Diophantine Approximation: Liouville’s Theorem

Diophantine approximation deals with the approximation of real numbers by rational numbers.

Liouville’s Theorem

In the 1840’s Liouville obtained the first lower bound for the approximation of algebraic numbers:

Let α ∈ R be an irrational algebraic number satisfying f(α) = 0 with non-zero irreducible (cannot be reduced) f ∈ Z[x] of degree d. Then there is a non-zero constant C such that for every fraction p/q

Screen Shot 2016-12-11 at 10.35.05 AM.png

Proof

The proof utilises the mean value theorem. By this theorem, given p/q, there is a real ξ between α and p/q such that

Screen Shot 2016-12-11 at 10.35.08 AM.png

Since f has integer coefficients and is of degree d, the value of f(p/q) is a rational number with denominator at worst q^d. Since f is irreducible, f(p/q) cannot be equal to 0. Thus

Screen Shot 2016-12-11 at 10.40.22 AM.png

and so

Screen Shot 2016-12-11 at 10.40.58 AM.png

A corollary of this result is that numbers that are well approximable by rational numbers, i.e. in for every d ≥ 1 and positive constant C, there is a rational p/q such that

Screen Shot 2016-12-11 at 10.43.32 AM.png

are transcendental.

Example

Letscreen-shot-2016-12-11-at-10-45-22-am

β is a real, transcendental number.

This is because there is a rational approximation

Screen Shot 2016-12-11 at 10.46.43 AM.png

with

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Analysing this inequality, the ratio

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is unbounded as N → +∞, and so β is well approximable by rationals.

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Transcendental Numbers

A transcendental number is a number that is real or complex, but it not algebraic, meaning that it is not the root of a polynomial with non-zero integer coefficients. For example, √2 is algebraic as it is the solution to the polynomial equation x– 2 = 0.

History

In 1844, Joseph Liouville proved the existence of transcendental numbers and in 1851 he gave the first example of such a number:

= 0.11000100000000000000000100……

(i.e. the nth digit after the decimal point is 1 if n = k! for some k, and 0 otherwise). This number is now known as the Liouville constant.

Only in 1873 was the first ‘non-constructed’ number shown to be transcendental when Charles Hermite proved that e was transcendental. Then, in 1882, Ferdinand von Lindemann proved that π was transcendental.

In fact, proving a number is transcendental is extremely challenging, even though they are known to be very common.

Why are they very common?

The algebraic numbers are countable (the set of algebraic numbers is the countable union of countable sets and so is therefore countable). However, the real numbers are uncountable. Therefore, since every real number is either algebraic or transcendental, the transcendentals must be uncountable. This implies that there are a lot more transcendental numbers than algebraic numbers.

Examples of Transcendental Numbers

  • ea if a is algebraic and non-zero
  • π
  • eπ
  • ab where a,b are algebraic, but a ≠ 0,1
    • in particular, 2^{\sqrt {2}}, the Gelfond-Schenider Constant
  • Continued Fraction Constant{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{4+{\cfrac {1}{5+{\cfrac {1}{6+\ddots }}}}}}}}}}}

If you want to find out more examples, click here.

 

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