ‘Critique of Practical Reason' (1788) he argued for a stern morality. His basic idea was in the form of a Categorical Imperative. This meant that humans should act so well that their conduct could give rise to a universal law. Kant died in Königsberg on Feb. 12, 1804. His last words were Es ist gut, “It is good.”

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Johann Peter Gustav Lejeune

1825- Johann Peter Gustav Lejeune Dirichlet and Arien-Marie Legendre Prove Fermat’s Last theorem for n = 5

Johann Peter Gustav Lejeune Dirichlet


Tender age.

Lejeune Dirichlet. A bizzare name. Its family come from a small town in so called Belgia of Richelet, where its grandfather remain the. Name given in fact " Le Jeune de Richelet" meaning " Young man from Richelet." Name of " smelling" French impressing invite the interpretation that Dirichlet come France. Its father is a post chief in Duren, a small town is which is located in midst of between town of Cologne and town Aachen. old age 12 year, Dirichlet very taking a fancy to of mathematics, what is proved with saving pocket money of just for buying mathematics book, before enter the Gymnasium, a school in town Bonn in the year 1817. When becoming student here, Dirichlet known as by a demure student with the attitude praised and very taking a fancy to of Iesson of mathematics dansejarah.

Only have time to go to school two year, before its father carry over the College Jesuit in Cologne. Here Dirichlet have time to be taught by Ohm. Age 16 year, pass and ready to step into the university. That moment is standard of university education in Germany still less so that Dirichlet set mind on to learn in Paris have. Stock to of Book of Disquisitiones arithmeticae of Gauss masterpiece which is always brought within reason a Bible, Dirichlet go to the Paris in May 1822. Have time to be attacked by a smallpox, but [do] not pursue the kuliah and lessen the enthusiasm learn the nya. Kuliah from notable lecturer like Biot, Francoeur, Hachette, Laplace, Lacroix, Legendre And Poisson make its eye is opened. Later this matter is brought to by come home to Germany and some years later;then university standard in Germany can reach for achievement as best in world.

Return to Germany.

At in the middle of the 1823, Dirichlet put hand to the General of Maximilien Sebastien Foy, who live in near by its house near by Paris. General Foy of is including especial figure of Napoleon in action, and retired after Napoleon surender in Waterloo. In the year 1819, Foy become the Liberal opposition party chief is which is taken hold of die it. Dirichlet treated by like within reason family member and given by the salary which is last for teaching German to wife and the general child. Year 1825, General Foy die and Dirichlet set mind on to [go/come] home to Germany to the pressure of Alexander von Humboldt to become the instructor. Constraint happened by because a instructor have to be titled of Doctor.

In fact Dirichlet can make thesis, as prerequisite to become the Doctor and is entitled to teach, but because unable to converse in Latin Ianguage, considered to be by of resistor factor. This constraint is overcome by university Cologne by lifting Dirichlet as honour Doctor and make the thesis of about polynomial to be defended in " special class" at university Breslau. Initially arising different idea of among professor in Germany, before finally alleviate by itself

Year 1827, Diriclet, for the first time nya, teaching in university Breslau and know that education standard in Germany still lower. Dissatisfy this fact, return, pass the aid Humboldt, Dirichlet move to have Line to in the year 1828 to teach at Military College. Seemingly this position become the " stepping point" to teach and become the professor of at university have Line. This Position is taken hold of from year 1828 until year 1855. Occupation as instructor and do the administrative duty in Military College remain to be defended.

Proving theorema Fermat

Moment still in Paris, Dirichlet try to solve the TTF ( Last Theorema of Fermat), that is: xn + yn = zn, where n 2. In case n = 3 and n = 4 have been proved by Euler and Fermat, and Dirichlet try to prove for the n of 5. If n = 5 and x, y and z is one is other and even number of number able to be divided 5. There is two case here: case 1 happened by if all divisible number 5 is anomalous number, and case 2 happened by if there is two even number and one divisible other five: both differing.

Dirichlet prove the case 1 and present its handing out at Academy Paris in the year 1825. At the moment Dirichlet have time to come in contact with the Abel which is residing in Paris ( read the: Abel). Legendre which that moment become the jury, later;then can prove the case 2. Description of the way of verification published [at] September 1825. In fact Dirichlet can prove the case 2 because only representing extension from case 1. There is separate note that Dirichlet later;then give the contribution in verification n = 14

In the field of mechanics, Dirichlet [do/conduct] the research [of] about balance of potential theory and system. Pass the handing out written in the year 1839 dipaparkan method to evaluate interdependent integral ( integral multiple) and application to solve the problem of ellipsoid gravitation. Other action is convergent of deret trigonometrik , first multiply weared by Fourier to finish the equation diferensial.

