weird math question

[kevinquinnyo]

This question is weird. It may be that the question itself doesn't make sense.

But, here it is:

Let's say you have an infinite number. The digits of this number are random. Does this mean that, because the number is infinite, and because the digits are randomized, there would have to be a section of digits within this infinite series, that is itself, an infinite amount of say, 7's?


like: 3837934877777777777777777777777.... and it just goes on infinitely?


Doesn't that have to be true, if the number is

a) truly random

and

b) infinite?

It seems like a paradox that I just can't wrap my mind around. Any thoughts?

[Alkatran]

An infinite number doesn't really have 'digits'. If you want to talk about an infinite string of digits, then talk about the digits in a real number after the decimal point, or just 'an infinite string of digits'.

To answer your question, an unending sequence of random digits will _almost surely_ contain any specified substring.

http://en.wikipedia.org/wiki/Almost_surely#.22Almost_sure.22_versus_.22sure.22

[Ziggurat]

It seems like a paradox that I just can't wrap my mind around. Any thoughts?

You should be able to find any finite sequence somewhere within an infinite and truly random sequence. So that means if you pick any number of 7's, you should find a string of them somewhere. And that number can be arbitrarily large, and you should still be able to find it somewhere. But infinity isn't a number, so there's no reason there needs to be an infinite number of consecutive 7's anywhere within the sequence. And if it's a truly random sequence, then there wouldn't be.

[kevinquinnyo]

You should be able to find any finite sequence somewhere within an infinite and truly random sequence. So that means if you pick any number of 7's, you should find a string of them somewhere. And that number can be arbitrarily large, and you should still be able to find it somewhere. But infinity isn't a number, so there's no reason there needs to be an infinite number of consecutive 7's anywhere within the sequence. And if it's a truly random sequence, then there wouldn't be.

An infinite number doesn't really have 'digits'. If you want to talk about an infinite string of digits, then talk about the digits in a real number after the decimal point, or just 'an infinite string of digits'.

To answer your question, an unending sequence of random digits will _almost surely_ contain any specified substring.

http://en.wikipedia.org/wiki/Almost_surely#.22Almost_sure.22_versus_.22sure.22

Ok thanks! That makes more sense.

So would it be correct to say that the arbitrarily high amount of 7's would approach infinity?

It still kind of messes with my head. :confused:

[drkitten]

Let's say you have an infinite number. The digits of this number are random. Does this mean that, because the number is infinite, and because the digits are randomized, there would have to be a section of digits within this infinite series, that is itself, an infinite amount of say, 7's?

No, it doesn't.

Part of the reason for that is that the "digits" for any number are dependent upon the base in which you express the number; I can express the same number in base 2, base 8, base 10, or base 16, and "obviously" I won't have a string of twenty-five nines, or even one nine, in the base-8 expression.

But "random" can also mean something much weaker than what you think it does; for example, I could generate a number by rolling a die (and getting a number from 1..6) and using that for the next digit (in base 10). So the number might look like 134256353425211523226... and again would not have any nines in it. While the digits are random, they're not random "enough."

The real concept you're looking for is "normal." A real number is "normal" (in this sense, there are few words more over(ab)used than "normal" in mathematics) to the base B if, when expressed in base B, every finite string appears somewhere in it. A number is "normal" generally if it is normal to all bases.

And, yes, normal numbers exist. In fact, "almost all" numbers are normal.


However, an infinite string of the name number woudn't be random (as was pointed out); in practical terms, it would mean a repeating decimal, which would be a rational number, which is hardly random at all.

[kevinquinnyo]

No, it doesn't.

Part of the reason for that is that the "digits" for any number are dependent upon the base in which you express the number; I can express the same number in base 2, base 8, base 10, or base 16, and "obviously" I won't have a string of twenty-five nines, or even one nine, in the base-8 expression.

But "random" can also mean something much weaker than what you think it does; for example, I could generate a number by rolling a die (and getting a number from 1..6) and using that for the next digit (in base 10). So the number might look like 134256353425211523226... and again would not have any nines in it. While the digits are random, they're not random "enough."

The real concept you're looking for is "normal." A real number is "normal" (in this sense, there are few words more over(ab)used than "normal" in mathematics) to the base B if, when expressed in base B, every finite string appears somewhere in it. A number is "normal" generally if it is normal to all bases.

And, yes, normal numbers exist. In fact, "almost all" numbers are normal.


However, an infinite string of the name number woudn't be random (as was pointed out); in practical terms, it would mean a repeating decimal, which would be a rational number, which is hardly random at all.

Right.

Well, I suppose I could have phrased the question differently. Something like, say you have a random and infinite line of apples, oranges, and bananas... just to avoid the confusion of whether its in base 10 or binary or whatever. I guess this example would be base 3, but that's irrelevant now that it's fruit, right?

