- 0 < |
*x*-*p*/*q*| < 1/*q*^{n}.

It is relatively easily proven that if *x* is a Liouville number, *x* is irrational. Assume otherwise; then there exists integers *c*, *d* with *x* = *c*/*d*. Let *n* be a positive integer such that 2^{n-1} > *d*. Then if *p* and *q* are integers such that *q*>1 and *p*/*q* ≠ *c*/*d*, then

- |
*x*-*p*/*q*| = |*c*/*d*-*p*/*q*| ≥ 1/*dq*> 1/(2^{n-1}*q*) ≥ 1/*q*^{n}

In 1844, Joseph Liouville showed that numbers of with this property are not just irrational, but are always transcendental (see proof below). He used this result to provide the first example of a provably transcendental number,

More generally, the **irrationality measure** of a real number *x* is a measure of how "closely" a number can be approximated by rationals. Instead of allowing any *n* in the power of *q*, we find the least upper bound of the set of real numbers μ such that

- 0 < |
*x*-*p*/*q*| < 1/*q*^{μ}

The Liouville numbers are precisely those numbers having infinite irrationality measure.

**Lemma:** If α is an irrational number which is the root of a polynomial *f* of degree *n* > 0 with integer coefficients, then there exists a real number *A* > 0 such that, for all integers *p*, *q*, with *q* > 0,

- |α -
*p*/*q*| >*A*/*q*^{n}

*A*< min(1, 1/*M*, |α - α_{1}|, |α - α_{2}|, ..., |α - α_{m}|)

- |α -
*p*/*q*| <=*A*/*q*^{n}<=*A*< min(1, |α - α_{1}|, |α - α_{2}|, ..., |α - α_{m}|)

By the mean value theorem, there exists an *x*_{0} between *p*/*q* and α such that

*f*(α) -*f*(*p*/*q*) = (α -*p*/*q*) ·*f*′(*x*_{0})

- |(α -
*p*/*q*)| = |*f*(α) -*f*(*p*/*q*)| / |*f*′(*x*_{0})| = |*f*(*p*/*q*)| / |*f*′(*x*_{0})|

- |
*f*(*p*/*q*)| = |∑_{i = 1 to n}*c*_{i}*p*^{i}*q*^{-i}| = |∑_{i = 1 to n}*c*_{i }*p*^{i}*q*^{n-i}| /*q*^{n}≥ 1/*q*^{n}

Thus we have that |*f*(*p*/*q*)| ≥ 1/*q*^{n}. Since |*f* ′(*x*_{0})| ≤ *M* by the definition of *M*, and 1/*M* > *A* by the definition of *A*, we have that

- |(α -
*p*/*q*)| = |*f*(*p*/*q*)| / |*f*′(*x*_{0})| ≥ 1/(*M**q*^{n}) >*A*/*q*^{n}≥ |(α -*p*/*q*)|

**Proof of assertion:** As a consequence of this lemma, let *x* be a Liouville number; as noted in the article text, *x* is then irrational. If *x* is algebraic, then by the lemma, there exists some integer *n* and some positive real *A* such that for all *p*, *q*

- |
*x*-*p*/*q*| >*A*/*q*^{n}

- |
*x*-*a*/*b*| < 1/*b*^{m}= 1/*b*^{r+n}= 1/(*b*^{r}*b*^{n}) ≤ 1/(2^{r}*b*^{n}) ≤*A*/*b*^{n}