Early Last verification Theorema of Fermat so that trigger the other mathematics to be continued by berkutat by and inspire the Gauss of about law of biquadratic reciprocity. Dirichlet give the difinisi of about function is such as those which weared now. ( y and x is dependent and independent variable). From all above most importantly is strong elementary meletakan to education in Germany in general and mathematics especially so that become the first one. Era predominate all French mathematics ( last of Cauchy) start replaced by kirprah of all Germany mathematics ( Jacobi, Weierstrass, Kummer, Kronecker, Dedekind, Riemann, Cantor, Steiner).

Sabtu, 29 November 2008

sejarah matematika

Nama : Ratna Satyawati
NIM : 05301244058
P. Matematika NR D ’05
Kuliah hari Senin jam ke 3
HISTORY OF MATHEMATICS
Mathematics is often defined as the study of quantity, magnitude, and relations of numbers or symbols. It embraces the subjects of arithmetic, geometry, algebra, calculus, probability, statistics, and many other special areas of research.
There are two major divisions of mathematics: pure and applied. Pure mathematics investigates the subject solely for its theoretical interest. Applied mathematics develops tools and techniques for solving specific problems of business and engineering or for highly theoretical applications in the sciences.
Mathematics is pervasive throughout modern life. Baking a cake or building a house involves the use of numbers, geometry, measures, and space. The design of precision instruments, the development of new technologies, and advanced computers all use more technical mathematics.
HISTORY
Mathematics first arose from the practical need to measure time and to count. Thus, the history of mathematics begins with the origins of numbers and recognition of the dimensions and properties of space and time. The earliest evidence of primitive forms of counting occurs in notched bones and scored pieces of wood and stone. Early uses of geometry are revealed in patterns found on ancient cave walls and pottery.
Ancient Periods
As civilizations arose in Asia and the Near East, the field of mathematics evolved. Both sophisticated number systems and basic knowledge of arithmetic, geometry, and algebra began to develop.
Egypt and Mesopotamia.
The earliest continuous records of mathematical activity that have survived in written form are from the 2nd millennium BC. The Egyptian pyramids reveal evidence of a fundamental knowledge of surveying and geometry as early as 2900 BC. Written testimony of what the Egyptians knew, however, is known from documents drawn up about 1,000 years later.
Two of the best-known sources for our current knowledge of ancient Egyptian mathematics are the Rhind papyrus and the Moscow papyrus. These present many different kinds of practical mathematical problems, including applications to surveying, salary distributions, calculations of the areas of simple geometric surfaces and volumes such as the truncated pyramid, and simple solutions for first- and second-degree equations.
Egyptian arithmetic, based on counting in groups of ten, was relatively simple. Base-10 systems, the most widespread throughout the world, probably arose for biological reasons. The fingers of both hands facilitated natural counting in groups of ten. Numbers are sometimes called digits from the Latin word for finger. In the Egyptians' base-10 arithmetic, hieroglyphs stood for individual units and groups of tens, hundreds, and thousands. Higher powers of ten made it possible to count numbers into the millions. Unlike our familiar number system, which is both decimal and positional (23 is not the same as 32), the Egyptians' arithmetic was not positional but additive.
Unlike the Egyptians, the Babylonians of ancient Mesopotamia developed flexible techniques for dealing with fractions. They also succeeded in developing a more sophisticated base-10 arithmetic that was positional, and they kept mathematical records on clay tablets. The most remarkable feature of Babylonian arithmetic was its use of a sexagesimal (base 60) place-valued system in addition to a decimal system. Thus the Babylonians counted in groups of sixty as well as ten. Babylonian mathematics is still used to tell time—an hour consists of 60 minutes, and each minute is divided into 60 seconds—and circles are measured in divisions of 360 degrees.
The Babylonians apparently adopted their base-60 number system for economic reasons. Their principal units of weight and money were the mina, consisting of 60 shekels, and the talent, consisting of 60 mina. This sexagesimal arithmetic was used in commerce and astronomy. Surviving tablets also show the Babylonians' facility in computing compound interest, squares, and square roots.
Because their base-60 system was especially flexible for computation and handling fractions, the Babylonians were particularly strong in algebra and number theory. Tablets survive giving solutions to first-, second-, and some third-degree equations. Despite rudimentary knowledge of geometry, the Babylonians knew many cases of the Pythagorean theorem for right triangles. They also knew accurate area formulas for triangles and trapezoids. Since they used a crude approximation of three for the value of pi, they achieved only rough estimates for the areas of circles.
Greece and Rome.
The Greeks were the first to develop a truly mathematical spirit. They were interested not only in the applications of mathematics but in its philosophical significance, which was especially appreciated by Plato (see Plato).
The Greeks developed the idea of using mathematical formulas to prove the validity of a proposition. Some Greeks, like Aristotle, engaged in the theoretical study of logic, the analysis of correct reasoning (see Aristotle). No previous mathematics had dealt with abstract entities or the idea of a mathematical proof.
