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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 the 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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