Is the number Pi's sequence of digits infinite and unpredictable?

[suffian]

I was looking at the number Pi yesterday, with its odd sequance of digits, and an plausible idea popped into my head: could any possible sequance of digits be found as a substring of the string of digits that make up pi? (sure seems so)

 

[Hurkyl]

The vast majority of real numbers have that property. As to whether &pi does, I don't think anyone knows.

Hurkyl

 

[selfAdjoint]

Originally posted by Hurkyl
The vast majority of real numbers have that property. As to whether &pi does, I don't think anyone knows.

Hurkyl

Hey, I noticed that in your post &pi doesn't come out as [pi]. Are they kidding us?

 

[Hurkyl]

Originally posted by selfAdjoint
Hey, I noticed that in your post &pi doesn't come out as [pi]. Are they kidding us?

Aha, so [pi] is made the same way on this forum!

I guess that we're faced with a tough decision...

Do we use &pi that fits in nicely with the font but is unrecognizable if you don't know what it's supposed to be, or [pi] which looks nice but is a frusterating half a line above the rest of the equation

I dug out Windows' character map, it seems that the culprit is the (default) Verdana font. (For those lucky ones without Verdana, you get arial as default!)

Arial looks like this: &pi.
Times New Roman looks like this: &pi
Courier looks like this: &pi
Century Gothic looks like this: &pi

Looks like I might be doing my posting in TNR from now on.

Hurkyl

 

[Ed Quanta]

Nope. I don't think there is any sequence which can describe the randomness of the successive digits of pi.

 

[suffian]

Originally posted by Ed Quanta
Nope. I don't think there is any sequence which can describe the randomness of the successive digits of pi.

No, not quite what I meant. I'm wondering whether given any natural number x, you could find the x embedded somewhere within the digits of pi (i still have to looked up how to do this math board notation).

Say for instance I chose "322", could I find "322" inside the digits of pi. What about any other natural number?

 

[climbhi]

Originally posted by suffian
No, not quite what I meant. I'm wondering whether given any natural number x, you could find the x embedded somewhere within the digits of pi (i still have to looked up how to do this math board notation).

Say for instance I chose "322", could I find "322" inside the digits of pi. What about any other natural number?

That's an interesting idea. It seems like since the digits of π are infinite that you should be able to as long as you looked for long enough.[/Size]

 

[FZ+]

Yep. I think that the same is true of any irrational number. If it does not repeat and goes on for infinity, I would think all possible combinations would be exhausted. At least, it seems so.
I'll be very interested if anyone can find a numerical proof for it though.

 

[lethe]

Originally posted by FZ+
Yep. I think that the same is true of any irrational number. If it does not repeat and goes on for infinity, I would think all possible combinations would be exhausted. At least, it seems so.
I'll be very interested if anyone can find a numerical proof for it though.

this is certainly not true for every irrational number. just most of them.

this property is called normality. it is strongly suspected that π is normal, and most other irrational numbers, but so far it is unproved

 

[climbhi]

Originally posted by lethe

this is certainly not true for every irrational number. just most of them.

this property is called normality. it is strongly suspected that π is normal, and most other irrational numbers, but so far it is unproved

Please explain more about this normality. This is really interesting. Are there any irrational numbers which are known not to be normal, are/or are their any irrational numbers that are known for sure to be normal? Could you point me in the direction of some papers on this or anything? Thanks!

ps: lethe, the times new roman does look better for the math symbols, but I found it works a little better if you increase the size to large too, just makes it easier to read, but for sure the pi looks better in the tnr then it does in the defualt where you can barely distinguish it!

 

[lethe]

Originally posted by climbhi
Please explain more about this normality. This is really interesting. Are there any irrational numbers which are known not to be normal, are/or are their any irrational numbers that are known for sure to be normal? Could you point me in the direction of some papers on this or anything? Thanks!

ps: lethe, the times new roman does look better for the math symbols, but I found it works a little better if you increase the size to large too, just makes it easier to read, but for sure the pi looks better in the tnr then it does in the defualt where you can barely distinguish it!

How s this look? π

OK, anyway, to answer: firstly, yes there are certainly numbers which are known to not be normal. i m going to show you one in a second.

secondly, as far as i know, there are no numbers which are known for sure to be normal, although it is experimentally verified for the first many millions of digits of π and √2 and some other irrationals.

apparently normality depends on what base you re in! so just because π is normal in base 10, does not mean it is normal in binary!

i think mathworlds page about normal numbers is pretty good, it has some tables showing statistics about digits of π. also, apparently there are websites that will search digits of π for your phone number. if it truly is a normal number, like it appears to be, then your phone number is in there.

now, let me show you a nonrepeating irrational (that s redundant, eh?) number which is not normal:

0.1010010001000010000010000001...

this number will obviously never repeat itself, so it is not a rational number, and yet you will never find your phone number here (unless you have a pretty funky phone number).

