infinite numbers

I have no clue if this is the right spot for this subject, so feel free to move it.

There is a question I had. If there is an infinite amount of numbers between two numbers than when does the next number start and the other one end?

I'm going to answer this, but move it to the Science section...

Originally posted by DirtyWater33
There is a question I had. If there is an infinite amount of numbers between two numbers than when does the next number start and the other one end?

I'm lucky here; my muchly adored spouse is a mathematics educator with a couple of masters' degrees. So I asked him.

Let's go back to your question and I'll reframe it for folks who may have not heard this one before: in math, there are an infinite number of numbers between any two ... numbers.

We could take any two numbers we liked (like between 1/4 and 2/4 or between .000001 and .00002 or between -44444 and -44444.00001) but for the sake of illustration we'll pick the numbers 1 and 2

So.. there are numbers between 1 and 2 -- 1.2, 1.4, 1.5, (etc etc etc)

But there's an infinite number of those numbers between each and every step. No matter how small you make the difference between two numbers, you can stick more numbers in there by packing another zero in: between .0000001 and .0000002 you can put in .00000011 and .0000001111 and .00000011111111 -- and so on until you get tired and bored of adding numbers.

These things get studied in "category and set theory" in mathematics, and it's actually all rather fascinating (I love reading about it, but some of the equations just give me the HIVES!)

Anyway, DW, the short answer is that the sequence will "approach" a number but won't actually BE that number... until it hits the number itself. Once it leaves that number, it starts "approaching" another number.

I hope that makes sense.

A famous (and favorite of mine) illustration of infinity is mathemetician David Hilbert's "Hotel"... I'll let Wikipedia explain it since they did a nice job:
en.wikipedia.org...'s_paradox_of_the_Grand_Hotel

thanks the link really helped clarify what you meant.

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**footnote to hilbert's hotel if you add guests to an already full hotel by moving them into room 1 and telling the current occupant to move to room 2 etc , in a sense you double the capacity of the hotel. In any given instant there are potentially an infinite number of guests moving to the next room and an infinite number of guests in the rooms giving 2*infinity guests.

In terms of infinite real numbers between any two numbers the set of numbers of any given limited number of digits (ie. 2 digits = 100) is always vastly smaller than the set of numbers with more digits. So the largest set of numbers must be the one with infinite digits (regardless of the base).
So i think it follows that those would be in fact irrational numbers. I believe the only way of having infinite digits for a rational number is to have repeating sequences of digits. Since the alteration of a single digit in repeating sequence disturbs that pattern the number of permutations variant from that pattern must vastly outnumber that pattern.

Slank's Thesis: The majority of real numbers are in fact irrational numbers.

Does that make sense? Does anyone have arguments or rationales against that or that support my idea?
.

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alternate rationale,

b is the number base (with b digits)

If you take a set of strings of length n of b digits and consider that gives (b^n) permutations, which is less than the number of permutations of n+1 length strings [ b^(n+1) ]

so string lengths of infinity-1 [treating those as rational numbers] has less permutations than string lengths of infinity by a factor of b.

(I think that makes sense, except that it makes it sound like the preponderance of irrational numbers varies depending on the base used.)

(not sure how you would subtract out the number of repeating sequences though
intuitively i would think those would be a small subset of any set of permutations)
.

hope no one minds while i mine this vein of thought.

on repeating sequences:
0 = 0.0_

1/9 = 0.1_
2/9 = 0.2_
3/9 = 0.3_ = 1/3
4/9 = 0.4_
5/9 = 0.5_
6/9 = 0.6_ = 2/3
7/9 = 0.7_
8/9 = 0.8_
9/9 = 1.0 = 3/3
1/99 = 0.01_
2/99 = 0.02_
3/99 = 0.03_
. . .
13/99 = 0.13_
. . .
98/99 = 0.98_
. . .
1/11 = 0.09_ = 9/99
2/11 = 0.18_ = 18/99
3/11 = 0.27_ = 27/99
4/11 = 0.36_ = 36/99
5/11 = 0.45_ = 45/99
6/11 = 0.56_ = 56/99
7/11 = 0.63_ = 63/99
8/11 = 0.72_ = 72/99
9/11 = 0.81_ = 81/99
10/11 =0.90_ = 90/99
11/11 =1.0 = 99/99

1/999 = 0.001_
. . .
123/999 = 0.123_

1/7 = 0.142857_ = 142857/999999
2/7 = 0.285714_ = 285714/999999
3/7 = 0.428571_ = 428571/999999
4/7 = 0.571428_ = 571428/999999
5/7 = 0.714285_ = 714285/999999
6/7 = 0.857142_ = 857142/999999
7/7 = 1.0

I think you can create any permutation repeating sequence by putting the repeating substring over a substring of the same length that is all 9s.

