Definition:Liouville's Constant

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Definition

Liouville's constant is the real number defined as:

\(\ds \sum_{n \mathop \ge 1} \dfrac 1 {10^{n!} }\) \(=\) \(\ds \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} + \frac 1 {10^{24} } + \cdots\)
\(\ds \) \(=\) \(\ds 0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots\)

This sequence is A012245 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also known as

Some sources refer to this as Liouville's number.


Also see


Source of Name

This entry was named for Joseph Liouville.


Historical Note

Liouville's Constant was created by Joseph Liouville in $1844$ as an example of a real number which is provably transcendental.

He constructed several such numbers, of which this is the simplest.


Sources