Search results
Results From The WOW.Com Content Network
In 1840, Liouville published a proof of the fact that e 2 is irrational [10] followed by a proof that e 2 is not a root of a second-degree polynomial with rational coefficients. [11] This last fact implies that e 4 is irrational. His proofs are similar to Fourier's proof of the irrationality of e.
This shows that any irrational number has irrationality measure at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.
Thus the accuracy of the approximation is bad relative to irrational numbers (see next sections). It may be remarked that the preceding proof uses a variant of the pigeonhole principle: a non-negative integer that is not 0 is not smaller than 1. This apparently trivial remark is used in almost every proof of lower bounds for Diophantine ...
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
However, there is a second definition of an irrational number used in constructive mathematics, that a real number is an irrational number if it is apart from every rational number, or equivalently, if the distance | | between and every rational number is positive. This definition is stronger than the traditional definition of an irrational number.
In the 10th century, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers. [11]
A more general proof shows that the mth root of an integer N is irrational, unless N is the mth power of an integer n. [7] That is, it is impossible to express the m th root of an integer N as the ratio a ⁄ b of two integers a and b , that share no common prime factor , except in cases in which b = 1.
The powers of two whose exponents are powers of two, , form an irrationality sequence.However, although Sylvester's sequence. 2, 3, 7, 43, 1807, 3263443, ... (in which each term is one more than the product of all previous terms) also grows doubly exponentially, it does not form an irrationality sequence.