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# Translations of Encyclopedia about Mathematics

## The History of Mathematics

Mathematics has been in people’s lives from the beginning of time. Even in prehistory was a person required to perform mathematical tasks. The number of hunters, tools or members of a family were probably shown by the fingers on one or more hands, which is probably why today’s number systems are based on the number 5 or 10.

The second step probably arose out of the need to organise greater quantities: such as between young and older hunters, light or heavy weaponry, large or small hides.

The first number appeared before the creation of letters. By making carvings into pieces of wood, traders and surveyors were able to make their calculations. Egyptian pyramids are in fact the result of the early use of mathematics.

In 3,000 B.C., people first started using theoretical tools and numbering systems in Mesopotamia (Babylon) and Egypt. The mathematician and scholar Euclides started his famous school of mathematics in Alexandria, as such laying the corner stone of mathematics. The Egyptians used a decadic numbering system, which is based on the number 10 and still in use today. They also introduced characters used to describe the numbers 10 and 100, making it easier to describe larger numbers. Geometry started to receive great attention and served in surveying land, cities and streets.

The Egyptians knew how to count in fractions and used defined rules to calculate geometric objects. For example, they managed to almost perfectly calculate the surface area of circles. The number p , which amounts to roughly 3.1415926, was calculated very closely by the Egyptians at about 3.16.

The Babylonians used a numbering system based on the number sixty and used wedge-shaped symbols and arrows when making numerical records. Using their mathematical tool, they even managed to solve quadratics, and their number tables helped them multiply, divide, involve or calculate interest. They even understood Pythagoras’ theorem 2,000 years before Pythagoras was born.

The Greeks also made great gains in science and mathematics, adopting knowledge in mathematics from both the Babylonians and Egyptians to create generally binding theorems. The Greeks founded everything on strict logic and systematic approaches, where mathematical proof was required to prove the truth of scientific assertions.

In the end, even the Chinese knew Pythagoras’ theory several centuries beforehand. Greek astronomers sped up developments of geometry, making calculations of heavenly bodies and the diameter of the earth and proving that the earth is in fact round. Euclid is a Greek mathematician who documented knowledge from his times in the most demanding method by using axioms (unprovable truth sentences) composed from the simplest of mathematical aids. This created the foundation from which other rules were based on.

In the fifth century before Christ, Democritos formulated an equation for calculating the volume of spires, and Hippocrates determined that crescent shapes with round edges have the same surface as certain triangles (Hippocrate’s half-moon), a phenomenon which relates to the proverbial "quadrature of the circle". In this manner and using rulers and compass pairs, they tried to draw a square which would have the same surface area as a particular circle (the Delsk problem). This mathematical problem led Archimedes to try and calculate the number p , although he did not manage to fully succeed yet.

The Greeks could not manage to solve these four problems: dividing an angle into three parts, doubling the volume of a cube, computing the quadrature of a circle, and the number p . None of these problems were solvable by using a ruler and pair of compasses alone.

During the time of Alexander the Great, Alexandria became the world’s city of culture where scientific power was also concentrated. Archimedes and Euclid are two mathematical researches who advanced this science greatly.

Because, at that time, most mathematicians were also interested in astronomy, optics or mechanics, theoretical discoveries often went hand in hand with practical use. In the third century B.C., these discoveries were conic sections, ellipses, hyperbolas and parabolas.

Around 476 B.C. in India, Aryabhata calculated the number p to its fourth decimal point, managed to correctly forecast eclipses and, when solving astronomical problems, used sinusoidal functions. His compatriot Brahmagupta worked with negative numbers and defined the quadratic equation. Baskar then defined theories of numbers, analysis and algebra.

Around the year 900, the Arabs continued in advancing the knowledge of mathematics, particularly in the area of arithmetics, translating and adopting existing knowledge from the most varied of cultures. The concept of algorithms is derived from the name of the researher: Muhammad al-Chwarismis (also named al-Choresmi). Algebra is named after the name of the book written about it, and the Persion Omar-al Chajjam managed to derive the second and third roots from the fifth root. These mathematicians were concerned with the special problems of trigonometry.

Over the Middle Ages, the Indians and the Mezopotamians took over the lead in arithmetics, once again defining the number zero. Although the Mayans already knew of the number zero a long time beforehand, it was first used in India (in the Old World).

However, important discoveries made in Asia were not known in the rest of the world until 200 years later.

Even so, the Europeans had their own accomplishments, translating from Arabian and Greek mathematicians. In Germany in the 15th century, many new mathematical symbols were introduced and which still apply today. Nikolaus von Oresme studied mathematical problems of infinity and Adam Riese (actually Ries) counted using logarithmic rulers. In the 16th century, the Italians made major advances in geometry. By seeking a certain algebraic equation, Geronimo Cardano increased interest for complex and irrational numbers.

In the 17th and 18th centuries, European scientists crumbled < founding pillars of mathematical systems which applied up to that time and made further advancements in this area.

Now, the number one was no longer indivisible, and fractions were allowed values less than one.
Fractions also began to be written as decimal points and how they were calculated with.
Analytical geometry was presented in terms of
curves and coordinates.
In 1680, Isaac Newton and Gottfried W. Leibnitz came up with differential and integral calculus.
At the start of the 19th century, non-Euclidean geometry was born, which relied on the discovery of parallel axioms, which where independent of Euclidean geometry up until that time.
The famous mathematic of Carl Friedrich Gauss solved the problem of complex numbers.
Bernoulliov introduced variation calculus and Gaspard Monge descriptive geometry.
The so-called Fermatov assumption advanced algebra further.

Sets
and logics led the way to Boolean algebra, after whose logical connection was it possible to use the first computer.
Lagrange
created the dynamic system equation, which is a continually changing system.
Laplace
studied the theory of probability and Leonhard Euler was one of the prominent mathematicians who made major gains in analyses.

As such, mathematics became more exact and effective. But even in light of all these gain, the science was still not made complete. Without the definite computation of basic assumptions (axioms), as was proved by Kurt Güdel, it was not possible to completely explain mathematics.

The 20th century was a period of abstract mathematics, a move away from the previous mathematicians, who studied the theoretical fundamentals of their discipline.

The theory of chaos sprang up in the second half of the 20th century and which studies dynamic or changing systems, as is evident in nature. These systems are able to change in swings, such as the weather or developments on the stock market, which also make them dynamic system. As soon as these systems behave with abrupt erraticness, chaos researches start to study them. Jules Henri Poincaré cautioned about this problem at the end of the 19th century. At the end of the 1970s, Benoit Mandelbrot developed fractal geometry, which made it possible to graphically depict chaotic systems.

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