Liouville's Constant is Transcendental
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Theorem
\(\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {10^{n!} }\) | \(=\) | \(\ds \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} + \frac 1 {10^{24} } + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0.11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots\) |
is transcendental.
Corollary
All real numbers of the form:
\(\ds \sum_{n \mathop = 1}^\infty \frac {a_n} {10^{n!} }\) | \(=\) | \(\ds \frac {a_1} {10^1} + \frac {a_2} {10^2} + \frac {a_3} {10^6} + \frac {a_4} {10^{24} } + \cdots\) |
where
- $a_1, a_2, a_3, \ldots \in \set {1, 2, \ldots, 9}$
are transcendental.
Proof
Let $q = 10^{n!}$ and write:
- $\ds L = \frac p q + \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }$
for some suitable $p \in \Z$.
Then:
\(\ds \size {L - \frac p q}\) | \(=\) | \(\ds \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \frac 2 {10^{\paren {n + 1}!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {q^{n + 1} }\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \frac 1 {q^n}\) | as $q \ge 10$ for all $n \in \N$ |
Thus, by definition, $L$ is a Liouville number.
Therefore, by Liouville's Theorem, $L$ is transcendental.
$\blacksquare$
Historical Note
Liouville's constant was proved to be transcendental by Joseph Liouville in $1844$ as a demonstration that there exist real numbers which are provably transcendental.
This was the simplest of several such numbers that he constructed.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 11000 10000 00000 00000 00010 00000 00000 00000 0 \ldots$
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00000 \, 00000 \, 00000 \, 0 \ldots$