Pi and Irrational Numbers

[Bob Klase]

Here is an interesting link, showing the appearance of sequences of numbers within [latex]\pi[/latex]:

http://www.angio.net/pi/piquery (Try your birthdate.)

Oh crap. Not only did it find my birthdate, it also knows my social security number, my address, and all of my credit card numbers. It's only a matter of time until identity thieves discover that site and they'll have all our information!

 

[Perpetual Student]

I think mathematics is discovered, not invented... and numbers are a fundamental part of that. Our notation for them, not so much.

A good example of this is the concept of continuity, which is so easily comprehended and quite intuitive for most people (which I would argue is a mathematical "discovery"). Yet the definitions developed (example: for continuous functions) took a long time historically, are laden with notation and are not easily understood by laymen.

 

[sol invictus]

Neat, huh?

Very.

Thank you.

 

[ravdin]

No, "irrational" means that it cannot be expressed as a ratio of integers, hence "irrational". 1/3 Is a rational number, but its decimal equivalent is 0.333333333333333 ... forever. It never ends either, but it is not irrational.

Exactly. If you think about it, all rational numbers have a nonterminating sequence when expressed in decimal form- the equivalent of 1/2 is 0.500000000...

The difference with an irrational number is that the sequence does not repeat itself. If it did, you could express it as the ratio of two integers.

 

[edd]

Actually, there's enough regularity in pi that it wouldn't astonish me to learn that pi is not, in fact, "normal" and you could prove things didn't appear in it.

For example, there's a theorem from the mid 90s that allows you to compute the x'th digit of pi without computing the previous (x-1) digits. The trick is that it only works in hexadecimal (base 16), not in base 10. But this suggests to me that pi might not be normal in base 16, meaning, of course, that it might not be normal is base-2, either.

I'm not that willing to follow that connection between the base-16 generating formula and normality myself. Although of course "might not be normal" is a pretty weak statement

A brief look around also brings up this interesting paper which hints at pi being base-2 normal.

http://crd.lbl.gov/~dhbailey/dhbpapers/baicran.pdf

 

[wexer9]

No, "irrational" means that it cannot be expressed as a ratio of integers, hence "irrational". 1/3 Is a rational number, but its decimal equivalent is 0.333333333333333 ... forever. It never ends either, but it is not irrational.

Thank you for the clarification.

 

[drkitten]

I'm not that willing to follow that connection between the base-16 generating formula and normality myself. Although of course "might not be normal" is a pretty weak statement

It was intended to (be a pretty weak statement). My argument is merely along the lines of "there is a surprisingly strong pattern in the base 16 digits, threfore it wouldn't astonish me to learn there was another surprisingly strong pattern," along with the idea that examining THIS pattern might be a good start to finding THAT one.

If I actually had a firm argument that pi was not normal to base 16, I wouldn't burn it here; there are better number theory journals than the JREF.

 

[CapelDodger]

Oh crap. Not only did it find my birthdate, it also knows my social security number, my address, and all of my credit card numbers. It's only a matter of time until identity thieves discover that site and they'll have all our information!

Not only that - they'll have all our future information! Before it's even happened!

What chance do we have against that?

 

[CapelDodger]

It was intended to (be a pretty weak statement). My argument is merely along the lines of "there is a surprisingly strong pattern in the base 16 digits, threfore it wouldn't astonish me to learn there was another surprisingly strong pattern," along with the idea that examining THIS pattern might be a good start to finding THAT one.

Where infinity is involved, any weakness is terminal.

If I actually had a firm argument that pi was not normal to base 16, I wouldn't burn it here; there are better number theory journals than the JREF.

I used to spend a lot of time in number theory lectures musing on the better things I could be doing with my time ...

All the same, a Pass was necessary and I really don't buy this special-base idea. I recall seeing it demolished : the details escape me (decades have passed), but it was a sound demolition. It's hardly a new idea, after all. I was probably getting it at fifth-generation pithiness.

 

[CapelDodger]

http://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml

Neat, huh?

But wrong.

... the numerator decreases very quickly as k gets large so that terms become negligible

This is where infinity and continuity kick in. There is no neglible in infinity; rounding error is always there, and you don't know what it is.

 

[69dodge]

But wrong.

This is where infinity and continuity kick in. There is no neglible in infinity; rounding error is always there, and you don't know what it is.

I'm not sure what you're trying to say here.

It's certainly possible, sometimes, to know an upper bound on rounding error, and often that's good enough.

There is no doubt, for example, that the first six digits of pi are 3.14159.

 

[sol invictus]

But wrong.

Certainly not. You can check in three lines that that sum equals pi.

This is where infinity and continuity kick in. There is no neglible in infinity; rounding error is always there, and you don't know what it is.

Again, not so. It's trivial to show that the error is bounded. Therefore if you want to compute the Nth digit, you only need to compute a finite number of terms (which turns out to be order N in this case).

