The golden ratio has been called as "the most irrational number", based on a particular method called a continued fraction method. Using this continued fraction method the golden ratio has been stated as "the most irrational number". My question is: If there's a number with the greatest irrationality, then what is irrational number with the lowest irrationality among all irrational numbers ?
[Math] irrational number with the least/lowest irrationality
elementary-number-theory
Best Answer
Theoretically, the concept measure of irrationality of a real number $\alpha$ is, technically speaking, the following specialized notion: it is the infimum of all real $\mu$ for which there is a positive constant $A$ such that for all rational $\frac pq\ne \alpha$ with $q>0$ one has $$|\alpha - \frac pq|>\frac {A}{q^{\mu}}$$ This inequality indicates how “far” of the real $\alpha$ is a rational “close” to $\alpha$; in other words, all rational “near” to $\alpha$ determines its “distance” from $\alpha$; Or even, how rational can not approach $\alpha$.
See as example the “striking inequality” (Baker) discovered by Mahler in 1953 and today improved, $$|\pi-\frac pq|>\frac{1}{q^{42}}$$ valid for every rational $\frac pq;\space q> 1$
There is a whole Epica about this topic of measure of irrationality, beginning with Dirichlet and his approximation theorem (1842), Liouville and his discovery of the first known transcendental number (1844), passing through Thue (1909), Siegel (1929), Dyson (1947) and the gold brooch finisher with Klaus Friedrich Roth (1955) and his deep result which earned him the Fields Medal.
Theorem (Roth).- For all algebraic irrational $\alpha$ and all $\epsilon > 0$ the inequation $$ |\alpha- \frac pq |< \frac{1}{q^{2+\epsilon}}$$ has only a finite number of solutions in irreducible rational $\frac pq$ i.e. for all $\epsilon>0$ there is a positive constant $C(\alpha,\epsilon)$ such that for all rational $\frac pq$; $q>0$ one has $$|\alpha- \frac pq|>\frac {C(\alpha, \epsilon)}{q^{2+\epsilon}}$$ “The achievement is one that speacks for itself: it closes a chapter, and a new chapter is now opened. Roth’s theorem settles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated” (Harold Davenport, in his presentation of Roth to the Fields Medal at the International Congress in Edinburgh,1958).
With Liouville, measure of irrationality of a real algebraic $\alpha$ was equal to its degree $n$ and $n$ was successively decreasing (with the discoveries of the above mentioned authors) till the optimal value $ 2 $ established by Roth.