Natural Blogarithms

Champernowne’s Constant

September 21st, 2006 by SplineGuy

During Calculus a few days ago, I covered an interesting little number called Champernowne’s Constant. We were in the middle of introducing the concept of infinite sequences of numbers and their convergence. We stated the theorem that states than any monotone, bounded sequence must converge. After review mathematical induction and proving the convergence of a couple example, I gave them the following example.

Let us construct a sequence in the following way:
$a_1 = 0.1$
$a_2 = 0.12$
$a_3 = 0.123$
$a_4 = 0.1234$
$a_5 = 0.12345$
$a_6 = 0.123456$
$a_7 = 0.1234567$
$a_8 = 0.12345678$
$a_9 = 0.123456789$
$a_{10}=0.12345678910$
$a_{11}=0.1234567891011$
…
Notice that each time we simply append to the the previous value the next integer to the end of the decimal expansion. Now notice that because each term is appended at the end of the expansion, then each term is necessarily larger than the last. This implies that this sequence is increasing (thus, monotonic). Also, this sequence is bounded. For example, it will never be larger than 0.2 nor will it be smaller than 0. So, by the previously mentioned theorem it is convergent. So, it converges to something and we call that something the Champernowne Constant.

Now, the interesting thing about this sequence is that it contains every possible finite seqeuence of numbers. That is, eventually, any number will appear somewhere in the Champernowne Constant. For example, 32084701283472 will appear somewhere, because of the nature of the constant. So if one were to take any book and convert it to a number using the code A=1, B=2, etc., that book appears somewhere in the Champernowne Constant. This book could be already written, e.g. Hamlet is in there, Harry Potter, too. The book might not have even been written yet. In other word’s, the Champernowne Constant contains the future hidden somewhere in its sequence.

Now don’t get too excited, the information is not really there since along with every book every written it contains every possible ordering of letters so information is not discernible from the rest of the gibberish that is there as well. But, it’s still an interesting concept. Nostradomus has nothing on Champernowne.

Share and Enjoy: