Clean Up All Materials 4. Get Started Right Away 6. Let the students guided you, you can always rephrase what they say to fit one of your core non-negotiable station rules.

Archimedes used the method of exhaustion to approximate the value of pi. The history of mathematics can be seen as O level math paper 2011 paper ever-increasing series of abstractions. The first abstraction, which is shared by many animals, [16] was probably that of numbers: Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

It is in Babylonian mathematics that elementary arithmetic additionsubtractionmultiplication and division first appear in the archaeological record. The Babylonians also possessed a place-value system, and used a sexagesimal numeral system, still in use today for measuring angles and time.

His textbook Elements is widely considered the most successful and influential textbook of all time. Other notable developments of Indian mathematics include the modern definition of sine and cosineand an early form of infinite series.

The most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. During the early modern periodmathematics began to develop at an accelerating pace in Western Europe.

The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gausswho made numerous contributions to fields such as algebraanalysisdifferential geometrymatrix theorynumber theoryand statistics.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both.

Mathematical discoveries continue to be made today.

According to Mikhail B. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. In Latin, and in English until aroundthe term mathematics more commonly meant "astrology" or sometimes "astronomy" rather than "mathematics"; the meaning gradually changed to its present one from about to This has resulted in several mistranslations.

For example, Saint Augustine 's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.

It is often shortened to maths or, in North America, math. Today, no consensus on the definition of mathematics prevails, even among professionals. Brouweridentify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other.

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In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". In formal systems, the word axiom has a special meaning, different from the ordinary meaning of "a self-evident truth".

In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

Mathematics as science Carl Friedrich Gaussknown as the prince of mathematicians The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".

The specialization restricting the meaning of "science" to natural science follows the rise of Baconian sciencewhich contrasted "natural science" to scholasticismthe Aristotelean method of inquiring from first principles.

The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as biologychemistryor physics. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the other sciences.

Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics. The opinions of mathematicians on this matter are varied. Many mathematicians [46] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts ; others[ who?

One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created as in art or discovered as in science.Cambridge GCE O Level Mathematics D Questions & Comments You are here: Home» Qualifications» International» Cambridge» O Level» Mathematics D "Our Mission is to help the youth excel at the CIE through extensive practice of Past Papers and Marking Scheme".

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