You have a numberline, and on the numberline you have plotted two distinct
irrational numbers, which forms a line segment. You bisect that segment to form
two more, bisect each of those, ad infinitum.
Question: Will every irrational number between the two endpoints be plotted
eventually?
Douglas, Let's see if I understand you correctly.
Pick two distinct irrational numbers, a0 and a1.
Define a00 = a0, a10 = a1, and
a01 = (a0+a1)/2
Define a000 = a00, a010 = a01, and
a001 = (a00+a01)/2
Define a010 = a01, a100 = a10, and
a011 = (a01+a10)/2
Continue defining abitstring as follows:
For each pair of "neighboring" abitstring_k numbers
(where where k is the length of the
bitstring and neighboring means differing in binary value by one),
construct new abitstring_k+1 numbers as follows:
Define afirststring_k||0 = afirststring_k,
asecondstring_k||0 = asecondstring_k, and
afirststring_k||1 = (afirststring_k + asecondstring_k)/2
When you have done this, you will find that abitstring = a0 + b
(a1
- a0), where b is the value obtained by treating the bitstring as a
binary number, with the "binary point" immediately following
the first digit. Using this expression, you can solve for b and
convert to binary in order to see that every real number
between a0 and a1 has a unique (up to length, for finite strings)
corresponding bitstring, and every bitstring has a unique
corresponding real value.