 

[lethe]

Originally posted by lethe

secondly, as far as i know, there are no numbers which are known for sure to be normal, although it is experimentally verified for the first many millions of digits of π and √2 and some other irrationals.

i should have read the link before i posted. yes, there are few numbers which are known to be normal, as you will see on mathworld, but not very many. two such numbers are the champernowne constant and the copeland-erdos constant which are both basically constructed in such a way as to ensure that they are normal.

no "naturally occurring" irrational numbers are known to be normal, however.

 

[lethe]

Originally posted by Hurkyl
The vast majority of real numbers have that property.

do you mean by this statement that the cardinality of the normal numbers is greater than the cardinality of the non-normal numbers? i.e. that nonnormal numbers are countable?

 

[climbhi]

For these numbers which are known to be normal, do you know if 1/the number is also normal? It seems like if that n is normal that 1/n should be too but I don't know much about number theory so...

 

[Hurkyl]

do you mean by this statement that the cardinality of the normal numbers is greater than the cardinality of the non-normal numbers? i.e. that nonnormal numbers are countable?

Well, non-normal numbers can't be countable; the set of all decimal numbers without any 8's in their representation is clearly nonnormal, but the set is uncountable.


http://mathworld.wolfram.com/AbsolutelyNormal.html

"Almost all" numbers in the set [0, 1) are absolutely normal, a stronger condition than normality.

Almost all is a technical term, though it apparently has more than one definition since the definition at mathworld at http://mathworld.wolfram.com/AlmostAll.html is different from the definition I have in my Measure Theory text, but they both mean roughly the same thing; the ratio of nonnormal numbers to normal numbers is zero.

Hurky

 

[lethe]

Originally posted by Hurkyl
Well, non-normal numbers can't be countable; the set of all decimal numbers without any 8's in their representation is clearly nonnormal, but the set is uncountable.


http://mathworld.wolfram.com/AbsolutelyNormal.html

"Almost all" numbers in the set [0, 1) are absolutely normal, a stronger condition than normality.

Almost all is a technical term, though it apparently has more than one definition since the definition at mathworld at http://mathworld.wolfram.com/AlmostAll.html is different from the definition I have in my Measure Theory text, but they both mean roughly the same thing; the ratio of nonnormal numbers to normal numbers is zero.

Hurky

the definition of "almost all" that i learned in school is "on all but a set of measure zero". is that what you learned too?

OK, so when you said "vast majority" you were not referring to cardinalities, but to measure.

got it. thanks.

 

[lethe]

almost everywhere

 

[lethe]

Originally posted by climbhi
For these numbers which are known to be normal, do you know if 1/the number is also normal? It seems like if that n is normal that 1/n should be too but I don't know much about number theory so...

well, it seems like this statement should be true. beyond that i can t comment.

 

[Hurkyl]

Good call! It's "almost everywhere" that I got from my measure theory text (yes, it was defined as the set of counterexamples having measure 0).

Hurkyl

 

[climbhi]

Anyone else have any thoughts on my question about if n is normal is 1/n normal too?

 

[STAii]

Wow ! it would be great if we knew [pi] was normal !
We would know that any file on your computer (represented in decimal) is found inside [pi], all the litrature of the world (when represneted (someway) is found in [pi]).
We wouldn't have to save files anymore (if we had super computers), we would only save the starting and ending digits of Pi where the file comes !
Very exciting (we would save million of bytes (but will need LOT more processing powers).

 

[climbhi]

well we know that some other numbers are normal, so why don't we just do what you described using those numbers instead? Interesting idea, though I'm not sure if its really practical, sounds pretty cool though.

And does anyone have any other thoughts on my question if n is normal is 1/n normal also?

 

[Hurkyl]

Unfortunately, your file coding scheme can't work.


Consider the following:

There are 2⁶⁴ possible programs with a length of 64 bits.

Therefore, you must be able to generate at least 2⁶⁴ distinct combinations of starting and ending markers.

It takes at least 64 bits to generate 2⁶⁴ distinct things.

So you've gained no space savings.

Hurkyl

 

[lethe]

Originally posted by climbhi
And does anyone have any other thoughts on my question if n is normal is 1/n normal also?

that s the third time you ve asked about that. why are you so anxious to find out if n normal => 1/n normal? do you have some use for this result?

just a few more thoughts: one way to calculate the digits of 1/n, is by long division into 1.00000... each digit of the quotient is then somehow related to a finite sequence of the digits of n. if i wanted to look for a proof of this conjecture, this is where i would start looking.

but right now, there are excedingly few numbers that are known to be normal. this leads me to infer that a proof of normality is very difficult, and not much is known about the properties of normality.

this doesn t surprise me, since most mathematicians aren t too interested in the properties of decimal (or other base) representations of numbers. they usually work with the numbers themselves, and their intrinsic (i.e., algebraic) properties.

but i m sure there are some mathematicians who like to think about decimal representations.

 

[climbhi]

Originally posted by lethe
that s the third time you ve asked about that. why are you so anxious to find out if n normal => 1/n normal? do you have some use for this result?

Absolutely no use for it, just very interested. I don't know why but it just piqued my interest. I see what you're saying about where to start with a proof, in fact I was already thinking that, perhaps I'll try and do a proof of it, but it probably won't happen so...thanks anyway.