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In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that e is not a root of a third-degree polynomial with rational coefficients, which implies that e 3 is irrational. [12] More generally, e q is irrational for any non-zero rational q. [13] Charles Hermite further proved that e is a transcendental number, in 1873 ...
The real number e is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate. [38] (See also Fourier's proof that e is irrational.) Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational ...
A stronger result is the following: [31] Every rational number in the interval ((/) /,) can be written either as a a for some irrational number a or as n n for some natural number n. Similarly, [ 31 ] every positive rational number can be written either as a a a {\displaystyle a^{a^{a}}} for some irrational number a or as n n n {\displaystyle n ...
In other words, the n th digit of this number is 1 only if n is one of the numbers 1! = 1, 2! = 2, 3! = 6, 4! = 24, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by rational numbers than can any irrational algebraic number, and this class of numbers is called the ...
The numbers π and e π are also known to be algebraically independent over the rational numbers, as demonstrated by Yuri Nesterenko. [3] It is not known whether e π is a Liouville number. [ 4 ] The constant was mentioned in Hilbert's seventh problem alongside the Gelfond-Schneider constant 2 √ 2 and the name "Gelfond's constant" stems from ...
In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...
The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root √2 = 1.414...; these are called algebraic numbers.
Any complex number = + can be represented by the point (,) on the complex plane. This point can also be represented in polar coordinates as ( r , θ ) {\displaystyle (r,\theta )} , where r is the absolute value of z (distance from the origin), and θ {\displaystyle \theta } is the argument of z (angle counterclockwise from the positive x -axis).