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90-485: The main responsibilities of the county municipality included the running of 29 upper secondary schools . It administered the county roadways, public transport , dental care , culture , and cultural heritage in the county. The administration was located in Skien . The county municipality had 1,544 employees, and in 2007, a revenue of 1,508 million  kr . The Telemark county council ( Norwegian : Fylkestinget )

180-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects

270-843: A curriculum used and recognized internationally or an alternative approach to teaching – either religious or pedagogical . These schools must be approved by the government and are heavily grant-aided, and cannot select students based on subjectivity, such as skill or intelligence. The Norwegian school system can be divided into three parts: elementary school ( barneskole , ages 6–12), lower secondary school ( ungdomsskole , ages 13–16), and upper secondary school ( videregående skole , ages 16–19). The barneskole and ungdomsskole levels are compulsory, and are commonly referred to as grunnskole ('foundation school'). Elementary and lower secondary school are mandatory for all children aged 6–16. Before 1997, mandatory education in Norway started at

360-547: A few weeks before the final examinations of the final year. The titles of educators in Norwegian schools vary with the degrees they have. Higher education is anything beyond upper secondary school , and normally lasts three years or more. To be accepted to most higher education schools, a student must have attained a general university admissions certificate ( generell studiekompetanse ). This can be achieved by taking general studies while in upper secondary school or through

450-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)

540-486: A given right. When primary and lower secondary education is completed, upper secondary schooling is entitled to students for enrollment, which prepares students for higher education or vocational studies . The school year in Norway runs from mid-August to late June the following year. The Christmas holiday from mid-December to early January historically divides the Norwegian school year into two terms . Presently,

630-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were

720-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of

810-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it

900-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes

990-520: A professionally independent agency under the Ministry of Education and Research , assures the quality of higher education in Norway. Where there are more applicants than students admitted, applicants are ranked based on their grades from upper secondary school. To gain access to studies commonly requiring a high GPA, like medicine, law and engineering, many students re-sit their upper secondary school examinations to improve their grades. Higher education

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1080-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as

1170-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of

1260-443: A single system. Among the goals of the reform was that all students should have a certain amount of general studies large enough to make them eligible for higher education later, meaning more theory in vocational studies, and that it should be possible to cross over from one education path to another without losing too much credit. In the old system, two years of carpentry would be wasted if one wanted to switch to general studies, but in

1350-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating

1440-463: A typical structure named the "2+2 model": after two years of school training (with workshops and a short internship in industry), the student does an apprenticeship for two years in an enterprise or a public institution. The apprenticeship is divided into one year of training and one year of practical work. Some vocational curricula are nonetheless entirely school-based, and others include three years of apprenticeship instead of two. The new reform makes

1530-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to

1620-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry

1710-473: Is broadly divided into: There is no formal distinction between vocational and non-vocational higher education. Before the 19th century the main source for higher education of Norwegians was the University of Copenhagen . Norway's first large-scale education institution for people with intellectual disabilities was founded in 1898 by educator Emma Hjorth . Since the 1970s, the government has legislated

1800-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example

1890-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of

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1980-675: Is eight weeks from the middle of June until the middle of August. They also have Christmas holidays after the first semester ends in December until the second semester begins in January. In addition, in the last week of October, pupils in primary, lower secondary and upper secondary schools have fall break, with another week-long break (winter break) in the middle of February. Students at universities or university colleges typically do not have fall and winter breaks. In Norway, there are thirteen national holidays that students are free from school. In

2070-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module

2160-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as

2250-471: Is made up of 35 representatives that were elected every four years. The council essentially acted as a Parliament or legislative body for the county and it met several times each year. The council is divided into standing committees and an executive board ( fylkesutvalg ) which meet considerably more often. Both the council and executive board are led by the County Mayor ( fylkesordfører ) who held

2340-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example

2430-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,

2520-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of

2610-547: Is often held to be Archimedes ( c.  287  – c.  212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and

2700-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it

2790-456: Is primarily based on public schools: In 2007, 93% of upper secondary school students attended public schools. Until 2005, Norwegian law held private secondary schools to be illegal unless they offered a "religious or pedagogic alternative", so the only private schools in existence were religious ( Christian ), Steiner/Waldorf , Montessori schools , and Danielsen  [ no ] . The first "standard" private upper secondary schools opened in