Pythagoras provided one of the first proofs in mathematics and discovered incommensurable magnitudes, or irrational numbers (see Pythagoras). The Pythagorean theorem relates the sides of a right triangle with their corresponding squares. The discovery of irrational magnitudes had another consequence for the Greeks: since the lengths of diagonals of squares could not be expressed by rational numbers of the form a/b, the Greek number system was inadequate for describing them. Due to the incompleteness of their number system, the Greeks developed geometry at the expense of algebra. The only systematic contribution to algebra was made much later in antiquity by Diophantus. Called the father of algebra, he devised symbols to represent operations, unknown quantities, and frequently occurring constants.
Ancient knowledge of the sciences was often wrong and wholly unsatisfactory by modern standards. However, the mathematics of Euclid, Apollonius of Perga, and Archimedes—the three greatest mathematicians of antiquity—remains as valid today as it was more than 2,000 years ago (see Apollonius of Perga; Archimedes; Euclid). Euclid's ‘Elements of Geometry' used logic and deductive reasoning to set up axioms, postulates, and a collection of theorems related to plane and solid geometry, as well as a theory of proportions used to resolve the difficulty of irrational numbers. Despite its flaws, the ‘Elements' remains a historic example of how to establish universally agreed-upon knowledge by following a rigorous course of deductive logic. Apollonius, best known for his work on conic sections, coined the terms parabola, hyperbola, and ellipse. Another great figure was Ptolemy, who contributed to the development of trigonometry and mathematical astronomy.
Roman mathematicians, in contrast to the Greeks, are renowned for being very practical. The Roman mind did not favor the abstract side of mathematics, which had so delighted the Greeks. The Romans cared instead for the usefulness of mathematics in measuring and counting. As the fortunes of the Roman Empire declined, a rising interest in mathematics developed elsewhere, in India and among Arab scholars.
The Middle Ages
Indian mathematicians were especially skilled in arithmetic, methods of calculation, algebra, and trigonometry. Aryabhata calculated pi to a very accurate value of 3.1416, and Brahmagupta and Bhaskara II advanced the study of indeterminate equations. Because Indian mathematicians were not concerned with such theoretical problems as irrational numbers, they were able to make great strides in algebra. Their decimal place-valued number system, including zero, was especially suited for easy calculation. Indian mathematicians, however, lacked interest in a sense of proof. Most of their results were presented simply as useful techniques for given situations, especially in astronomical or astrological computations.
One of the greatest scientific minds of Islam was al-Khwarizmi, who introduced the name (al-jabr) that became known as algebra. Consequently, the numbers familiar to most people are still referred to as Arabic numerals. Arab mathematicians also translated and commented on Ptolemy's astronomy before it was brought to the attention of Europeans. Islamic scholars not only translated the works of Euclid, Archimedes, Apollonius, and Ptolemy into Arabic but advanced beyond what the Greek mathematicians had done to provide new results of their own.
By the end of the 8th century the influence of Islam had extended as far west as Spain. It was there, primarily, that Arabic, Jewish, and Western scholars eventually translated Greek and Islamic manuscripts into Latin. By the 13th century, original mathematical work by European authors had begun to appear.
Renaissance Period
Most of the early mathematical activity of the Renaissance was centered in Italy, where the mathematician Luca Pacioli wrote a standard text on arithmetic, algebra, and geometry that served to introduce the subject to students for generations. The solution of the cubic equation instigated great rivalries and priority claims between Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano. Among the advances in algebra made during the 16th century, the use of letters of the alphabet to denote constants, variables, and unknowns in equations is notable. This symbolic algebra later proved to be the key to advances in geometry, algebra, and the infinitesimal calculus.
17th Century
Mathematics received considerable stimulus in the 17th century from astronomical problems. The astronomer Johannes Kepler, for example, who discovered the elliptical shape of the planetary orbits, was especially interested in the problem of determining areas bounded by curved figures (see Kepler). Kepler and other mathematicians used infinitesimal methods of one sort or another to find a general solution for the problem of areas. In connection with such questions, the French mathematician Pierre de Fermat investigated properties of maxima and minima. He also discovered a method of determining tangents to curves, a problem closely related to the almost simultaneous development of the differential and integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz later in the century (see Fermat; Leibniz; Newton).
Of equal importance to the invention of the calculus was the independent discovery of analytic geometry by Fermat and René Descartes (see Descartes). Of the two, Descartes used a better notation and devised superior techniques. Above all, he showed how the solution of simultaneous equations was facilitated through the application of analytic geometry. Many geometric problems could be translated directly into equivalent algebraic terms for solution.
Developed in the 17th century, projective geometry involves, in part, the analysis of conic sections in terms of their projections. Its value was not fully appreciated until the 19th century. The study of probability as related to games of chance had also begun.
The greatest achievement of the century was the discovery of methods that applied mathematics to the study of motion. An example is Galileo's analysis of the parabolic path of projectiles, published in 1638. At the same time, the Dutch mathematician Christiaan Huygens was publishing works on the analysis of conic sections and special curves. He also presented theorems related to the paths of quickest descent of falling objects (see Huygens).
