The Paradox of 1 = 0.9 repeating
Sven Gelbhaar
06:35 05.10.2018
The problem is: 1/3 = 0.3 repeating. 3 * 1/3 = 1 = 0.9 repeating
0.9 repeating cannot exist in empirical/objective/physical world.
Irrational/rational (decimal vs fractions) don’t exist. Let’s explore some
possible reconciliations.
Maybe it has to deal with the base of our calculations (10 in the above case).
It turns out that it isn’t our base-ten number scheme, for 0.3 (repeating,
decimal: 1 / 3) comes out to 0.307692307 (repeating, octal, 04 / 015) and
0x1388888 (8 repeating, hexadecimal, 0x5 / 0x24) So that didn’t pan out.
However, there might be an alternative — philosophical — solution yet.
For all intents and purposes, 1/3 (decimal) does not exist in nature. Reality
would have to be infinitely divisible, and that flies in the face of Planck who
stated that the smallest divisors of physical space were 1.6 x 10^-35 m. This
is the quantized perspective of dimensions, but it doesn’t seem right because we
presume the universe to be infinite in size so why limit (in this case the
scope) of dimensions arbitrarily? As the measurements get smaller and smaller
these little things cease to matter as much, so we can safely assume that actual
irrational numbers probably don’t exist. The same can be said of 1.000 (zero
repeating). Infinite resolution of numbers, in this case an unbounded bottom
scope, are unusable in our rendition of mathematics.
The problem with 1 = 0.9 repeating has to do with approximations inherent in
(how we process) set theory. We approximate 1/3 to be .3 repeating and the rest
follows suit.