But yeah I think it makes more sense now. It was really screwing with my head the other day.

[Alkatran]

Ok thanks! That makes more sense.

So would it be correct to say that the arbitrarily high amount of 7's would approach infinity?

It still kind of messes with my head. :confused:

I'm not sure what you mean by the "arbitrarily high amount of 7's".

The expected number of occurrences of any finite substring, such as "777", is infinite.

If you're increasing the number of 7s as you go, so that you only match "7" for the first 100 digits, then "77" for the next thousand, etc, then the expected number will not necessarily be infinite.

[Dr. Trintignant]

However, an infinite string of the name number woudn't be random (as was pointed out); in practical terms, it would mean a repeating decimal, which would be a rational number, which is hardly random at all.

Hmmm. I think that infinite repeating substrings must exist in a normal number.

Take one proof that there are an infinite number of primes as an example. Suppose there were a "largest" prime N. But then I could construct a new number Q which is the product of N and all numbers less than that, minus 1. Q does not divide any number from 2 to N, and therefore has factors which are all larger than N. Thus, there is no largest prime, and the set of primes is infinite.

Now, suppose there were a longest repeating substring of length N in some normal number. But if N is finite, then so is N+1, and since our number is normal, then a repeating substring of length N+1 must appear somewhere. Therefore there is no longest repeating substring, and the substrings can grow to infinite size.

What's wrong with this argument as compared to the prime number one?

- Dr. Trintignant

[GreedyAlgorithm]

Hmmm. I think that infinite repeating substrings must exist in a normal number.

Take one proof that there are an infinite number of primes as an example. Suppose there were a "largest" prime N. But then I could construct a new number Q which is the product of N and all numbers less than that, minus 1. Q does not divide any number from 2 to N, and therefore has factors which are all larger than N. Thus, there is no largest prime, and the set of primes is infinite.

Now, suppose there were a longest repeating substring of length N in some normal number. But if N is finite, then so is N+1, and since our number is normal, then a repeating substring of length N+1 must appear somewhere. Therefore there is no longest repeating substring, and the substrings can grow to infinite size.

What's wrong with this argument as compared to the prime number one?

- Dr. Trintignant
Simple: There is no "infinitely large prime". The set of primes is infinitely large, but every prime is finite. Similarly there is no "infinitely large substring" of a repeated digit: the set of distinct-length repeat substrings {7, 77, 777, 7777, 77777, ...} is infinitely large, but every number 7777777 is finite.

[gnome]

@Dr. Trintigiant -- You're describing infinity as if it were something "reachable"... as if since the size of the strings must approach infinity, that a string of "infinite" length must somehow exist in it.

I can give an intuitive counterexample.

1.01001000100001...

There is no largest string of 0's... but it does not contain any infinitely long strings of zeros... no matter how far you go, there's always a "1" coming up.

[arthwollipot]

OK, I'm far from an expert - I can just about do moderately complex arithmetic in my head - but something is confusing me about the OP.

If you have an infinite string of digits, can that string really be said to contain another infinite string of digits? Is one infinity "larger" than another infinity such that one can contain another? Or are all infinities equally infinite?

[Dr. Trintignant]

Simple: There is no "infinitely large prime". The set of primes is infinitely large, but every prime is finite. Similarly there is no "infinitely large substring" of a repeated digit: the set of distinct-length repeat substrings {7, 77, 777, 7777, 77777, ...} is infinitely large, but every number 7777777 is finite.

This is why abstract math goes over my head :-). I find that a somewhat weird distinction to make, though I might agree that it's necessary for infinity to make any sense at all. On one hand, any one element of our set is finite in length, but on the other, there is no largest element. I can see why the OP has problems...

- Dr. Trintignant

[Dr. Trintignant]

Or are all infinities equally infinite?

That much I can safely say is false. The infinity of the integers is smaller than the infinity of the real numbers, for instance. There is a very slick argument as to why this must be true, although it gives no indication as to "how much" bigger the larger infinity is.

- Dr. Trintignant

[Dr. Trintignant]

There is no largest string of 0's... but it does not contain any infinitely long strings of zeros... no matter how far you go, there's always a "1" coming up.

And yet, the next string of zeroes you reach is larger than any you've seen before.

I understand the distinction, really, but I find it bizarre to apply "infinity" to some things and not others. GreedyAlgorithm agreed that the set of {7, 77, 777, etc.} substrings is infinitely large. But if the set were composed solely of finite elements, then it must be itself only finite in size. So there must be infinite elements in there.

Ok, so we're probably not allowed to make leaps of reasoning like this. It's just difficult to know when and where it's allowable to use concepts held over from finite sets and such. Why is it that we can call a set infinite, but not make the same claim about any of its elements?

- Dr. Trintignant

[sol invictus]

But if the set were composed solely of finite elements, then it must be itself only finite in size. So there must be infinite elements in there.

