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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 ...
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
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.
If we don't allow √ 2 then we can increase the number on the right hand side of the inequality from 2 √ 2 to √ 221 /5. Repeating this process we get an infinite sequence of numbers √ 5, 2 √ 2, √ 221 /5, ... which converge to 3. [1] These numbers are called the Lagrange numbers, [2] and are named after Joseph Louis Lagrange.
The greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, [3] earliest and most notably by J. J. Sylvester [4] A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative ...
In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply ...
Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots, that is, expressions of the form , in which and are integer numerals and the whole expression denotes an irrational number. [6] Irrational numbers of the form , where is rational, are called pure quadratic ...
Farey sequences are very useful to find rational approximations of irrational numbers. [15] For example, the construction by Eliahou [ 16 ] of a lower bound on the length of non-trivial cycles in the 3 x +1 process uses Farey sequences to calculate a continued fraction expansion of the number log 2 (3) .