Download How REAL is the Real Line?

Aum Sri Sai Ram
How REAL is the Real Line?
Natural numbers were natural to our ancestors as they needed to count their sheep. But the
discovery of zero had to wait till the concept of “Shunya” of Hindu Philosophy could sink into
human psyche. The negative numbers emerged so that a + b = 0 could be solved for every natural
number a . This created an Integral domain. In an integral domain there are balances and counter
balances, pluses and minuses, where punya and papa co-exist, just as day and night, bright and
darkness – equal but opposite.
A bitter truth:
The development did not stop there. The next rational step was the Rational numbers.
Rational numbers were introduced to solve the multiplication problem a.b = 1 , for any integer
a≠0.
Caution: DO NOT divide by zero. But rationals could not solve other multiplication
problems such as x 2 = 2 , exploding the myth of rational thinking!
“What is the reality that we need to realize from the real line?”
muses Prof. C. Jagan Mohan Rao
in the fifth of the issues under this monthly column.
Thus irrationals came flooding in, filling in every bit of gap in the real line. The two
together became REAL. Now we have a real number for each point on the real line ℜ and a point
on the real line for each real number. But in this world of the real line there are overwhelmingly
more irrationals than rationals! A bitter truth, we are forced to live with.
Algebraic Completeness:
The next logical question is “Are the real numbers complete?”, i.e., are these real numbers
sufficient to solve all problems. The real world still does not have a solution to the problem
x2 + 1 = 0 ,
where
as square of every real number is non-negative. So imagine an imaginary number a + ib ,
i 2 = −1 .
Strange as it may sound, it solves all problems. Every polynomial has a root and in
fact, a polynomial of degree n has exactly n roots, if a root is counted to its multiplicity. This is
algebraic completeness. This put a stop to the extensions of real numbers. Every type of
completeness that needs to be attained has been attained.
So is the real world Incomplete?
Does that mean the real numbers are not so real after all? Well, not if we tackle the issue of
completeness in the following two perfectly mathematical ways.
Topological Completeness
Order -Theoretic Completeness
We introduce a metric on the real numbers We introduce an order on the real numbers
through the norm
x = x
. Under this metric as
a≥b
if
a + (−b)
is positive. The order is
real numbers are complete: Every Cauchy reflexive, anti-symmetric and transitive. It
sequence of real numbers converges.
is a total order which is complete: Every
bounded subset has a supremum.
A Cauchy sequence is one whose members Supremum is an upper bound of any
ultimately get closer to each other.
bounded set.
A sequence converges if all the members Supremum is closest to the bounded set
ultimately get closer to some ONE.
among all upper bounds.
In essence topological completeness says: In essence Order-theoretic completeness
Loving thy neighbour is loving GOD.
says: If one remains within bounds, he is
protected by the Supreme, the ONE
beyond him but closest to him.
For the real numbers the above two types of completeness are equivalent, which is to say,
“You can love your neighbour if and only if you remain within your bounds”
This is the reality that we need to realize from the real line.
If "Number rules the universe" as Pythagoras asserted, Number
is merely our delegate to the throne, for we rule Number.
Bell, Eric Temple (1883-1960)
Cantor’s Ternary Set:
An interesting subset of the real numbers is the Cantor’s Ternary Set. Divide the closed
interval [0,1] into three equal parts and delete the middle portion. Divide each of the two
remaining portions into three equal parts and delete from each the open middle part. Continue this
process.
0
1/3
2/3
1
0
1/3
2/3
1
1/3
2/3
1
0
1/9
2/9
When you have finished ( Will you ever finish? No! We take a quantum jump of
imagination and imagine how it will be if we finish the job – a job which in reality can never be
finished!), what remains of the interval [0,1] is the Cantor’s Ternary set. It has some very
interesting properties:
a) It is an uncountable subset of [0,1] .
b) It is a closed subset.
c) Every point of it is a limit point.
d) It includes some rational numbers but not all of those in [0,1] .
e) It includes some irrational numbers but not all of those in [0,1] .
f) Its interior is empty.
g) It is a nowhere dense set.
h) Its measure is zero.
i) It is a perfect set.
j) Ternary expansions of each member of this set will have the form 0 ⋅ b1 b 2 b3 K , where each
bi
is either a
0
or 2 , i.e. bi ≠ 1 for all i .