You would agree that the set of rational (or real, if you prefer) numbers q that satisfy 0<q<1 is a set with infinitely many elements, right? Yet surely you would also agree that all the elements are finite?

The number of elements in a set and the size of those elements are two totally different things.

[Dr. Trintignant]

The number of elements in a set and the size of those elements are two totally different things.

I certainly agree in most cases, but this particular instance contains one element for every possible length. And among the natural numbers represented in a given base, a longer number is always a greater number.

Let's try this, instead. Imagine that instead of our set being {7, 77, 777, ...}, we store sets of pairwise numbers representing the digit and its position, like this: {{{0, 7}}, {{0, 7}, {1, 7}}, {{0, 7}, {1, 7}, {2, 7}}, ...}.

Does this new set contain other infinite sets, or not? And if so, what's the distinction between those sets and the decimal numbers that they represent?

- Dr. Trintignant

[sol invictus]

I certainly agree in most cases, but this particular instance contains one element for every possible length. And among the natural numbers represented in a given base, a longer number is always a greater number.

Let me explain why I asked you that. The set of rationals between 0 and 1 includes as a subset the numbers 1/2, 1/3, 1/4, etc. I can map those to the positive integers in an obvious - and one-to-one - way. And I can also map them to the set {7, 77, 777...} in an equally obvious way. When two sets can be mapped to each other 1-1 like that, they are the same set, really - you can think of one as a set of unique names or labels for the elements of the other. And here the map is always between finite numbers (like 1/n) and finite numbers (like 777...7, with n 7s).

So by agreeing that the set of rationals between 0 and 1 is infinite but contains no infinite elements, you've agreed the same for the set {7, 77, 777...}. Same goes for your "new" set (which is really just a new set of labels for the old one).

[LostAngeles]

OK, I'm far from an expert - I can just about do moderately complex arithmetic in my head - but something is confusing me about the OP.

If you have an infinite string of digits, can that string really be said to contain another infinite string of digits? Is one infinity "larger" than another infinity such that one can contain another? Or are all infinities equally infinite?

The set of real numbers is infinite.

The set of natural numbers is infinite.

The size of the set of real numbers is larger than the size of the set of the natural numbers.

The natural numbers are a strict subset of the rational numbers.

However, there are as many rationals as there are natural numbers.

How can this be? Simply put, the rationals are, "countable," meaning that it's possible to map them one to one to a natural number. http://en.wikipedia.org/wiki/Countable has a rather nice demonstration of this.

Infinities are fun.

This is where I stop since that's about all I have of set theory.

[LostAngeles]

I feel like I should talk more since I was thinking about this yesterday morning.

See, we had someone come into IRC asking for a, "shortcut," to find Shakespeare's works in pi. Setting aside the fact that depending on how you decide to encode the English alphabet, you can probably find anything you damn well please, I started wondering if an infinite sequence like pi (no patterns) would contain within it every finite subsequence.

My intuition says yes, but if you're familiar with the infinities above, intuition can confuse the hell out of you out here.

I'm certain this has been done by someone *coughGausscough*, but my simplistic reasoning assumed that the digits of pi was countable and considering non-overlapping subsequences. Why? Because I was thinking that this is the simplest case and I can "inductively,*" expand it to any infinity for the digits of pi. Plus, I feel as though I want to enumerate all the subsequences and then find them in pi. (i.e. find 0, 1, 2, ...,9, 00, 01,...)

That's about as far as I'm probably going to get anytime soon (or ever with my attention span and impending finals), but feel free to pick it apart or play with it more.

[Dr. Trintignant]

When two sets can be mapped to each other 1-1 like that, they are the same set, really - you can think of one as a set of unique names or labels for the elements of the other. And here the map is always between finite numbers (like 1/n) and finite numbers (like 777...7, with n 7s).

Right, but that map also changes the notion of what it means for one number to be greater than another. In your example, the function is exactly reversed: A > B implies that 1/A < 1/B.

In a way this makes it a much easier problem than before: there is no single number greater than all finite numbers, but there is a single non-negative number smaller than all rationals of the form 1/N, where N is finite--that number is zero.

So the problem is changed to whether our remapped set properly contains zero. Intuitively, I see no reason why it shouldn't.

- Dr. Trintignant

[gnome]

Right, but that map also changes the notion of what it means for one number to be greater than another. In your example, the function is exactly reversed: A > B implies that 1/A < 1/B.

In a way this makes it a much easier problem than before: there is no single number greater than all finite numbers, but there is a single non-negative number smaller than all rationals of the form 1/N, where N is finite--that number is zero.

So the problem is changed to whether our remapped set properly contains zero. Intuitively, I see no reason why it shouldn't.

- Dr. Trintignant

Ahh I can help with that one. It most definitely does not contain zero proper. It can get as close to zero as you'd like, but never contains actual zero.