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2880-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after

2970-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,

3060-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of

3150-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with

3240-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It

3330-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during

3420-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity

3510-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of

3600-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object

3690-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry

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3780-600: The 1970s and 1980s, the folkeskole was abolished, and the grunnskole ('foundation school') was introduced. Traditionally poorer counties like Finnmark and Hedmark have the highest shares of inhabitants who only have completed the compulsory primary education , with numbers as high as 38%. Under the Independent Schools Act of 2003, private schooling in Norway has become available. However, very few such schools exist in comparison to Norwegian public schools. Private schools must be founded upon

3870-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not

3960-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and

4050-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,

4140-703: The age of 7. Students often have to change schools when they enter lower secondary school and almost always have to change schools when they enter upper secondary school, as many schools only offer one of the levels. In the first year of primary school, students spend most of their time playing educational games and learning social structures, the alphabet, basic addition and subtraction, and basic English skills. In grades 2–7, they are introduced to mathematics , English , science , religion (focusing not only on Christianity but also on all other religions , their purpose, and their history), aesthetics , and music , complemented by geography , history , and social studies in

4230-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During

4320-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,

4410-543: The concept: "subjects students may be chosen to sit an exam for." In Norway's schools, there are two semesters. The new academic year begins in the middle of August. The first semester begins in August and ends in December. The second semester begins in January and ends in June. In Norway's school, there are several long vacations. For example, after the academic year ends in June, students in Norway have summer vacation, which

4500-402: The country will determine whether they get accepted to their upper secondary school of choice or not. From eighth grade, students can choose one elective ( valgfag ) and one language. Typical offered languages are German, French, and Spanish as well as additional English and Norwegian studies. Before the educational reform of August 2006, students could choose a practical elective instead of

4590-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is

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4680-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely

4770-556: The description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of

4860-523: The elementary/middle school curriculum of that subject. In 2009, Norwegian fifteen-year-olds performed better in OECDs Programme for International Student Assessment than other Scandinavian countries, with significant improvement since 2006. In mathematics; however, the top 10% were estimated to lag three years behind the top-scoring students in Shanghai . Secondary education in Norway

4950-791: The executive powers of the county. From 2017 until its dissolution in 2020, Gunn Marit Helgesen of the Conservative Party was the county mayor. The party breakdown of the council is as follows: The county government ran these schools: Education in Norway Education in Norway is mandatory for all children aged from 6 to 16. Schools are typically divided into two divisions: primary and lower secondary schooling. The majority of schools in Norway are municipal , where local governments fund and manage administration. Primary and lower secondary schools are available free of charge for all Norwegian citizens as

5040-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of

5130-403: The fall of 2005. Videregående is the equivalent to high school. As of 2017, graduation from videregående skole was at 73%. Prior to 1994, there were three branches of upper secondary schooling: general (language, history, etc.), mercantile (accounting, etc.), and vocational (electronics, carpentry, etc.) studies. The high school reform of 1994 ("Reform 94") merged these branches into

5220-494: The fifth grade. No official grades are given at this level. However, the teacher often writes a comment, analysis, and sometimes an unofficial grade on tests. Tests are to be taken home and shown to parents. There is also an introductory test to let the teacher know if the student is above average or is in need of some assistance at school. When the students enter lower secondary school, at age 12 or 13, they begin being graded on their work. Their grades together with their location in

5310-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",

5400-555: The grades are often postfixed with + or - (except 6+ and 1-) and it is also common to use grades such as 5/6 or 4/3 indicating borderline grades. However, the grades students get on their final diploma are either 1, 2, 3, 4, 5 or 6. At the conclusion of their school careers, upper secondary pupils aspire to graduate with a diploma. This is obtained by those who have passed all their subjects, and lists grades which are "based on teachers' determination of overall achievement marks of their own students." As well as these teacher-assigned grades,

5490-401: The holidays are included in the long break. For example, New Year's day and Christmas are national holidays in Norway, but students have Christmas holidays at that time. Official authorities Reports from international organizations Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for

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5580-475: The incorporation of IT into the schooling mandatory, and many counties (responsible for the public high schools) offer laptops to general studies students for free or for a small fee. Kunnskapsløftet also makes it harder to switch between electives that are taken in the second and third year in the general studies path. Students graduating upper secondary school are called russ in Norwegian. They often celebrate with parties and festivities, which are held

