I suggested earlier that the real number set was infinitely dense,
i.e. between any two numbers, however close to each other they are, there
are an arbitrarily large number of numbers in between them. This fact was
recognized 2000+ years ago by Pythagoras and others when it was thought
that all numbers could be represented by rational fractions.
A rational fraction is one in which the numerator and the
denominator are whole numbers with no prime factors in common, i.e. a
fraction reduced to its lowest terms. It is easy to show that all such fractions reduce to a
repeating decimal. Consider
the rational fraction 2/7 reduced to decimal form using the standard
methods of long division:
By the rules of long division, the remainder, 6 in this case, must
be less than the divisor, 7. There
are only 6 positive integers less than 7, so sooner or later we will
exhaust the list of possibilities and the decimal will repeat.
This is obviously true for all sizes of divisors.
I once read somewhere that the Louisiana Legislature had passed a
law making it legal for contractors doing business with the state to use
the fraction 22/7 = 3.1428571428... etc. instead of PI = 3.141592654...
In any case, there were mathematicians who believed that, if they
could but do the sums, numbers like PI and ÷2
= 1.4142135... could eventually by reduced to rational fractions.
The idea had a strong appeal to their intuition regarding the
grandeur and divine order of the rational world.
The following proof of the existence of irrational
numbers, due to Euclid, settled the matter for some. First we need to convince ourselves that the squares of odd
numbers are odd while the squares of even numbers are even, and visa
versa. If N is a positive
integer, 2N+1 is odd and 2N+2 is even, while (2N+1)^2 = 4N^2 + 4N + 1 is
odd and (2N+2)^2 = 4N^2 + 8N + 4 is even.
= A/B where A and B are integers with no prime factors in common.
Then square both sides and find A^2 = 2 B^2.
Thus A^2 is even and A is even, or A = 2 A' where A' may be odd or
even and A^2 = 4A'^2. But A^2 = 4A'^2 = 2 B^2, so B^2 = 2 A'^2.
Thus B^2 and B are also even.
This is contrary to the original proposition that A and B have no
prime factors in common, but if both A and B are even, they do have a
common prime factor, namely 2. The
can not be expressed as a rational fraction.
This doesn't mean that PI and e and all the rest can not be
expressed as rational fractions, but it does prove the existence of
irrational numbers. Of
course, if we think about the decimal representation of a number, it is
pretty clear that we can construct such a number in which each entry is
random and independent of all other entries and thus non-repeating.
The irrational number set is, in fact, infinitely more dense than
the rational number set and the probability of picking a rational number
at random from any interval on the real number line is zero.
Zeno's Paradox, of similar antiquity, is somewhat related to this
issue. Zeno proposed a
thought experiment in which Achilles, a swift runner, is matched in a race
with Tortuga, a tortoise. Tortuga
is given a head start and his original position is noted very precisely on
the real number line, while Achilles starting point is at the zero. The starting gun is fired and Achilles and Tortuga begin the
race along the real number line. When
Achilles reaches the starting point of Tortuga, he finds that Tortuga is
ahead of him, regardless of how fast each is moving.
The size of the interval between them at this point may be shorter,
but is this really relevant in abstract number space? Now we note the new position of Tortuga and see that when
Achilles reaches this new position he finds Tortuga still ahead of him.
How, asks Zeno, can Achilles ever get ahead?
Let us try to set up the problem on a digital computer.
The registers for keeping accounts of the positions of Achilles and
Tortuga are discrete and finite in length, unlike the real number line,
but we can perhaps use this model to reframe the issue.
We set the register representing Achilles position to zero and the
register representing the position of Tortuga to some number larger than
zero. A clock moves Achilles position along one step at each tick.
Tortuga moves slower, so his position is moved along less
frequently, say every third tick of the clock.
When Achilles reaches Tortuga's original starting point, Tortuga is
ahead, but by fewer positions. We
note Tortuga's new starting point and proceed with the race.
Sooner or later Achilles will have reduced the distance to less
than 3 spaces after which point he passes Tortuga before the tortoise can
take another step.
Clearly, Zeno's Paradox doesn't hold up in a discrete digital
universe regardless of how many columns there are in the registers, but
what about the infinitely dense real number line? We can start Achilles at the zero position and Tortuga at
some other well defined position on the real number line. Now as Achilles begins the race, is it clear that his
continuous motion can be correlated exactly to each and every number on
the continuous real number line? Is
it clear that he will eventually come exactly to the starting point of
Tortuga? If we note the
position of Achilles by the location of some specific atom in his body, we
can find his position to roughly 8 or 9 significant figures in decimal
notation, but this is hardly accurate enough for the problem at hand.
Tortuga's new starting point is even more ambiguous.
Perhaps Zeno's Paradox comes down to the question of whether the
real universe is discrete or continuous.
I have no problem with the number line being continuous and doubly
infinitely dense, but my intuition tells me that the universe is discrete.