If you want an intuitive idea of that... based on reasoning you've used before -- think of any spot "n" in the set... for whatever "n" you choose, you can be sure that none of the numbers before "n" are zero. Which means that zero can never appear in the sequence.

[Dr. Trintignant]

Ahh I can help with that one. It most definitely does not contain zero proper. It can get as close to zero as you'd like, but never contains actual zero.

If you want an intuitive idea of that... based on reasoning you've used before -- think of any spot "n" in the set... for whatever "n" you choose, you can be sure that none of the numbers before "n" are zero. Which means that zero can never appear in the sequence.

Hmmm. I suppose I find that argument... acceptable :-). I don't have to like it, but...

Rewinding a bit: what do we call arbitrarily large numbers if not infinite? There is no largest prime, and there are an infinite number of them, but we can't say there is an infinitely large prime--unless, perhaps, we decide to use a different number system like the surreal numbers. It seems like there's a hole in the vocabulary, or something.

- Dr. Trintignant

[GreedyAlgorithm]

Hmmm. I suppose I find that argument... acceptable :-). I don't have to like it, but...

Rewinding a bit: what do we call arbitrarily large numbers if not infinite? There is no largest prime, and there are an infinite number of them, but we can't say there is an infinitely large prime--unless, perhaps, we decide to use a different number system like the surreal numbers. It seems like there's a hole in the vocabulary, or something.

- Dr. Trintignant
You are conflating the value and the pointer to the value. None of the values are "arbitrarily large". An integer cannot be "arbitrarily large". But a spot where we will put an integer, a placeholder for an unknown integer, a pointer to the value - that we can describe as "arbitrarily large" when it is the case that the set of integers from which we are allowed to choose an integer has no upper bound.

[Dr. Trintignant]

You are conflating the value and the pointer to the value. None of the values are "arbitrarily large". An integer cannot be "arbitrarily large". But a spot where we will put an integer, a placeholder for an unknown integer, a pointer to the value - that we can describe as "arbitrarily large" when it is the case that the set of integers from which we are allowed to choose an integer has no upper bound.

Perhaps he was being imprecise, but Ziggurat said above "And that number can be arbitrarily large". What do we call a placeholder value, as opposed to a concrete number?

And for that matter, doesn't making this distinction prevent us from saying, for example, that lim(x->1, inf) 9/10^n is equal to 1.0? That really the limit just gives us a placeholder value that's not the same as the number it converges to?

- Dr. Trintignant

[GreedyAlgorithm]

Perhaps he was being imprecise, but Ziggurat said above "And that number can be arbitrarily large". What do we call a placeholder value, as opposed to a concrete number?
In English? Most phrasings are just imprecise, AFAIK. Or they're not imprecise once you are aware that concrete integers can't be "arbitrarily large". :D Seriously, it's hard for me to tell whether a given phrasing is imprecise or not because one of the readings just doesn't make any sense to me.

How's this as a paraphrase that is definitely correct: "And we can choose any integer number of 7s, no matter how large an integer that may be".

And for that matter, doesn't making this distinction prevent us from saying, for example, that lim(x->1, inf) 9/10^n is equal to 1.0? That really the limit just gives us a placeholder value that's not the same as the number it converges to?
Nope. The limit really is equal to 1. Limits are very rigorously defined mathematical objects whose values just so happen to correspond to intuition in almost all real-world scenarios. We're not choosing the largest number from {.9, .99, .999, ... } or any such silliness, that's just one approximate way of thinking about it which comes close to making sense a lot of the time.

[Dr. Trintignant]

How's this as a paraphrase that is definitely correct: "And we can choose any integer number of 7s, no matter how large an integer that may be".

I suppose that works, though it's a bit verbose.

Nope. The limit really is equal to 1.

There's a reason why I chose that example :-). I'm well aware that the limit is exactly 1, and that anybody who knows a bit of math wouldn't disagree... (but that a lot of fun can be had on forums defending that position).

We're not choosing the largest number from {.9, .99, .999, ... } or any such silliness, that's just one approximate way of thinking about it which comes close to making sense a lot of the time.

Well, I suppose that's just it. "Pick the largest of 0.9, 0.99, 0.999, etc." makes fine intuitive sense, even if it's not mathematically rigorous. "Pick the largest" is just not a sensible operation for some sets.

- Dr. Trintignant

[AntiTelharsic]

what do we call arbitrarily large numbers if not infinite?

We call them "finite". Every natural number, no matter how large it happens to be, is finite.

[arthwollipot]

My head hurts.

[Soapy Sam]

Your head is fine.
I hear a lot about infinities (they seem to have dropped the term "infinite number"), but I have yet to see one.
They seem to have much in common with invisible pink unicorns, of which I understand there are an infinity of species.

[kevinquinnyo]

So, let me know if I have got this straight. Can we come to a consensus on this?