5670-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before

5760-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and

5850-454: The languages. Teens born in 1999 and later could once again choose a practical elective known as ( arbeidslivsfag ) or career studies upon starting lower secondary school, thus getting the option to choose two electives. The electives vary significantly between schools, even in the same municipality. A student may take the grade 10 exam in a particular subject early as long as he or she has been granted an exemption from further instruction in

5940-621: The law of 23/5 where a person must be above 23 years of age, have five years of combined schooling and work experience and have passed exams in Norwegian, mathematics, natural sciences, English and social studies. Some degrees also require special electives in second and third grade (e.g. maths and physics for engineering studies.) The majority of higher educational institutions are run by the state and take responsibility for their own instruction, research, and dissemination of knowledge. The Norwegian Agency for Quality Assurance in Education (NOKUT),

6030-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term

6120-489: The middle of April, there are four days of Easter break. Norway celebrates International Workers' Day on May 1 and its Constitution Day on May 17. Furthermore, Norway also celebrates Ascension Day in May, with the exact date depending on the each year. Ten days after Ascension Day, the country celebrates Pentecost , and Whit Monday the next day. Students do not have to attend school on these national holidays. However, some of

6210-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to

6300-409: The needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves

6390-400: The new Bologna system was introduced implies that students who had started their studies while the old system still was in effect will graduate with transcripts containing grades from both systems (i.e. both numbers and letters). Lower levels of education use a scale running from 1 through 6, with 6 being the highest and 2 the lowest passing grade. For non-final tests and mid-term evaluations

6480-526: The new system one could keep credit for at least half of it. Since the introduction of the reform Kunnskapsløftet ('the knowledge promise' or 'the lifting of knowledge', the word løfte having two meanings) in the fall of 2006, a student can apply for a general studies ( studieforberedelse ) or a vocational studies ( yrkesfag ) path. Inside these main paths, there are many sub-paths to follow. An upper secondary school usually offers general and vocational curriculum. Vocational studies usually follow

6570-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains

6660-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000  BC , when

6750-647: The policy that all children should be educated in local schools. Since then, special education has taken place mostly in ordinary schools . The need for special education in school depends on the individual student's abilities and capabilities. Pupils who cannot manage to yield learning outcomes from the typical teaching style have a right to special education . In special education, the Educational/Psychological Service maintains highly qualified specialists, educational psychologists , social welfare workers, and kindergarten teachers, playing

6840-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been

6930-420: The role of safety net in society. The Educational/Psychological Service visits school, provides students with help when needed, and assists their family members. In addition to the society system, special educators play a significant role, particularly in the content of their instructions. According to The Act relating to Universities and Colleges, universities and university colleges must take responsibility for

7020-527: The second term begins in January. Organized education in Norway dates as far back as 2000 B.C . Shortly after Norway became an archdiocese in 1153, cathedral schools were constructed to educate priests in Trondheim , Oslo , Bergen and Hamar . After the Reformation of Norway in 1537, following the unification with Denmark in 1536 , cathedral schools were turned into Latin schools , and it

7110-406: The students' diplomas may feature one or more examination grades. A minority of students will be chosen at random to sit an exam, a concept called trekkfag in Norwegian. The word fag means 'subject', whilst the verb å trekke can be translated as 'to draw, to pick', as one would describe picking a card from a pack. The dictionary translation of trekkfag is given as a description of

7200-472: The students' learning environment. Institutions are expected to create an inclusive and flexible learning environment that represents universal design . Norway has multiple different grading systems, both unique ones and ones that have been based on foreign grading systems. The former most common system of grades used at university level was based on a scale running from 1.0 (worst) through 6.0 (best) with 4.0 or above being considered passing grades. The way

7290-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become

7380-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and

7470-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,

7560-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in

7650-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in

7740-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until

7830-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"

7920-408: Was made mandatory for all market towns to have such a school. In 1736 training in reading was made compulsory for all children, but was not effective until some years later, when ambulatory schools ( omgangsskoler ) were also established. In 1827, Norway introduced the folkeskole ('people's school'), a primary school which became mandatory for seven years in 1889 and nine years in 1969. In

8010-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to

8100-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In

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