Translations of Encyclopedia about 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 haands, 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 aand 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 aand 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 aand numbering systems in Mesopotamia (Babylon) aand Egypt. The mathematician aand scholar Euclides started his famous school of mathematics in Alexaandria, as such laying the corner stone of mathematics. The Egyptians used a decadic numbering system, which is based on the number 10 aand still in use today. They also introduced characters used to describe the numbers 10 aand 100, making it easier to describe larger numbers. Geometry started to receive great attention aand served in surveying laand, cities aand streets.

The Egyptians knew how to count in fractions aand 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 aand used wedge-shaped symbols aand arrows when making numerical records. Using their mathematical tool, they even managed to solve quadratics, aand 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 aand mathematics, adopting knowledge in mathematics from both the Babylonians aand Egyptians to create generally binding theorems. The Greeks founded everything on strict logic aand 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 beforehaand. Greek astronomers sped up developments of geometry, making calculations of heavenly bodies aand the diameter of the earth aand proving that the earth is in fact round. Euclid is a Greek mathematician who documented knowledge from his times in the most demaanding 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, aand 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 aand using rulers aand 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 aand 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, aand the number p . None of these problems were solvable by using a ruler aand pair of compasses alone.

During the time of Alexaander the Great, Alexaandria became the world’s city of culture where scientific power was also concentrated. Archimedes aand 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 haand in haand with practical use. In the third century B.C., these discoveries were conic sections, ellipses, hyperbolas aand parabolas.

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

Around the year 900, the Arabs continued in advancing the knowledge of mathematics, particularly in the area of arithmetics, translating aand 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, aand the Persion Omar-al Chajjam managed to derive the second aand third roots from the fifth root. These mathematicians were concerned with the special problems of trigonometry.

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

Now, the number one was no longer indivisible, aand fractions were allowed values less than one.
Fractions also began to be written as decimal points aand how they were calculated with.
Analytical geometry was presented in terms of curves aand coordinates.
In 1680, Isaac Newton aand Gottfried W. Leibnitz came up with differential aand integral calculus.
At the start of the 19^th 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 aand Gaspard Monge descriptive geometry.
The so-called Fermatov assumption advanced algebra further.
Sets aand 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 aand Leonhard Euler was one of the prominent mathematicians who made major gains in analyses.