example of transcendental number

The following is a classical application of Liouville’s approximation theorem. For completeness, we state Liouville’s result here:

Theorem 1.

For any algebraic number $\alpha$ with degree $m\mathrm{>}\mathrm{1}$, there exists a constant $c\mathrm{=}c\mathit{}\mathrm{(}\alpha \mathrm{)}\mathrm{>}\mathrm{0}$ such that:

$$|\alpha -\frac{p}{q}|>\frac{c}{q^m}$$

for all rationals $p\mathrm{/}q$ (with $q\mathrm{>}\mathrm{0}$).

Next we use the theorem to construct a transcendental number.

Corollary 1.

The real number

$$\psi =\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{10}^{n!}}=0.1100010\mathrm{\dots }$$

is transcendental.

Proof.

Clearly, the number $\psi$ is well defined, i.e. the series converges. Indeed,

and ${\sum }_{n=1}^{\mathrm{\infty }}{10}^{-n}=1/9$. Thus, by the comparison test, the series converges and .

Suppose, for a contradiction, that $\psi$ is algebraic of degree $m$. We will construct infinitely many rationals $p/q$ such that

where $c=c(\psi )$ is the constant given by the theorem above. Let $k\in \mathbb{N}$ be such that . Then, in fact, we will show that there are infinitely many rationals $p/q$ with $q\ge 2$ such that

For all $j>k+m$ we define a rational number $p_j/q_j$ by:

$$p_j={10}^{j!}\sum _{n=1}^j{10}^{-n!},q_j={10}^{j!}$$

then $p_j$ and $q_j$ are relatively prime integers and we have:

$|\psi -{\displaystyle \frac{p_j}{q_j}}|$$=$${\displaystyle \sum _{n=j+1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{{10}^{n!}}}$

${\displaystyle \frac{1}{{10}^{(j+1)!}}}(1+{\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{{10}^2}}+\mathrm{\dots })$

$=$$10/9\cdot {\displaystyle \frac{1}{q_j^{(j+1)}}}$

${\displaystyle \frac{1}{q_j^j}}$

${\displaystyle \frac{1}{q_j^{(k+m)}}}$

where in the last inequality we have used the fact that $j>k+m$. Therefore, all rationals ${\{p_j/q_j\}}_{j=k+m+1}^{\mathrm{\infty }}$ satisfy the desired inequality, which leads to the contradiction with the theorem above. Thus $\psi$ cannot be algebraic and it must be transcendental.∎

Many other similar transcendental numbers can be constructed in this fashion.