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From a Point on a Line, to a Point in the Plane, to ... the Axiom of Choice

Date: 01/26/2011 at 17:26:31
From: Alex
Subject: One-to-One Correspondence Between Real Line and Real Plane

I'm a high-school freshman enrolled in an Algebra 2 class exploring
concepts in set theory and topology. I was wondering if it was possible to
establish a one-to-one correspondence from a point on the real number line
to a point on the two-dimensional real plane. That is, can you create a
function that will transform any number (or point) on the real line into
an ordered pair (or point) on the real plane and vice versa?

I can't think of a tiling or correspondence that would transform the line
into a plane. I know that the real plane is a two-dimensional manifold
while the real line is one-dimensional, and infinitely thin, so I can't
imagine how you can create a homeomorphism between them.

I've considered

   - concentric circles formed from the real line, with a fractal-like
     tiling of them, thinking that it might produce the plane at infinity
   - an infinitely dense spiral from the origin
   - creating an ordered pair from the numerator and denominator of
     rational numbers -- but this creates duplicate points, so it's not
     one-to-one (e.g., 1/2 maps to (1,2); but 2/4 which is equal to 1/2
     maps to (2,4))

I have a hunch this may be connected to Cantor sets and cardinality (from
what I've read), but I haven't found a clear answer to my question.

Date: 01/26/2011 at 23:09:31
From: Doctor Vogler
Subject: Re: One-to-One Correspondence Between Real Line and Real Plane

Hi Alex,

Thanks for writing to Dr. Math. 

Yes, this can be done. For example, one way to do this is to start with a
correspondence between the integers Z and two copies of the integers
{0,1}xZ, such as

  even n -> (0, n/2)
   odd n -> (1, (n - 1)/2)

Then every real number r can be written in decimal as a sequence of
digits, so r = the sum of d_i * 10^i where each d_i is a digit (integer
between 0 and 9) and the sum runs over all integers i. So you write the
number as


Now you just map your digits to the digits of a pair of real numbers by
taking the d_n digit and mapping it to either the d_(n/2) digit of the
first number or the d_((n - 1)/2) digit of the second number, according to
whether n is odd or even. For example, it would map ...

        123456789 -> (13579, 2468)
   1234.567891234 -> (24.6813, 13.57924)
  pi (3.14159...) -> (3.4525599286638..., 0.11963873342433...)

... and so on.

Of course, this is not the only way to do it.

If you have any questions about this or need more help, please write back
and show me what you have been able to do, and I will try to offer further

- Doctor Vogler, The Math Forum 

Date: 01/26/2011 at 23:38:09
From: Alex
Subject: Thank you (One-to-One Correspondence Between Real Line and Real Plane)

Thanks so much for your help!

I'm assuming that this can be generalized to higher n dimensions using
mod-n arithmetic.

It's very counterintuitive that this sort of bijection exists and you did
an excellent job of helping me understand it. 

Thank you.

Date: 01/27/2011 at 23:01:55
From: Doctor Vogler
Subject: Re: Thank you (One-to-One Correspondence Between Real Line and Real 

Hi Alex,

Doctor Jacques points out that there is a catch to my answer; it isn't
exactly a bijection because there are some real numbers that can be
written in decimal in two ways. 

For example, my map would say to do the following:

               1 -> (1, 0)
  0.999999999... -> (0.9999999..., 0.99999999...)

But this mapping is not quite well-defined:

  0.999999999... = 1 

Furthermore, you can occasionally get two numbers that go to the same
point, as in

                         1 -> (1, 0)
  1/11 = 0.090909090909... -> (0.9999999..., 0)

There are ways to fix up this kind of problem -- by making special cases
for those kinds of numbers. For example, with numbers that can be written
in decimal in two ways (which are only rational numbers the denominators
of which are divisible by no primes other than 2 or 5; that is, numbers of
the form a/10^b for integers a and b), if you insist that these have to
use the form that ends in repeating zeros rather than repeating nines,
then you solve the problem of one number going to two possible pairs of
numbers. You also avoid ever going to a pair (x, y) where both x and y end
in repeating nines. But you still have to make an exception for the
possibility of one of them ending in repeating nines and the other in
repeating zeros, like 1/11 shown above. 

