In details

Origin of Irrational Numbers

The historical origin of the need for the creation of irrational numbers is closely linked with facts of a geometrical and arithmetic nature. Geometric ones can be illustrated with the problem of measuring the diagonal of the square when compared to its side.

This geometric problem drags another of an arithmetic nature, which consists in the impossibility of finding known - rational - numbers for square roots of other numbers, such as square root of 2.

These problems were already known to the Pythagorean School (5th century BC), which considered irrational heretics. Greek science has succeeded in deepening the whole theory of rational numbers, geometrically - "Euclid's Elements" - but has not advanced, for essentially philosophical reasons, in the field of the concept of number.

For the Greeks, the entire geometrical figure was formed by a finite number of points, these being conceived as tiny corpuscles - "the monads" - all the same; As a result, when measuring a length of n monads with another of m, that measure would always be represented by a ratio between two integers n / m (rational number); such length was then included in the commensurable category.

In finding the irrational, to which they cannot form fraction, the Greek mathematicians are led to conceive immeasurable quantities. The line where all rationals were marked was, for them, perfectly continuous; admitting the irrational was imagining it full of "holes". It is in the century. XVII, with the creation of Analytic Geometry (Fermat and Descartes), which establishes the symbiosis of geometric with algebraic, favoring the arithmetic treatment of the commensurable and the incommensurable. Newton (1642-1727) first defines "number", both rational and irrational.

ø = 1.6180339887… or ø = (1 + sqr (5)) / 2 is considered a symbol of harmony. Greek artists used it in architecture; Leonardo da Vinci, in his artistic works; and, in the modern world, the architect Le Corbusier, based on it, presented, in 1948, The modulor. The gold number is found in metric relations:
- in nature: in animals (as in the Nautilus shell) flowers, fruits, the arrangement of the branches of certain trees;
- in geometrical figures such as the gold rectangle, regular hexagon and decagon and regular polyhedra;
- in numerous monuments, from the Cheops Pyramid to various cathedrals, in sculpture, painting and even in music.

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