The unsurpassed master of the application of mathematics to problems of physics was Isaac Newton, who used analytic geometry, infinite series, and calculus to make numerous mathematical discoveries. Newton also developed his method of fluxions and fluents—the differential and integral calculus. He showed that the two methods—derivatives and integrals—were inversely related to one another (see Newton). Newton and Leibniz were studying similar problems of physics and mathematics at the same time. Having made his own discovery of the calculus in 1674, Leibniz published a rather obscure version of his methods in 1684, some years before Newton published a full version of his own methods. The sequence of mathematical developments that flows out of the discovery of the calculus is called analysis.
Although the new calculus was an immediate success, its methods were sharply criticized because infinitesimals were sometimes treated as if they were finite and, at other times, as if they were zero. Doubts about the foundations of the calculus were unresolved until the 19th century.
18th Century
The discovery of analytic geometry and invention of the calculus made possible the application of mathematics to a wide range of problems in the 18th century. The Bernoullis, a Swiss family of mathematicians, were pioneers in the application of the calculus to physics. However, they were not the only ones to advance the calculus in the 18th century. Mathematicians in France and England also tried to extend the range of the work of Newton and Leibniz.
The greatest development of mathematics in the 18th century took place on the Continent, where monarchs such as Louis XIV, Frederick the Great, and the Empress Catherine the Great of Russia provided generous support for science, including mathematics. The most prolific 18th-century mathematician was Leonhard Euler of Switzerland (see Euler). He published hundreds of research papers, and his major books dealt with both the differential and integral infinitesimal calculus as well as with algebra, geometry, mechanics, and the calculus of variations.
Joseph-Louis Lagrange contributed to mechanics, foundations of the calculus, the calculus of variations, probability theory, and the theories of numbers and equations (see Lagrange). While analysis was being developed by some French mathematicians, others were turning to geometry and probability theory. The French astronomer Pierre-Simon Laplace succeeded in applying probability theory and analysis to the Newtonian theory of celestial mechanics. He was thereby able to establish the dynamic stability of the solar system (see Laplace).
19th Century
The 19th century witnessed tremendous change in mathematics with increased specialization and new theories of algebra and number theory. The entire scope of mathematics was enriched by the discovery of controversial areas of study such as non-Euclidean geometries and transfinite set theory. Non-Euclidean geometries, in showing that consistent geometries could be developed for which Euclid's parallel postulate did not hold, raised significant questions pertaining to the foundation of mathematics.
In Germany, Carl Friedrich Gauss discovered the law of quadratic reciprocity, proved the fundamental theorem of algebra, and developed the theory of complex numbers. The Norwegian mathematician Niels Henrik Abel also made great strides during the 19th century, particularly with his theory of integrals of algebraic functions and a theorem that led to the Abelian functions, later advanced by Karl Gustav Jacobi. (See also Abel; Gauss.)
The German mathematician Karl Weierstrass brought new levels of rigor to analysis by reducing its elements to arithmetic principles and by using power series as a foundation for the theory of complex functions. August Möbius, also from Germany, worked in the area of analytic geometry and was a pioneer in topology. He discovered the Möbius strip, a topological space obtained by twisting one end of a rectangular strip and pasting it to the other.
Mathematicians in England slowly began to take an interest in advances made on the Continent during the previous century. The Analytic Society was formed in 1812 to promote the new notation and ideas of the calculus commonly used by the French. A form of noncommutative algebra called quaternions was discovered by William Rowan Hamilton, and other mathematical forms were applied to the theory of electromagnetism (see Hamilton, William Rowan).
In the United States indigenous groups of mathematicians were beginning to form, particularly in the areas of linear associative algebra and logic. In France mathematicians made significant contributions to work in geometry and analysis, especially analysis of elliptic functions. Other advances were made in complex analysis, modular functions, number theory, and invariant theory. Augustin-Louis Cauchy advanced nearly every branch of mathematics, but especially real and complex analysis. Henri Poincaré made significant contributions to mathematical physics, automorphic functions, differential equations, topology, probability theory, and the foundations of mathematics (see Poincaré). Italian mathematics in the 19th century tended to stress geometry and analysis.
Two related areas of mathematics established in the 19th century proved to be of major significance in the 20th century: set theory and mathematical logic. These were closely related to questions concerning the foundations of mathematics and the continuum of real numbers as investigated by Richard Dedekind and Georg Cantor (see Cantor). It was Cantor who created set theory and the theory of transfinite numbers.
Modern Times
Twentieth-century mathematics is highly specialized and abstract. The advance of set theory and discoveries involving infinite sets, transfinite numbers, and purely logical paradoxes caused much concern as to the foundations of mathematics. In addition to purely theoretical developments, devices such as high-speed computers influenced both the content and the teaching of mathematics. Among the areas of mathematical research that were developed in the 20th century are abstract algebra, non-Euclidean geometry, abstract analysis, mathematical logic, and the foundations of mathematics.