 

[Almo]

No, "irrational" means that it cannot be expressed as a ratio of integers, hence "irrational". 1/3 Is a rational number, but its decimal equivalent is 0.333333333333333 ... forever. It never ends either, but it is not irrational.

0.333333333333333 approaches 1/3 as a limit as the number of digits increases. With an infinite number of digits, it is in fact = 1/3.

Irrationals can't be expressed as repeating decimals.

 

[cyborg]

0.12345678910111213141516171819202122...

Which could be expressed as the concatenation of the symbols in the set of integers in base 10. And of course any other normal number in base 10 would be some permutation of this set.

And I'm pretty sure determining the permutation is incomputable - since the permutation could have infinite entropy.

Adding in the task of having to check this for every possible integer base... well, it's just a staggeringly huge task to say the least.

It's odd to think that this number is merely a single specific quantity, and yet it contains all the books in Borges' Babylon Library, and infinitely more.

The problem is extracting this information of course.

 

[Myriad]

But there are ways to define numbers using (say) geometry that have nothing to do with symbols... pi as a geometrical ratio, 0 as the genus of a torus, 1 as the generator of the 3rd homotopy group of a 3-sphere, etc.

True, but it works both ways. Pi as a geometrical ratio seems very simple in concept has a complex numerical expression. On the other hand, the binary number 0.101100111000111100001111100000... has a very simple numerical expression, but I very much doubt that it can be expressed as any property of simple geometric objects.

I think mathematics is discovered, not invented... and numbers are a fundamental part of that. Our notation for them, not so much.

Our specific notation systems and axiom systems are of course invented. But what is being discovered (not invented; no one designed this behavior or even expected to find it) is that they all behave fundamentally the same. Some are sufficiently limited so as to have only rather simple behavior (e.g. the two-body problem), and the rest have behavior that becomes intractably complex when given inputs outside specifically constrained domains (e.g. the three-body problem). Of the latter type, a large number have been proven to be computationally equivalent -- that is, transformable into known computationally universal systems such as Turing machines, proofs from the axioms of arithmetic, lambda calculus, symbolic logic, solvability of Diophantine equations, the Rule 110 cellular automaton, and everyday computers (when idealized to have infinite memory), all of which are capable of emulating one another and are therefore fundamentally equivalent.

It remains to be seen whether any systems with complex behavior will turn out to be exceptions. For instance, gravitational movement of three bodies in space has not yet been proven to be computationally universal, but it's looking very likely that it eventually will be, and that would explain (or at least put into a more consistent context) the surprising complexity of the behavior of arbitrary three-body systems.

Numbers are looking less fundamental than they used to.

Respectfully,
Myriad

 

[Alkatran]

Which could be expressed as the concatenation of the symbols in the set of integers in base 10. And of course any other normal number in base 10 would be some permutation of this set.

And I'm pretty sure determining the permutation is incomputable - since the permutation could have infinite entropy.

Adding in the task of having to check this for every possible integer base... well, it's just a staggeringly huge task to say the least.



The problem is extracting this information of course.

In binary you can permute any irrational number into any other irrational number. If you need a 1 you just scan ahead for the next 1, then swap and go to the next bit. Do the same for 0. Because the number is irrational there will always be a 1 or 0 left to choose from.

I'm defining permute as the limit of permuting more and more of the bits, with the constraint that each bit's destination must eventually become fixed [otherwise you can turn any irrational binary number into any real number, but not vice versa].

 

[IMST]

Oh crap. Not only did it find my birthdate, it also knows my social security number, my address, and all of my credit card numbers. It's only a matter of time until identity thieves discover that site and they'll have all our information!

Actually, that site doesn't have anything to do with pi at all. It just collects sensitive information for future identity theft because humans naturally put that kind of stuff in first.

joking, of course. Though that would be a decent scam

 

[cyborg]

In binary you can permute any irrational number into any other irrational number.

This would imply all binary infinite sequences are normal.

ETA: On second though it wouldn't. 101101110111101111101111110... wouldn't be normal but the set of 1's would have the same cardinality as the set of 0's since you could map all the zeros to all the 1's. I'm a little unsure at this point.

 

[linusrichard]

0.333333333333333 approaches 1/3 as a limit as the number of digits increases. With an infinite number of digits, it is in fact = 1/3.

Or with a finite number of digits and three dots after them, thus:
.33...
Or with a finite number of digits and a horizontal line over the last one.

 

[CapelDodger]

Certainly not. You can check in three lines that that sum equals pi.

I wasn't referring to the formula, but to the line I quoted

"... the numerator decreases very quickly as k gets large so that terms become negligible "

I'm not sure that follows.

Again, not so. It's trivial to show that the error is bounded.

Trivially, it's bounded by the difference between the current estimate and pi. The problem arises when the bound - which is itself indeterminate - coincides closely enough with a digit-boundary that it never delivers a definite answer. Which is to say, the pi-digit problem has merely been replaced with the error-bound-digit problem. Which is no great help.

Therefore if you want to compute the Nth digit, you only need to compute a finite number of terms (which turns out to be order N in this case).

I'm not yet convinced.