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
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,1100010000$
- 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): $0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00000 \, 00000 \, 00000 \, 0 \ldots$