Within said infinite string, if it was apples, oranges, and bananas:

There would be subset(s) of bananas that approaches infinite, likewise the same would be true of both oranges, and apples, but they wouldn't actually be infinite in length, within the infinite set.

Is that correct? I made it fruits because it takes away the math and makes it more conceptual.

Mmmmmm.. infinite bananas... :)

[Perpetual Student]

So, let me know if I have got this straight. Can we come to a consensus on this?

Within said infinite string, if it was apples, oranges, and bananas:

There would be subset(s) of bananas that approaches infinite, likewise the same would be true of both oranges, and apples, but they wouldn't actually be infinite in length, within the infinite set.

Is that correct? I made it fruits because it takes away the math and makes it more conceptual.

Mmmmmm.. infinite bananas... :)

Im not sure what you're saying. You can certainly define a set to have infinite subsets. The set {(A,A...)(B,B...)(C,C,...)} can be defined to be an infinite set with infinite subsets of A's, B's and C's. It depends on what you are trying to define.
If, on the other hand you defined the set of an infinite sequence of random entries of A, B and C then you would expect to find finite sequences of A's, B's or C's arbitrarily long.

[Beerina]

This question is weird. It may be that the question itself doesn't make sense.

But, here it is:

Let's say you have an infinite number. The digits of this number are random. Does this mean that, because the number is infinite, and because the digits are randomized, there would have to be a section of digits within this infinite series, that is itself, an infinite amount of say, 7's?


like: 3837934877777777777777777777777.... and it just goes on infinitely?


Doesn't that have to be true, if the number is

a) truly random

and

b) infinite?

It seems like a paradox that I just can't wrap my mind around. Any thoughts?


Such a number would have a "countably infinite" number of digits. I.e. you can number them 1, 2, 3, 4, 5, ...

This basically means that, for any finite string of digits you care to produce, no matter how long, you will find it in your infinite string somewhere, assuming the digits are truly random.

Indeed, if they are truly random, you will find a countably infinite number of copies of that finite string!

[Perpetual Student]

Such a number would have a "countably infinite" number of digits. I.e. you can number them 1, 2, 3, 4, 5, ...

This basically means that, for any finite string of digits you care to produce, no matter how long, you will find it in your infinite string somewhere, assuming the digits are truly random.

Indeed, if they are truly random, you will find a countably infinite number of copies of that finite string!

Nonsense!

[69dodge]

If you have an infinite string of digits, can that string really be said to contain another infinite string of digits?

I guess it depends on what exactly you mean by "contain". If you mean something like "surround on both sides", then no. But I think it's reasonable to say, for example, that...111110022222...contains22222...and that it contains...11111.But it can't also contain infinitely many 0s, because it has only two sides that go on forever---a left side and a right side---and they're already taken up by the 1s and the 2s.

[Vorticity]

Since I am a ****-stirrer, I cannot resist posting a link to this old, acrimonious, and related thread:

http://forums.randi.org/showthread.php?t=19218

[sol invictus]

Nonsense!

Eh? What Beerina said was perfectly correct.

[GreedyAlgorithm]

Your head is fine.
I hear a lot about infinities (they seem to have dropped the term "infinite number"), but I have yet to see one.
They seem to have much in common with invisible pink unicorns, of which I understand there are an infinity of species.
Should I assume you're joking or should I ask whether you've ever seen a perfect sphere?

[Perpetual Student]

Eh? What Beerina said was perfectly correct.

Yes, I misread the comment. Sorry Beerina.

[kevinquinnyo]

The next logical question for me is:

Is "truly random" just as difficult to quantify as "infinity?"

Or rather, is the concept of "truly random" subjective? Is it definable in any way?

[drkitten]

The next logical question for me is:

Is "truly random" just as difficult to quantify as "infinity?"

More difficult.

Infinity is actually a very well-defined concept; it's just counterintuitive. (A set is infinite if it cannot be finitely enumerated; i.e. if there isn't a specific whole number corresponding to its size. This can be proven to be equivalent to "a set is infinite if it can be placed in 1-1 correspondence with a proper subset of itself.")

"Truly random" is a collection of meaningless syllables. Even "random" is difficult to describe formally. "True random" is as hard as "True Scotsman," and for the same reason.

[Perpetual Student]

One way to define "random" for a particular case is as follows:
Starting with the letters A, B and C, one can define a random sequence of these letters by stating that for each entry of a letter, A, B and C have an equal probability of occurring.
Obviously one then can ask, "what is meant by 'equal probability'"? That's why axiomatic systems require a small collection of undefined terms. Trying to define every concept in terms of other defined concepts is not possible.

[drkitten]

One way to define "random" for a particular case is as follows:
Starting with the letters A, B and C, one can define a random sequence of these letters by stating that for each entry of a letter, A, B and C have an equal probability of occurring.

