[2013-May-19] The Balls Constant

[OskarS]

This comic is mathematically inaccurate!!!

If the constant ends with "BALLS BALLS BALLS" repeated forever, then presumably there's some sequence of digits that corresponds to "BALLS BALLS BALLS". If that sequence repeats itself forever, then it's not an irrational number, it's a rational number!

EDIT: ahh, crap, just reread the comic again, and the lady says "Shouldn't there be some non-normal number that contains...". So ok, yes, that is true. Carry on.

[Tajun]

Hmm... I wonder if, for any given rational number, you can design a dictionary to show that it is a balls constant?

Should be tricky to get the works of literature in chronological order prior to the balls part, but perhaps not impossible. Any pure mathies out there?

[anyothergamma]

Hmm... I wonder if, for any given rational number, you can design a dictionary to show that it is a balls constant?

Should be tricky to get the works of literature in chronological order prior to the balls part, but perhaps not impossible. Any pure mathies out there?

If you want one for any given rational number you'll have to cover numbers like 5. That might be a bit of a stretch to encode that to balls. So lets focus on numbers that repeat forever (all rational numbers either terminate or repeat forever). In base 10, the numbers that repeat forever are any numbers that when expressed as reduced fractions, the denominator has prime factors outside of just 2 and 5. You still would have to address numbers like .22222222... Without changing the base, if all letters had the same length of encode, say 2, then you'd have to have 22 map to B, 22 map to A, 22 map to L, and 22 map to S. Not a very fun mapping.

If you decided to have the lengths of letters vary then you could have different lengths correspond uniquely to different letters, so 22 is B, 2 is A, 222222 is L, and 2222 is S. While the mappings are now unique, the ultimate reading of the word would still be ambiguous since you would have to arbitrarily decided when the current letter ends. This can be extended to any repeating decimal by saying 2 of the repeats is B, 1 of the repeats is A, 6 of the repeats is L, and 4 of the repeats is S, or anything you want.

Converting the base doesn't really help because if you take a number like 1/2 it'll always convert into .X where X=b/2 if b, the base, is even or .XXXXXXXX... where the digit X is X=(b-1)/2 where b, the base, is odd, so the number 1/2 will never convert into anything interesting under an integer base. So the only final recourse is using a non-integer base. Perhaps using a normal base would turn any rational number into a normal number? Actually, no, because then you're left with 0, which is always just 0 in any base, integer or otherwise.

[brianberns]

This comic is mathematically inaccurate!!!

If the constant ends with "BALLS BALLS BALLS" repeated forever, then presumably there's some sequence of digits that corresponds to "BALLS BALLS BALLS". If that sequence repeats itself forever, then it's not an irrational number, it's a rational number!

EDIT: ahh, crap, just reread the comic again, and the lady says "Shouldn't there be some non-normal number that contains...". So ok, yes, that is true. Carry on.

You're still technically right. In the first panel, the comic defines "normal" as applying only to irrational numbers.

[taliesin9]

This comic is mathematically inaccurate!!!

If the constant ends with "BALLS BALLS BALLS" repeated forever, then presumably there's some sequence of digits that corresponds to "BALLS BALLS BALLS". If that sequence repeats itself forever, then it's not an irrational number, it's a rational number!

EDIT: ahh, crap, just reread the comic again, and the lady says "Shouldn't there be some non-normal number that contains...". So ok, yes, that is true. Carry on.

You're still technically right. In the first panel, the comic defines "normal" as applying only to irrational numbers.

right, but a non-normal number can be rational or irrational in accordance with basic set logic. normal is defined as a subset of irrational. rational and irrational have no intersections. therefore rational and normal have no intersection. therefore all rational number are members of the set normal' aka non-normal.

[Peon]

Zach needs to get over obsessing about this idea because it wasn't a very good joke a few months ago either.

It's all in the specific choice of real number and the way you wish to decode that number. But since there is a whole continuum of real numbers, as well as a nigh endless list of ways to interpret it, that implies that while the assertion in the comic is technically correct, it's COMPLETELY UNINTERESTING because of the vast quantities of real numbers and decoding methods which would result in meaningless noise or, worse, nearly correct text with hidden errors/inaccuracies.

