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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.
Namely, Lehmer showed that for relatively prime integers k and n with n > 2, the number 2 cos(2πk/n) is an algebraic number of degree φ(n)/2, where φ denotes Euler's totient function. Because rational numbers have degree 1, we must have n ≤ 2 or φ(n) = 2 and therefore the only possibilities are n = 1,2,3,4,6.
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 ...
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 ...
In number theory, a Liouville number is a real number with the property that, for every positive integer, there exists a pair of integers (,) with > such that < | | <. The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations.
So equivalence is defined by an integer Möbius transformation on the real numbers, or by a member of the Modular group (), the set of invertible 2 × 2 matrices over the integers. Each rational number is equivalent to 0; thus the rational numbers are an equivalence class for this relation.
For instance, the continued fraction representation of 13 / 9 is [1;2,4] and its two children are [1;2,5] = 16 / 11 (the right child) and [1;2,3,2] = 23 / 16 (the left child). It is clear that for each finite continued fraction expression one can repeatedly move to its parent, and reach the root [1;] = 1 / 1 of ...
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also ...