And here you have the problem, in a nutshell. Your definition is both too limited and not limited enough.

In particular, is the sequence "ABCABCABCABCABCABC..." random? Most people would say "no," for obvious reasons. So your definition isn't limited enough, because it includes too much.

At the same time, the requirement of equal probability is too limiting; if I generated a sequence by independently selecting symbols from with probability A:1/6, B:1/3 C:1/2, I would get a sequence that any mathematician would accept as "random," but not "uniformly random." Since most cases in the real world aren't uniform anyway, that's not a major problem. A pair of dice, for example, are "random" but not equiprobable -- 7 comes up much more than either 2 or 12.

Most mathematicians define random as a process description : something is "random" if it was generated by reference to a probability distribution (you can emend that to a non-trivial probability distribution, if you like). The problem then becomes that it is provably impossible by looking at an object to determine whether or not it was determined "randomly."

[Perpetual Student]

And here you have the problem, in a nutshell. Your definition is both too limited and not limited enough.

In particular, is the sequence "ABCABCABCABCABCABC..." random? Most people would say "no," for obvious reasons. So your definition isn't limited enough, because it includes too much.

At the same time, the requirement of equal probability is too limiting; if I generated a sequence by independently selecting symbols from with probability A:1/6, B:1/3 C:1/2, I would get a sequence that any mathematician would accept as "random," but not "uniformly random." Since most cases in the real world aren't uniform anyway, that's not a major problem. A pair of dice, for example, are "random" but not equiprobable -- 7 comes up much more than either 2 or 12.

Most mathematicians define random as a process description : something is "random" if it was generated by reference to a probability distribution (you can emend that to a non-trivial probability distribution, if you like). The problem then becomes that it is provably impossible by looking at an object to determine whether or not it was determined "randomly."

Excellent points!

[Dr. Trintignant]

We call them "finite". Every natural number, no matter how large it happens to be, is finite.

But I'm not talking about individual finite numbers here; I'm talking about what GreedyAlgorithm called "placeholder" values. I'll guess that the answer is just "you aren't allowed to talk about numbers in this way; you have to use the language of set theory for that". And at this point, everyone is agreed that each set of repeating substrings (like {7, 77, 777, ...}) in a normal number is infinite.

As an aside: obviously, it's possible to reason about repeated fractions, like 0.3333.... Is it possible to reason about repeated digits on the left side of the decimal point? I realize that this goes outside the realm of the natural numbers, but I wonder if there's any sensible way in which you could claim, for instance, that ...4444.0 is greater than ...3333.0.

- Dr. Trintignant

[Alkatran]

But I'm not talking about individual finite numbers here; I'm talking about what GreedyAlgorithm called "placeholder" values. I'll guess that the answer is just "you aren't allowed to talk about numbers in this way; you have to use the language of set theory for that". And at this point, everyone is agreed that each set of repeating substrings (like {7, 77, 777, ...}) in a normal number is infinite.

As an aside: obviously, it's possible to reason about repeated fractions, like 0.3333.... Is it possible to reason about repeated digits on the left side of the decimal point? I realize that this goes outside the realm of the natural numbers, but I wonder if there's any sensible way in which you could claim, for instance, that ...4444.0 is greater than ...3333.0.

- Dr. Trintignant

Of course you can extend things in that way. All the rules in math are mutable, as long as you're careful not to ruin everything by introducing contradictions.

So what happens if we allow infinite digit strings? I can't remember the name of what you get, but let's just say things get a bit weird. For example you find that ...9999 = -1, because ...9999 + 1 = ...0000 = 0 = -1 + 1. [Assuming you don't change how carries and stuff work for addition, and keep the rule that a+b=c+b => a=c]

[Dr. Trintignant]

So what happens if we allow infinite digit strings? I can't remember the name of what you get, but let's just say things get a bit weird. For example you find that ...9999 = -1, because ...9999 + 1 = ...0000 = 0 = -1 + 1. [Assuming you don't change how carries and stuff work for addition, and keep the rule that a+b=c+b => a=c]

Not weird at all for anyone familiar with two's-complement arithmetic on a computer :-). Still, that's interesting. Having infinite strings to the left and right of a decimal means I can create a mapping function that allows storing an infinite number of infinite strings in a decimal number, while still being able to extract any of them into a contiguous sequence.

- Dr. Trintignant

[Myriad]

Not weird at all for anyone familiar with two's-complement arithmetic on a computer :-). Still, that's interesting. Having infinite strings to the left and right of a decimal means I can create a mapping function that allows storing an infinite number of infinite strings in a decimal number, while still being able to extract any of them into a contiguous sequence.


But you can do that anyhow.

