## 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

### 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.

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