Modern abstract algebra includes the study of groups, rings, algebras, lattices, and a host of other subjects developed from a formal, abstract point of view. This approach formed the cornerstone of the work of a group of mathematicians called Bourbaki. Bourbaki uses abstract algebra in an axiomatic framework to develop virtually all branches of higher mathematics, including set theory, algebra, and general topology.
The significance of non-Euclidean geometry was realized early in the 20th century when the geometry was applied in mathematical physics. It has come to play an essential role in the theory of relativity and has also raised controversial philosophical questions about the nature of mathematics and its foundations.
Another area of mathematics, abstract analysis, has produced theories of the derivatives and integrals in abstract and infinite-dimensional spaces. There are many areas of special interest in the field of abstract analysis, including functional analysis, harmonic analysis, families of functions, integral equations, divergent and asymptotic series, summability, and the study of functions of a complex variable. In recent years, analysis has advanced with the introduction of nonstandard analysis. By developing infinitesimals this theory provides an alternative to the traditional approach of using limits in the calculus.
The most notable development in the area of logic began in the 20th century with the work of two English logicians and philosophers, Bertrand Russell and Alfred North Whitehead. The object of their three-volume publication, ‘Principia Mathematica' (1910–13), was to show that mathematics can be deduced from a very small number of logical principles. In the 1930s questions about the logical consistency and completeness of axiomatic systems helped to spark interest in mathematical logic and concern for the foundations of mathematics. Since the 1940s mathematical logic has become increasingly specialized.
The foundations of mathematics have many “schools.” At the beginning of the 20th century, David Hilbert was determined to preserve the powerful methods of transfinite set theory and the use of the infinite in mathematics, despite apparent paradoxes and numerous objections (see Hilbert, David). He believed it was possible to find finite means of establishing the truth of mathematical propositions, even when the infinite was involved. To this end Hilbert devoted considerable effort to developing a metamathematical theory of proofs. His program was virtually abandoned in the 1930s when Kurt Gödel demonstrated that for any general axiomatic system there are always theorems that cannot be proved or disproved (see Gödel, Kurt).
Hilbert's followers, known as formalists, view mathematics in terms of abstract structures. The axioms are developed as arbitrary rules. When applied to the unspecified elements of the theory, they can be used to establish the validity of theorems. Mathematical “truth” is thus reduced to the question of logical self-consistency. Those opposed to the formalist view, called intuitionists, believe that the basic truths of mathematics present themselves as fundamental intuitions of thought. The oldest philosophy of mathematics is usually ascribed to Plato. Platonism asserts the existence of eternal truths, independent of the human mind. In this philosophy the truths of mathematics arise from an abstract, ideal reality.
Math In 2004
Mathematics Figure Which Have Meritorious During The Year 2004.
1. Hans Wospakrik
Hans Jacobus Wospakrik ( Serui, Papua, 10 September 1951 - Jakarta, 11 January 2005) is a Indonesia physics representing lecturer of physics teoritik in Technological Institute of Bandung.
Hans is a getting best physics appreciation by Glorious University Atma of Jakarta at the service of, consistency, and its dedication is high in research in area of theory physics. He give the contribution mean to community of world physics in the form of mathematics method utilize to comprehend the physics phenomenon in elementary particle and public Relativity of Einstein. its Research Pickings this publicizing of in notable international journal, like Physical Review D, Journal Of Mathematical Physics, Modern of Physics Letters A, and Modern International Journal of Physics. He die at 11 January 2005 effect leukimia.
Education history
Year 1971 Hans enter the ITB by taking majors of Mining Technique, what [do] not enthuse of so that move next in the year to Physics majors. Year 1976 he finish the its master education. By the end of year 1970-an, he go to Dutch in order to continuing study pascasarjana in area of physics teoritik. Semenjak Year 1999 Hans go to University Durham, English. But newly year 2002 he take the doctor program in same university. Early year 1980-an, beside continuing study pascasarjananya, Hans have performed to research into with Martinus JG Veltman ( year 1999, Veltman reach for the Nobel Physics), in Utrecht, Dutch, and [in] Ann Arbor, Michigan, United States.
From Atomos Till Quark is a book of result of masterpiece Hans narrating to regarding the seeking of human being as long as history of concerning smallest compiler from this natural items. Early Greek where all that moment philosopher philosophize to regarding the smallest compiler each;every items, Jazirah Arab touched by Hans as owner " knowledge torch" next after Greek, science alkemi, nuclear reaction " narrating" at us about atom existence, proton and neutron, until finding in this time regarding set of smaller items, that is quark.
2. Christa Lorenzia
Christa Lorenzia Soesanto born in Jakarta, 21 October 1996, Math Club is a Indonesia student having many achievement in the field of mathematics. and Computer. First Child from couple of Edy Soesanto Prawirohardjo Intelbc International and this Betsy Eliane Rahardjo, succeeding well reputed of Indonesia with the champion of various competition of mathematics and Computer of either in national storey;level and also international. SINCE CHILDHOOD, Christa Lorenzia always install the goals. Either in school and also moment follow the championship, he always want to be best. Proven, he/she always become the public champion from TK till pass the SD. Even, in various ajang competition, he also reach for the predikat champion, especially mathematics race. Christa hope can be well reputed of nation and state, so that Indonesia people do not is always stamped stupid. According to him, oftentimes Indonesian nation stamped to by lower by other nation even by nation by xself. This matter of course bother its mind as Indonesia citizen capable to show its achievement. Since October 2008, Christa have made the new champion in world of Mathematic Olympiad, where Christa have succeeded to get FOUR Gold Medal to Alone and TWO Gold Medal for the Beregu of alternately. This champion is first multiply made by a Indonesian nation child.