What's that short story by the spanish dude about all the librarians in the library with books with random text? Whatever Zach tries to do with this idea, it's already been done there.

[00011010101]

Pardon my pedantry, but I think the math Slide 3 (1-indexed) isn't implied by Slide 2. If the only criteria for a normal number is that each digit is randomly distributed, you could just take concatenations of permutations of "0123456789". Clearly, mankind in its infinite capacity for breaking rules will create or has created works of art whose digital representation has the same digit in three consecutive positions - something which clearly isn't present in the number I created.

What you also want to insist on is independence.

Edit: What I said above isn't probably true. Zach does insist that the property hold in all base systems - which trivially breaks my counter-example.

[Lethal Interjection]

I think I know what kind of balls we are going to have in this thread ere long.

[Parcival]

What's that short story by the spanish dude about all the librarians in the library with books with random text? Whatever Zach tries to do with this idea, it's already been done there.

Jorge Luis Borges. He was Argentinian. "The Library of Babel" is available on the net. Just google it. Each book had 410 pages with 40 lines of 80 characters on each page. There were 25 symbols consisting of 22 letters, the space, the period and the comma.

The set of balls constants doesn't seem to be clearly defined because new members are being added on a regular basis. Works of literature not yet written are coded into future balls constants.

[Irrelevancy]

The last panel silohette has nipple in it

[Greg]

I think I know what kind of balls we are going to have in this thread ere long.

Metric balls?

[Apocalyptus]

Seeing as ThePeople seems to be busy, I will fulfill his duty this time.

[ton]

This is interesting...... It makes pi into a family member of Borges' Library of Babel

[Wilko]

Hmm... I wonder if, for any given rational number, you can design a dictionary to show that it is a balls constant?

Should be tricky to get the works of literature in chronological order prior to the balls part, but perhaps not impossible. Any pure mathies out there?

If you want one for any given rational number you'll have to cover numbers like 5. That might be a bit of a stretch to encode that to balls. So lets focus on numbers that repeat forever (all rational numbers either terminate or repeat forever). In base 10, the numbers that repeat forever are any numbers that when expressed as reduced fractions, the denominator has prime factors outside of just 2 and 5. You still would have to address numbers like .22222222... Without changing the base, if all letters had the same length of encode, say 2, then you'd have to have 22 map to B, 22 map to A, 22 map to L, and 22 map to S. Not a very fun mapping.

If you decided to have the lengths of letters vary then you could have different lengths correspond uniquely to different letters, so 22 is B, 2 is A, 222222 is L, and 2222 is S. While the mappings are now unique, the ultimate reading of the word would still be ambiguous since you would have to arbitrarily decided when the current letter ends. This can be extended to any repeating decimal by saying 2 of the repeats is B, 1 of the repeats is A, 6 of the repeats is L, and 4 of the repeats is S, or anything you want.

Converting the base doesn't really help because if you take a number like 1/2 it'll always convert into .X where X=b/2 if b, the base, is even or .XXXXXXXX... where the digit X is X=(b-1)/2 where b, the base, is odd, so the number 1/2 will never convert into anything interesting under an integer base. So the only final recourse is using a non-integer base. Perhaps using a normal base would turn any rational number into a normal number? Actually, no, because then you're left with 0, which is always just 0 in any base, integer or otherwise.

For any rational number other than 0, this problem is trivial using dictionary arithmetic coding. While arithmetic coding traditionally uses the interval [0,1], this constraint can be modified to whatever interval you intend to use. The recurring
digits (0, if the number has finite significant digits) are set as "BALLS", and the non-recurring digits as all the works of literature in chronological order concatenated. If you choose a good number, you can split your literature string into smaller strings, each with their own dictionary entry, reducing the total size of the dictionary (which is the way the arithmetic coding is supposed to work).

The problems come with irrational numbers, which, by definition, have no recurring digits, and 0, which has only recurring digits (for other fully recurring numbers, eg 10/9, we can just set the interval to a suitable value to give us a 0 as the first digit). The dictionary for these numbers are infinitely long, as they must contain both the number, the literature string, and an inifinite repition of "BALLS".