Transcendental number

In mathematics, ie a real number (or more generally a complex number ) transcendent, if they can not occur as a zero of a polynomial with integer coefficients. Otherwise, it is an algebraic number. Each transcendental number is also irrational.

Degree of an algebraic number

Furthermore, it will also be important to have a definition of the so-called degree of an algebraic number. Is a complex number will be referred to as algebraic number of degrees, when an algebraic equation

But no such equation lesser degree sufficient. If they do not satisfy such an equation, so is not algebraic, then it is called transcendent.

Historical development of the concept of transcendence

The notion of mathematical transcendence came during the 18th century gradually in the reflections great mathematician Gottfried Wilhelm Leibniz ( omnem rationem transcendunt ) and Leonhard Euler, which although not a strict definition of this term had, but were nevertheless sure that it must give such mathematically " elusive " numbers of which Euler wrote that she " exceed [ ... ] the effectiveness of algebraic methods." 1748 claimed Euler in his textbook Introductio in Analysin Infinitorum even that with a positive rational and natural, which is not a square number, the number is not rational, but also " not irrational" is (where he " irrational numbers " today algebraic numbers mentioned speed range understood ). In fact, this transcendence conjecture in 1934 was confirmed as a special case of a result of the Russian mathematician Alexander Ossipowitsch Gelfond and Theodor Schneider of the German mathematician in their accuracy. Their proofs differ in essential points.

Joseph Liouville in 1844 could prove first the existence of transcendental numbers and provide explicit examples through its constructive method of proof. In his work he was able to show that there is a constant for each algebraic number of degree, so that for each rational approximation:

Applies. It follows that the number Liouville

Is transcendent.

See the proof of the approximation theorem of Liouville in the proof Archives

In 1874, Georg Cantor could not only once again prove the existence of transcendental numbers, but also show that it is transcendental "more" than algebraic numbers. Unlike Liouville Cantor's proof of the existence of transcendental numbers used any number-theoretic properties of algebraic numbers, but is ( from today's perspective ) amount of purely theoretical nature. However, the exact mathematical formulation of the concept of 'more' was certainly the most important result of Cantor's work, because it in-depth knowledge of the real number system revolutionary. However, his new ideas to influential conservative critics could not prevail as Leopold Kronecker long time. Cantor proved that the set of algebraic real numbers is countable ( in a modern manner of speaking), while the set of all real numbers uncountable ( infinite but not countable ) is. It also follows easily that the set of all transcendental numbers equally powerful with the set of all real numbers (in particular, also uncountable ) is.

This fact can amount linguistically be formulated as follows:

If the amount of transcendental numbers and the set of real numbers called, then:

Here is the set-theoretic symbol for the cardinality of; (pronounced " aleph null") is the set-theoretic symbol for the cardinality of a countably infinite set, in particular.

Transcendence of e and π evidence

The original proofs of the transcendence of and come from Charles Hermite or by Ferdinand von Lindemann. The evidence is, however, very difficult to understand. Over time, however, there were always simplifications of this evidence. A very " elegant " proof published the famous mathematician David Hilbert (1862-1943) in 1893 in his essay "On the transcendence of numbers and ".

See proof of the transcendence of and in the proof Archives

Examples of transcendental numbers

• The transcendence of π, which was proved by Carl Louis Ferdinand von Lindemann, is also the reason for the insolubility of the quadrature of the circle by ruler and compass.
• For algebraically. See also set of Lindemann - Weierstrass.
• . General could Gelfond 1934 and Schneider 1934 independently show with different methods: If, algebraic, algebraic and irrational, then is a transcendental number. This is a partial solution of Hilbert's seventh problem. For transcendent obviously this theorem does not apply because, for example (see also the set of Gelfond -Schneider).
• The sine of 1
• Ln ( ) for rational positive.
• And (see gamma function )
• . The bracket is in this case the floor function.

Generalization

In the context of general field extensions L / K is also considered elements in L which are algebraic or transcendental over K. See Algebraic element.