3. Frank Wilczek
Franc Wilczek ( born 15 May 1951) is a United States physics. H. David Politzer and David Gross he is awarded by the Appreciation Nobel in Physics 2004.
Research
In the year 1973, Wilczek, a laboring master student by David is Jonathan Gross in University Princeton, finding freedom asimtot, mentioning that more and moreing the near by quark one another, more and more to weaken the strong interaction ( or colour payload) among both; for a while quark of in back part far, nuclear energy of among both so weaken so that berperilaku almost loo like the free particle. Teori--Yang found independently by Hugh David Politzer--Penting for the development of kromodinamika quantum.
Wilczek have assisted to express and develop the aksion, anyon, freedom asimtot, and other aspect of field theory of quantum in general, and have checked the solid goods physics, astrophysics, and particle physics.
4. Yohanes Surya
Yohanes Surya born in Jakarta on 6 November 1963. He start to deepen the physics of at majors of Physics of MIPA of Indonesia University till year 1986, teaching in SMAK I of Dredger Jakarta till year 1988 and hereinafter go through the master program and its doctor [in] College of William and Mary, Virginia, United States. its Master Program [is] finished in the year 1990 and its doctor program in year 1994 by predikat is cum laude. After getting title Ph.D., Yohanes Surya become the Consultant of Theoretical Physics in TJNAF / cebaf ( Continous Electron Beam Accelerator Facility) Virginia - United States ( 1994). Although have had the Greencard remain and work in United States, Yohanes Surya home to Indonesia with a purpose to wish well reputed [of] Indonesia through the physics olympiad ( its password at that time is " Go Get Gold") and also develop the physics in Indonesia.
Come home from America, beside train and lead the Team of Olympiad of Indonesia Physics ( TOFI), Yohanes Surya become the instructor and researcher of at program of pasca of master UI for the area of nuclear physics ( year 1995 - 1998). From year 1993 till 2007 student binaannya succeed well reputed [of] nation by menyabet 54 gold medal, 33 silver medal and 42 bronze medal in so many Science competition / International physics. In the year 2006 a student binaannya, Jonathan Pradana Mailoa, succeeding to reach for the predikat " The Absolute Winner" ( World Champion) in International Physics Olympiad ( Ipho) XXXVII in Singapore.
Since 2000, Yohanes Surya of many performing a training for the teachers of Physics and Mathematics in most of all metropolis in Indonesia, in capital of sub-province, to countryside in all pelosok Nusantara from Sabang till Merauke, including pesantren-pesantren. To place this training is Yohanes Surya found the Surya Institute. Surya Institute nowadays is developing building of TOFI center to become the training center learn and also student to contest in various science championship.
Yohanes Surya represent the productive writer for the area of mathematics. There is 68 book have been written for the student of SD until SMA. Besides writing book, he also write hundreds of Physics article in erudite journal of national goodness and also international, daily of COMPASS, TEMPO, Indonesia Media and others. He also pencetus of term MESTAKUNG and three law Mestakung, and also pencetus of Whirligig study Easy, Besotted, Pleasing.
Besides as writer, Yohanes Surya also as guest speaker [of] various program of Physics instruction [pass/through] the CD ROM for the SD OF, SMP And SMA. He also follow to produce various program of TV education among others " Adventure in Fantasy World", and " Tralala-Trilili" in RCTI.
Outside aktifitasnya of above, Yohanes Surya act in so many international organization as Board member of the International Physics Olympiad, Vice of President of The First step to Nobel Prize 1997; conceptor And President Asian Physics Olympiad 2000; Chairman Of The first Asian Physics Olympiad, in Karawaci, Tangerang ( 2000); officer of Member of the World Physics Federation Competition; Chairman Of The International Econophysics Conference 2002; Chairman The World Conggress Physics Federation 2002; Board Of Experts [in] magazine of National Geographic Indonesia and also become the Chairman of Asian Science Camp 2008 in Denpasar, Bali. During have career to in area of physics development, Yohanes Surya have got various award / fellowship for example CEBAF / sura award ACE ' 92-93 one of the best student in the field of nuclear physics at American south-east region, creativity appreciation 2005 from Institution of Creativity Development, Loyal Badge award [of] Masterpiece Hero ( 2006) from President of RI Susilo Bambang Yudhoyono. is same in the year, he chosen as Indonesia proxy in the field of education to come in contact with the United States President, George W. Bush. In the year 2007, he write the book " Mestakung: Successful Secret of World Champion" getting appreciation as quickest writer Best Seller in Indonesia.
Yohanes Surya is physics professor from Loyal Christian University of Discourse, Salatiga. He have become the Dean of Faculty of Science and Mathematics of University of Pelita Expectation; Lead the Promotion and Cooperation of Gathering of Indonesia Physics ( 2001-2004), jury of various science race / mathematics XL-COM, L'Oreal, UKI, member of Council of Curator of Beautiful Museum Iptek Indonesia Miniature, one of [the] founder The Mochtar Riady Institute, member of Council of Sponsor of Trust of College of Islam of Assalamiyah Banten and nowadays Prof. Yohanes Surya take hold of as Rector of University of Multimedia Nusantara ( Compass of Gramedia Group) active to and also campaign the Physics Love Bali Love The Physics, Kalbar Love the Physics in all Indonesia.