Imagine the real number 0.abacbadcbaedcbafedcba...

where the a's are the locations of the sequence of digits encoding the first string, the b's are where you find the digits encoding the second string, and so forth. Each string is infinitely long (made so, if necessary, by appending an indefinite number of placeholder characters at the end) and an infinite number of strings are included.

The math to extract the nth digit of the mth string is pretty straightforward.

Repsectfully,
Myriad

[Meridian]

As an aside: obviously, it's possible to reason about repeated fractions, like 0.3333.... Is it possible to reason about repeated digits on the left side of the decimal point? I realize that this goes outside the realm of the natural numbers, but I wonder if there's any sensible way in which you could claim, for instance, that ...4444.0 is greater than ...3333.0.

To the first part: yes there is, and people do - google "p-adic numbers". To the second: as far as I know there is no "sensible" way. If anything, the way p-adic integers work mean that it would be most natural to make the last digit most important, so, for example, ...4 > ..3333.

[Dr. Trintignant]

The math to extract the nth digit of the mth string is pretty straightforward.

Yes, that's more or less what I had in mind (actually, the sequence I was thinking of is 0.a ab abc abcd abcde, but the exact encoding doesn't matter). What I was curious about is whether it's possible to put all the a's in a row, for instance. If all the digits are on one side of the decimal, it's not possible: you can't create a map that has an infinite number of a's, then an infinite number of b's, and so on. But if you can use the left and right, you can create mappings like this, to extract one number at a time:

...aaaaa.bbcbcdbcde...
...bbbbb.aaacacdacde...
...ccccc.aabababdabde...

I don't think there's necessarily anything profound about contiguous substrings; I just find it curious.

- Dr. Trintignant

ETA: Of course, I could just use the mappings 0.aaaaa...., 0.bbbbb..., etc., but the point is that the above mappings are one-to-one, and thus invertible.

[Blackadder]

That much I can safely say is false. The infinity of the integers is smaller than the infinity of the real numbers, for instance. There is a very slick argument as to why this must be true, although it gives no indication as to "how much" bigger the larger infinity is.

- Dr. Trintignant


don't go there... :) you will end up with a lot of tourists, busses and hotel rooms.

[arthwollipot]

One way to define "random" for a particular case is as follows:One pithy definition of a random sequence I have seen is "a sequence that can not be described in a manner that is shorter than the sequence itself."

[AntiTelharsic]

http://web.archive.org/web/20011027002011/http://dilbert.com/comics/dilbert/archive/images/dilbert2001182781025.gif

[Soapy Sam]

Should I assume you're joking or should I ask whether you've ever seen a perfect sphere?
No I'm not joking.
No, I have never (to my knowledge) seen a perfect sphere.
Should I assume that the putative non-existence of the one in some way implies the existence of the other?

I suspect (and I claim no more than that), that "infinity" is nothing more than a human concept- an artifact of the mind- in exactly the same way as an invisible pink unicorn.
Concrete evidence of the existence of either would change my suspicion. Until I see any, I remain sceptical.

[Soapy Sam]

One pithy definition of a random sequence I have seen is "a sequence that can not be described in a manner that is shorter than the sequence itself."

But you just defined a whole class of the things in 17 words! Mirabile dictu!

[69dodge]

But you just defined a whole class of the things in 17 words!

There's no contradiction. Sometimes it's easier to describe a whole class of things than to describe a single one of them in enough detail to distinguish it from all the rest.

Saying "all million-digit numbers" is quick and easy; telling you which particular one I have in mind requires, most likely, just about a million digits.

[GreedyAlgorithm]

No I'm not joking.
No, I have never (to my knowledge) seen a perfect sphere.
Should I assume that the putative non-existence of the one in some way implies the existence of the other?

I suspect (and I claim no more than that), that "infinity" is nothing more than a human concept- an artifact of the mind- in exactly the same way as an invisible pink unicorn.
Concrete evidence of the existence of either would change my suspicion. Until I see any, I remain sceptical.
Oh okay, sure. No problems there. Even if you're right, though, it's nothing more than a human concept in exactly the same way that a perfect sphere or Fourier transform is nothing more than a human concept... namely incredibly useful to progress of math & science.

[Myriad]

Describing numbers using words can be tricky, indeed.

For instance, I can "prove" that any integer, regardless of how many digits it contains, can be uniquely described in fourteen or fewer English words. Because it that were not the case, there must exist "the smallest integer that cannot be uniquely described in fewer than fifteen English words" -- which is a contradiction.

Respectfully,
Myriad

[Soapy Sam]

There's no contradiction. Sometimes it's easier to describe a whole class of things than to describe a single one of them in enough detail to distinguish it from all the rest.

Saying "all million-digit numbers" is quick and easy; telling you which particular one I have in mind requires, most likely, just about a million digits.

Oi! That was a joke!

[Soapy Sam]

Oh okay, sure. No problems there. Even if you're right, though, it's nothing more than a human concept in exactly the same way that a perfect sphere or Fourier transform is nothing more than a human concept... namely incredibly useful to progress of math & science.