In fact, the problem is precisely the rational numbers of the form ...

   (11*a + 1)/(11*10^b)
... for integers a and b.

You could fix up this problem by choosing to change the images of those
numbers so that the images of both sets of numbers are interleaved among
the images that used to be for the second set. Similarly with the images
of another infinite subset of real numbers, such as ones of the form

   (11*a + 1)/(11*10^b) + ln(2)

Or you could use a strategy similar to the one used for the
Bernstein-Schroeder Theorem:

I don't know if there is a cleaner way to make a one-to-one correspondence
between R and R*R without having to fix up some subset of the points.

- Doctor Vogler, The Math Forum 

Date: 01/27/2011 at 23:45:06
From: Alex
Subject: Thank you (One-to-One Correspondence Between Real Line and Real Plane)

Dear Dr. Vogler,

Thanks! That's a lot to think about, but it's very interesting. 

By the way, could you please explain the Axiom of Choice to me and why
it's so important in set theory? I saw it mentioned while I was reading
about the Banach-Tarski Paradox.

Thanks so much!

Date: 01/28/2011 at 22:58:50
From: Doctor Vogler
Subject: Re: Thank you (One-to-One Correspondence Between Real Line and Real 

Hi Alex,

Oh, where to begin?

If you've ever done any proofs, you'll find that they generally start with
some assumptions or a hypothesis, and they frequently use other theorems
proved previously, and they end with the conclusion of the theorem.
Euclid's Elements was an early (perhaps the first) book that did this.

But if you use smaller theorems to prove bigger ones, then where do you
start?  How can you prove the first theorem?

The answer is that you have to start with some kind of initial assumptions
or hypotheses. The name for those initial assumptions is "axioms." Euclid
started with five axioms about geometry, and then he went on to prove lots
of theorems about geometry.

Well, most concepts in mathematics can be described in terms of set
theory, so to build up all of mathematics in this way, the hard part is
starting with the basic building blocks in logic and set theory. So there
have been several attempts to do this, and it always turns out to be much
more complicated than it would at first seem: someone comes up with some
simple set of axioms, and then someone else proves something ridiculous
from them; or someone shows that certain basic concepts can't be proven
from your axioms.

Well, the most widely-accepted set of axioms for set theory are the
Zarmelo-Fraenkel (ZF) axioms, with or without the Axiom of Choice (with
it, you call them ZFC). See also 

Most of the axioms seem relatively basic and straightforward, but the
Axiom of Choice seems obvious in some situations and almost ridiculous in
others. So while it is usually accepted as an axiom and freely used in
proofs where needed, sometimes mathematicians will point out when the
Axiom of Choice is necessary for a proof or theorem to hold.

- Doctor Vogler, The Math Forum 

Date: 01/29/2011 at 08:24:39
From: Alex
Subject: One-to-One Correspondence Between Real Line and Real Plane

Thanks, but what does the Axiom of Choice actually mean? In other words,
what does it state and how can it lead to ridiculous situations if it is
so simple?

Thanks again!

Date: 01/29/2011 at 13:52:31
From: Doctor Vogler
Subject: Re: One-to-One Correspondence Between Real Line and Real Plane

Hi Alex,

I'm sorry. I thought you were reading about the Banach-Tarski paradox. You
don't consider that to be ridiculous? Well, then maybe the Axiom of Choice
isn't so silly after all.

On the other hand, if the Axiom of Choice is false, then an infinite
product of infinite sets might be empty, which I find ridiculous.

As for the precise statement of the Axiom, see: 

- Doctor Vogler, The Math Forum 
Associated Topics:
High School Sets

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