SUBDIVISIONS OF MATHEMATICS
Throughout history mathematics has become increasingly complex and diversified. At the same time, however, it has become increasingly general and abstract. Among the major subdivisions of modern mathematics are the following:
Arithmetic
Arithmetic comes from the word arithmos, meaning “number” in Greek. It is the study of the nature and properties of numbers. It includes study of the algorithms of calculation with numbers, namely the basic operations of addition, subtraction, multiplication, and division, as well as the taking of powers and roots. Arithmetic is often applied in the calculation of fractions, ratios, percentages, and proportions.
Algebra
Algebra has often been described as “arithmetic with letters.” Unlike arithmetic, which deals with specific numbers, algebra introduces variables that greatly extend the generality and scope of arithmetic. The algebra taught in high schools involves techniques for solving relatively simple equations.
Modern algebra, or abstract algebra, is a more general branch of mathematics that analyzes algebraic axioms and operations with arbitrary sets of symbols. Special areas of abstract algebra include the study of groups, rings, fields, the algebra of matrices, and a large variety of nonassociative and noncommutative algebras. Special algebras of sets and vectors and Boolean algebras arise in the study of logic (see Boole). Algebra is used in the calculation of compound interest, in the solution of distance-rate-time problems, or in any situation in the sciences where the determination of unknown quantities from a body of known data is required.
Geometry
The word geometry is derived from the Greek meaning “earth measurement.” Although geometry originated for practical purposes in ancient Egypt and Babylonia, the Greeks investigated it in a more systematic and general way.
In the 19th century, Euclidean geometry's status as the primary geometry was challenged by the discovery of non-Euclidean geometries. These inspired a new approach to the subject by presenting theorems in terms of axioms applied to properties assigned to undefined elements called points and lines. This led to many new geometries, including elliptical, hyperbolic, and parabolic geometries. Modern abstract geometry deals with very general questions of space, shape, size, and other properties of figures. Projective geometry, for example, is an abstract geometry concerned with the geometric properties that remain invariant under the projection of figures onto other figures, as in the case of mathematical perspective.
A very useful approach to geometry is found in topology, the study of the properties of a geometric figure that remain the same when a figure is subjected to continuous transformation without loss of identity of any of its parts. Differential geometry is the study of geometry in terms of infinitesimals.
Analytic Geometry and Trigonometry
Analytic geometry combines the generality of algebra with the precision of geometry. It is sometimes called Cartesian geometry, after Descartes, who was the first to exploit the methods of algebra in geometry. Analytic geometry addresses geometric problems from an algebraic point of view by associating any curve with variables by means of a coordinate system. For example, in a two-dimensional coordinate system, any point on a curve can be associated with a pair of points (a,b). General properties of such curves can then be studied in terms of their algebraic properties.
Trigonometry is the study of triangles, angles, and their relations. It also involves the study of trigonometric functions. There are six trigonometric ratios associated with an angle: sine, cosine, tangent, cotangent, secant, and cosecant. These are especially useful in determining unknown angles or the sides of triangles based upon known trigonometric ratios. In antiquity, trigonometry was used with considerable success by surveyors and astronomers.
Calculus
The calculus discovered in the 17th century by Newton and Leibniz used infinitesimal quantities to determine tangents to curves and to facilitate calculation of lengths and areas of curved figures. These operations were found to be inversely related. Newton called them “fluxions” and “fluents,” corresponding to what are now termed derivatives and integrals. Leibniz called them “differences” and “sums.”
In the 19th century, in response to questions about its rigorous foundations, the calculus was developed in terms of a theory of limits. Analysis—differential and integral calculus—was subsequently approached even more rigorously by those who sought to establish its results by strictly arithmetic means. This required an exact definition of the continuity of the real numbers. Others extended the power of analysis with very general theories of measure.
