Transcendental number

Number π.

A transcendental number, also called a transcendental number, is a number that is not a root of any algebraic equation with integer coefficients that are not all zero. A transcendental real number is not an algebraic number, since it is not a solution of any algebraic equation with rational coefficients. Nor is it a rational number, since these solve algebraic equations of the first degree; being real and not being rational, it is necessarily an irrational number. In this sense, transcendent number is antonym of algebraic number. The definition does not come from a simple algebraic relationship, but is defined as a fundamental property of mathematics. The best known transcendental numbers are π and e.

In general, if we have two bodies (K,+,⋅ ⋅ ){displaystyle (K,+,cdot)} and (L,+,⋅ ⋅ ){displaystyle (L,+,cdot)} so that the second is extension of the first, we will say that α α 한 한 L{displaystyle alpha in L} It's transcendent about K{displaystyle K} if there is no polynomial p한 한 K[chuckles]x]{displaystyle pin K[x]} of the α α {displaystyle alpha ,} is root (p(α α )=0{displaystyle p(alpha)=0,}).

The set of algebraic numbers is countable, while the set of real numbers is uncountable; therefore, the set of transcendental numbers is also uncountable. Or has the power of the continuum.

However, there are very few known transcendent numbers, and demonstrating that a number is transcendent can be extremely difficult. For example, it is not yet known if the constant of Euler (γ γ {displaystyle gamma ,}It is, being

γ γ ={displaystyle gamma =} 1+12+13+14+ +1n− − ln (n),{displaystyle 1+{frac {1}{2}}}+{frac {1}{3}+}{frac {1}{4}}}} +cdots +{frac {1}{n}}}-ln(n),} When n→ → +∞ ∞ {displaystyle nto +infty }.

In fact, you don't even know if γ γ {displaystyle gamma } is rational or irrational.

• Natural logarithms of positive reals, except powers of the number e,{displaystyle e,} are transcendent numbers; in the same way the values of trigonometric functions, except in some cases; there is a way of giving a transcendent number through continuous fractions, such as the case of the number of Archimedes or π. The difficulty is to prove whether the proposed number is transcendent or not.

The normality property of a number can help to prove whether it is transcendental or not.

History

The denomination «Transcendental» he coined it Leibniz when in an article of 1682 he showed that the function without (x){displaystyle textstyle sin(x)} It's not an algebraic function. x{displaystyle textstyle x}. Subsequently, Euler defined the transcendent numbers in the modern sense. The existence of the transcendental numbers was finally proved in 1844 by Joseph Liouville, who in 1851 showed some examples among those who were the "constant of Liouville":

␡ ␡ k=1∞ ∞ 10− − k!=0,11000100000000000000001000...... {displaystyle {sum _{k=1}^{infty }}}10^{-k!

where the nth digit after the decimal point is 1 if n is a factorial (i.e. 1, 2, 6, 24, 120, 720, etc.) and 0 elsewhere case. The first number to be shown to be transcendental without having been specifically constructed for it was e, by Charles Hermite in 1873. In 1882, Carl Louis Ferdinand von Lindemann published a proof that π is transcendental. In 1874 Georg Cantor found the argument described above establishing the ubiquity of transcendental numbers.

The discovery of these numbers has allowed the demonstration of the impossibility of solving several old geometry problems that only allow using straightedge and compass. The best known of them is that of squaring the circle, and its impossibility lies in the fact that π is transcendental. The same is not true of the other two "Greek problems" most famous, the doubling of the cube and the trisection of the angle, which are due to the impossibility of constructing numbers derived from polynomials of degree greater than two with a straightedge and compass (see Constructible Number) it is significant that these other two problems can be solved with modifications relatively simple method (allowing to mark the ruler, an action that Euclidean geometry did not tolerate) or with methods similar to the ruler and compass, such as origami, while the squaring of the circle, as it depends on the transcendence of π, is not solvable with those methods.

Examples

Here is a list of the most common transcendental numbers:

• e
• π
• 22{displaystyle 2^{sqrt {2}}}} or, more generally, ab{displaystyle a^{b},} where aI was. I was. 0,1{displaystyle aneq 0.1} is algebraic and b It's algebraic but irrational. The general case of Hilbert's seventh problem, that is, the determination of whether ab{displaystyle a^{b},} is transcendental when aI was. I was. 0,1{displaystyle aneq 0.1} is algebraic and b is irrational, it is partially demonstrated as true according to the Gelfond-Schneider theorem.
• ln (a){displaystyle ln(a),} Yeah. a is positive, rational and different from 1. See natural logarithm
• Interpreter Interpreter (13){displaystyle Gamma left({frac {1}{3}}right)} and Interpreter Interpreter (14){displaystyle Gamma left({frac {1}{4}}right)} (see Gamma function).
• Champernowne Number: C₁₀ = 0.123456789101112131415161718192021...
• Ω Ω {displaystyle Omega ,}Chaitin's constant.
• 1;,}" xmlns="http://www.w3.org/1998/Math/MathML">␡ ␡ k=0∞ ∞ 10− − β β k ;β β ▪1,{displaystyle sum _{k=0}^{infty }10^{-lfloor beta ^{krfloor };qquad beta ;}1;,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e151039b40b68a0213ce32763caac599045a815a" style="vertical-align: -3.171ex; width:23.407ex; height:7.009ex;"/>

where β β β β {displaystyle beta mapsto lfloor beta rfloor } It's the whole part function. For example, if β = 2 the number results

• ␡ ␡ k=1∞ ∞ 10− − k!=0,11,0000000000000000000001000.{displaystyle sum _{k=1}^{infty }10^{-k!}=0,110001000000000000000000001000... ! Liouville's constant.