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*

**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

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

and so

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

are** transcendental**.

**Example**

Let

β is a real, transcendental number.

This is because there is a rational approximation

with

Analysing this inequality, the ratio

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

M x

Sorry to point out, but the proof is wrong. You are working with ring of polynomials with integer coefficients, so you are allowed to use “formal derivative” but you can’t use mean value theorem. It appears that you learnt the proof from online noted by Paul Garrett. The idea is correct (I guess) but you are mixing algebra with analysis. Astonishingly, I couldn’t find a simple exposition using Google. You can look at the proof discussed on pp. 19 of one of my write-ups: https://gaurish4math.files.wordpress.com/2015/12/diophantine-approximation-gaurish.pdf

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