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Liouville's Number is defined as $L = \sum_{n=1}^{\infty}(10^{-n!})$. Does it have other applications than just constructing a transcendental number?

(Personally, I would have defined it (as "Steven's Number" :-)) as binary: $S = \sum_{n=1}^{\infty}(2^{-n!})$, since each digit can only be "0" or "1": the corresponding power of 2 (instead of 10) included or not. Since according to Cantor most number are transcendental one can conjecture that this is also the case for Steven's Number. Can a proof for this be devised based on the proof for Liouville's number?)

I'm not a mathematician, so please type slowly! :-)

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My understanding is that Liouville's number is of primarily historical importance: it was the first well-known explicit example of a transcendental number. That's about all you can say about it. –  Qiaochu Yuan Aug 28 '10 at 12:41
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Steven's Number is also a Liouville Number and indeed it is trancendental! Liouville argument proves that all numbers satisfying a certain criteria are transcendental and $L$ is just one example. –  anon Sep 11 '10 at 7:34
    
I guess I should have said "Liouville's constant" instead of "Liouville's number". –  stevenvh Sep 11 '10 at 8:39

2 Answers 2

up vote 11 down vote accepted

the number $S$ is also transcendental, for precisely the same reason as $L$: they both have rational approximations that are far too good to hold for an algebraic number. See here for more details.

If the specific number $L$ is useful for anything else, I'm not aware of it. It's really just a simple example, one of many which can be dreamed up, of a number which fails to be algebraic since it defies Liouville's Lemma in diophantine approximation.

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Liouville's Number makes it's next appearance in Making Transcendence Transparent - Edward B. Burger, Robert Tubbs. The transcendence of $e + L$ is proved. (This might be the end of it's career though).

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