No argument. Things can be very useful , yet not necessarily correct.
Navigation by assuming stars are on fixed spheres works adequately, despite being wrong- but it's that sort of wrongness where the details don't matter to the end result.

I'm deeply suspicious of the very idea of "number". I seriously think we are missing something fundamental.
I have not the slightest idea what, sadly.:o

[Soapy Sam]

Describing numbers using words can be tricky, indeed.


Respectfully,
Myriad

Spot on! This is one reason predominantly verbal thinkers (like me) have huge conceptual difficulty with mathematics.
I console myself with the wishful thought that it also may occasionally give a perspective whose differences may (however rarely) be useful.

[LostAngeles]

don't go there... :) you will end up with a lot of tourists, busses and hotel rooms.

And I already talked about that hotel's bar in another thread.

[JoeTheJuggler]

This reminds me of a bit in Carl Sagan's Contact. The random, infinite string of numbers that is Pi ended up containing messages from the uber-alien. Some insanely large number of digits to the right of the decimal point, there appeared a string of numbers (I think this was all in binary) that was a sequence of the first so many prime numbers, then other marvels.

I often wondered about this, because if it's a truly random and infinite string, then these highly unlikely sequences are in fact certain. (And as pointed out, certain to occur an infinite number of times even!)

Speaking of which, you can search the first 200 million digits of Pi for interesting strings here: http://www.angio.net/pi/piquery

My phone number appears, but not my social security number. Shorter strings, of course, appear many times (my 4 digit street address appears 3361 times).

[Soapy Sam]

http://www.angio.net/pi/piquery

How sad that anyone would devote time to creating such a web page.

Even sadder that I just bookmarked it! :o

[GreedyAlgorithm]

No argument. Things can be very useful , yet not necessarily correct.
Navigation by assuming stars are on fixed spheres works adequately, despite being wrong- but it's that sort of wrongness where the details don't matter to the end result.
(emphasis added)

But if you want to be precise, that's not what we do. We navigate by assuming that if we model stars as fixed spheres in the map, our calculation will correspond closely enough for the given purpose to the counterfactual precise calculation in the territory. And this assumption is not wrong.

The map is not the territory, but that doesn't mean that all statements about degrees of correspondence between the map and the territory are automatically wrong.

[kevinquinnyo]

Does it make sense to say that

A) Infinity contains all things

and

B) If this is true, is infinity the only 'thing' that is random?

I realize that this is bordering on nonsense, but I wonder if there is any validity to this, because something about that makes sense. If something goes on finitely and stops, it would theoretically be possible to make a pattern out of it, but if it goes on forever, a pattern can't emerge because it doesn't have an end.

Again, I realize this may just sound like what a coked up hipster on the back porch at a party at 4 in the morning would say, but I don't know..

Any thoughts on that?

[JoeTheJuggler]

B) If this is true, is infinity the only 'thing' that is random?

No. Infinite and random are two very different things.

You can make a random choice (among finite options), by tossing a coin or rolling a die, for example.

You could also have an infinite series of digits that are not random (for example, the repeating decimal 3.333. . . that is the result of dividing 10 by 3, or the number you get from dividing 3227 by 555: 5.8144144144. . . where the digits 144 repeat forever).

[HansMustermann]

Well, it's not infinity itself, it's the infinite number of random digits that, yes, makes it contain every single sub-string of digits imaginable.

Let's put it like this, since Pi is a normal number, if you generated Pi in binary to an infinite number of digits, you'd now have on your (hypothetical) infinite had drive as substrings of it

- all the copyrighted texts in existence, or which will ever be produced

- plus some quite original novels that nobody wrote yet

- all the copyrighted music as MP3s, OGG, WAV and a few other formats, in every imaginable bit rate

- ditto for movies

- an infinite number of derivative works thereof

- an impressive collection of kiddie porn

- your digitally signed confession for every crime that ever was committed

Etc.

So better don't do it, ok? ;)

[JoeTheJuggler]

if you generated Pi in binary to an infinite number of digits, you'd now have on your (hypothetical) infinite had drive as substrings of it

- all the copyrighted texts in existence, or which will ever be produced

- plus some quite original novels that nobody wrote yet

- all the copyrighted music as MP3s, OGG, WAV and a few other formats, in every imaginable bit rate

- ditto for movies

- an infinite number of derivative works thereof

- an impressive collection of kiddie porn

- your digitally signed confession for every crime that ever was committed

Etc.

So better don't do it, ok? ;)

Not to mention the secret messages (that look like they're from God or the uber-aliens in Sagan's Contact).

ETA: And infinitely many copies of each of these things.

[gnome]

To be fair, I think the premise in "Contact" was that the "creator's" message appeared far earlier in the sequence than statistically conceivable.