Das Kalenderblatt 110430



Emile Borel introduced the concept of a normal real number in 1909.
His idea was to provide a test as to whether the digits in a real
number occurred in the sort of way they would if we chose each one at
random. First assume that we have a real number written in base 10,
that is a decimal expansion. Then if it is a "random" number the digit
1 should occur about 1/10 of the time […] a specific 2-digit number,
say 47, should occur among all two digit blocks about 1/100 of the
time etc. Borel called a number normal (in base 10) if every k-digit
number occurred among all the k digit blocks about 1/10k of the time.
{{Nach Cantor gibt es genau so viele einstellige Ziffernblöcke wie
zehnstellige oder tausendstellige. Die Anzahl der zweistelligen und
der dreistelligen Ziffernblöcke ist genau so groß wie die der
fünfstelligen. Was ist das für ein Kalkül? Es erinnert stark an 1/2 +
1/3 = 1/5. Achtung nicht abschreiben oder anwenden! Diese Kalkulation
ist falsch! Andererseits ... Wenn Stammbrüchler brachial die
Untersuchung von Kehrwerten verböten, so wie Matheologen beim Banach-
Tarski-Paradoxon die realistische Interpretation verbieten - wer
weiß?}} He called a number absolutely normal if it was normal in every
base b.
Now Borel was able to prove that, in one sense, almost every real
number was normal. His proof of this involved showing that the non-
normal numbers formed a subset of the reals of measure zero. There
were still an uncountable number of non-normal numbers, however, which
was easily seen by taking the subset of all real numbers with no digit
equal to 1. These are uncountable as can be seen using Cantor's
diagonal argument, but clearly they are all non-normal. Clearly no
rational is normal since eventually it ends in a repeating pattern.
However despite proving these facts, Borel couldn't show that any
specific number was absolutely normal. This was achieved first by
Sierpinski in 1917.
In 1933 David Champernowne, who was an undergraduate at Cambridge
University and a friend of Alan Turing, devised Champernowne's number.
Write the numbers 1, 2, 3, ..., 9, 10, 11, ... in turn to form the
decimal expansion of a number
0.12345678910111213141516171819202122232425262728293031323334353637383940 ....
In 1946 Copeland and Erdős proved that the number
0.2357111317192329313741434753596167717379838997101103107109113127131137139 ...
obtained in a similar way to Champernowne's number, but using primes
instead of all positive integers, was normal. Neither Champernowne's
number nor the Copeland and Erdős number is absolutely normal.
It is reasonable to ask whether pi, Wurzel 2, e etc are normal. The
answer is that despite "knowing" that such numbers must be absolutely
normal, no proof of this has yet been found. In fact although no
irrational algebraic number has yet been proved to be absolutely
normal nevertheless it was conjectured in 2001 that this is the case.
[J.J. O'Connor and E.F. Robertson: "The real numbers: Attempts to
understand"]
http://www-history.mcs.st-and.ac.uk/HistTopics/Real_numbers_3.html
Gruß, WM
.



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