Experiment: Mapping real numbers between 0 to 1 to natural numbers

[gus_massa]

You mapped the numbers between 0 and 1 that have a finite number of decimals, like 0.123123425736289

But there are many more real numbers between 0 and 1 that have an infinite number of decimals, like 0.1231234736876578632786278365782635879281987087420...

As jgreen says in a sibling comment, which number do you assign to 1/3 = 0.333333333333333333333333333333333333333333... ?

The result you got is well known but interesting, because it's unintuitive. Moreover with some tricks you can try to extend your method to all fractions, like 1/3, 5/7, 27/127, ...

The problem is that there are irrational numbers like pi, sqrt(2), ... and many other numbers that we have assigned no friendly name. Those "unfriendly" numbers are the actual problem.

[smac__]

Again, excellent point! These numbers clearly show why* my reasoning is missing a rather large portion of this set.

[edit] - fixed typo

[jjgreen]

Which element of you tree maps to 1/π ?

[smac__]

Excellent point! Thank you for showing the glaring flaw more to explore.

[smac__]

Each time I hear the argument, that real numbers between zero to one are a "larger" (nuanced) infinity than natural numbers, I want explore and see if it is possible to make a one to one mapping between these sets. This post is my attempt. It was a fun thought experiment.

[GTP]

Interesting approach, but it has been proven by mathematicians (I'm not one) that the cardinality of the set of the real numbers between 0 and 1 is the same as the set of all the real numbers. I would suggest discussing your approach and your doubts with a mathematician, as when talking about infinities there is a lot counterintuitive stuff going on. Another commenter already pointed out a possible problem of your "mirror" method: you're relying in being able to read a number backwards. To do that, you need to know all the digits. But it doesn't work well with numbers with infinite decimal digits like pi or its inverse.

[onesphere]

If real numbers are more applicable to reality, then computer science should look for all the ways to calculate these numbers. Would this look like a “path integral” of real numbers?

[neonskies]

Not sure what's on the vid. Didn't watch it. But if I understand you correctly, the following is impossible as shown in Cantor's simple diagonalization argument:

> Follow each branch as deeply as possible counting all sub branches until there are no sub branches left.

[smac__]

Good point, the depth first search algorithm is a bit lack luster for this case: 1 -> .1 2 -> .01 3 -> .001 4 -> .0001 5 -> .00001 6 -> .000001 . -> .0∞1

I would definitely recommend not judging the quality of the video based upon the quality of my reasoning in the experiment.

[josephcsible]

You've mapped from the decimal numbers (the subset of rational numbers where the denominator is a power of 10) between 0 and 1 to the natural numbers, not from the real numbers between 0 and 1 to the natural numbers.

[onesphere]

In order for each level to exclude numbers that have already been counted, each level would only be 9 numbers:

So if level = lambda n: 1-int(math.log10(1./n))

We only ever generate [‘.’ + ‘0’ * (level(n) – 1) + str(i + 1) for i in range(9)]