Analysis gives primary emphasis to functions, convergence of sequences, series, continuity, differentiability, and questions about the completeness of the real numbers. Introductory courses in calculus generally include study of logarithms, exponential functions, trigonometric functions, and transcendental functions.
Complex Analysis
Complex analysis extends the methods of analysis from real to complex variables. Complex numbers first arose to permit general solutions to algebraic equations. They take the form a+bi, where a and b are real numbers. The variable a is called the real part of the number; b, the imaginary part of the number; and i represents the complex, or “imaginary,” number signified by the square root of –1. Because complex numbers have two independent components, a and b, they are especially useful in applications whenever two variables must be treated simultaneously. For example, complex analysis has proven particularly valuable in applications to fluid dynamics, where both pressure and velocity vary from point to point. Complex numbers were made more acceptable to many in the 19th century when they were given a geometric interpretation.
Number Theory
It has been said that any unsolved mathematical problem that is over a century old and is still considered interesting belongs to number theory. This branch of mathematics involves the study of the properties of numbers and the structure of different number systems. It is concerned with integers, or whole numbers. Many problems in number theory deal with prime numbers. These are integers larger than 1 that have only themselves and 1 as factors.
Questions about highest common factors, least common multiples, decompositions into primes, and the representation of natural numbers in certain forms as well as their divisibility are all the province of number theory. Computers have recently been applied to the solution of certain number-theory problems.
Probability Theory and Statistics
The branch of mathematics concerned with the analysis of random phenomena is called probability theory. The entire set of possible outcomes of a random event is called the sample space. Each outcome in this space is assigned a probability, a number indicating the likelihood that the particular event will arise in a single instance. An example of a random experiment is the tossing of a coin. The sample space consists of the two outcomes, heads or tails, and the probability assigned to each is one half.
Statistics applies probability theory to real cases and involves the analysis of empirical data. The word statistics reflects the original application of mathematical methods to data collected for purposes of the state. Such studies led to general techniques for analyzing data and computing various values, drawing correlations, using methods of sampling, counting, estimating, and ranking data according to certain criteria.
Set Theory
Created in the 19th century by the German mathematician Georg Cantor, set theory was originally meant to provide techniques for the mathematical analysis of the infinite. Set theory deals with the properties of well-defined collections of objects. Sets may be finite or infinite. A finite set has a definite number of members; such a set might consist of all the integers from 1 to 1,000. An infinite set has an endless number of members. For example, all of the positive integers compose an infinite set.
Cantor developed a theory of infinite numbers and transfinite arithmetic to go along with them. His ‘Continuum Hypothesis' conjectures that the set of all real numbers is the second smallest infinite set. The smallest infinite set is composed of the integers or any set equivalent to it.
Early in the 20th century certain contradictions of set theory concerning infinite sets, transfinite numbers, and purely logical paradoxes brought about attempts to axiomatize set theory in hopes of eliminating such difficulties. When Kurt Gödel showed that, for any axiomatic system, propositions could be devised that were neither true nor false, it seemed that the traditional certainty of mathematics had been suddenly lost.
In the 1960s Paul Cohen succeeded in showing the independence of the ‘Continuum Hypothesis', namely that it could be neither proved nor disproved within a given axiomatization of set theory. This meant that it was possible to contemplate non-Cantorian set theories in which the ‘Continuum Hypothesis' might be negated, much as non-Euclidean geometries treat geometry without assuming the necessary validity of Euclid's parallel postulate.
Logic
Logic is the study of the way in which valid conclusions may be drawn from given premises. It was first treated systematically by Aristotle and later developed in terms of an algebra of logic. Symbolic logic arose from traditional logic by using symbols to stand for propositions and relations between them. Modern logicians use algebraic and formal methods to study the relations between logical propositions. This has led to model theory and model logic.