If we were using Hexidecimal I would imagine the same thing put over a string of Fs.
Or in base 8 (octal) we would use a string of 7s.

(will have to think about what this means interms of the number of repeating substrings compared to all other permutations)

Originally posted by slank

Slank's Thesis: The majority of real numbers are in fact irrational numbers.

Does that make sense? Does anyone have arguments or rationales against that or that support my idea?
.

A few numbers between 1 and 2

I don't think so Slank... though I have no theories to back it up. Most numbers ere indeed rational

1 is a irrational
1/2 is rational
2/1 is irrational.

If you started plotting the numbers between 1 and 2, I think that you would find that most of the numbers were rational as a opposed to irrational. Though, you may see someting I don't see. Math is definitly NOT my strong point. What was making you believe this, I AM interested in your thoughts

[edit on 25-11-2004 by farhyde]

A number doesn't start and doesn't end, just like a point doesn't start or end. I guess you could say that 2 lies between lim(x->infinity) (2-1/x) and lim(x->infinty) (2+1/x).

What may interest you is that there are different kinds of infinity. The amount (or the cardinality of the set) of integer numbers is aleph-0, while the amount of real number is aleph-1. If you find this interesting, you should search for information about transfinite numbers, continuum hypothesis and Cantor's diagonal argument.

slank: actually, your thesis is right, in a certain sense. proof a:

let's call N the set of all positive integers (0,1,2,3,4,...)

let's say if we've got set A and set B then they have the same size if and only if you can "pair up" members of A and B so that every member of A has a unique matching item in B, and every member of B has some member in A so that A is paired with B (for the mathematicians, you want the "pairing" to be an injective and surjective map from A to B, or alternatively a bijective map from A to B, but if you already know those terms you've probably seen all that's coming next).

So, are there any sets that are the same size as N? Sure; how about we call

Z the set of all positive and negative integers, ie, (...-2,-1,0,1,2,...)

is Z bigger than N? You might be inclined to say yes, because everything in N is in Z but not everything in Z is in N, but using our definition of same size I can show you that N and Z are the same size:

match:

0 with 0
1 with 1
2 with -1
3 with 2
4 with -2
5 with 3
6 with -3

etc.,

and you can check that every member of N gets a unique member of Z, whereas every member of Z gets a unique member of N, so in fact using the "same size" definition above N and Z are the same size.

There's also a way to show that the rationals -- let's call them Q -- are the same size as N, but it's a bit trickier to write down. For now, take my word that the size of the rationals and the size of the integers is the same.

So, what about irrationals/the rest of the real numbers? follow amantine's advice and google the diagonalization argument, or look at this summary:

every number between 0 and 1 can be expressed as an infinite binary string:

0.1 = one half
0.11 = 3/4
0.01 = 1/4, etc.,

and since every number between 0 and 1 can be expressed like that, let's let R = the set of all infinite binary strings (padding finite strings like 0.1 with 0000's all the way to infinity). We need to be careful here:
0.11111111111111111111111111111111111...etc
is actually just 1,

and more generally, if we've got a string that ends like this

.....0111111111111111 (all 1s)

then it's the same as

----100000000000000

so we need to be careful about that; it's easiest just to throw away all strings that end in infinite 1s. Having done that, what would it mean if R was the "same size" as N? we'd have to be able to "pair off" strings from N and R, right? So, we could write a table like this:

0 , 0.100110101001
1, 0.101110100110
2, 0.101001101010

etc., (i'm just using random strings on the right side), and every string in R would be paired with a number from N. But, what if we did this:

the nth digit of x is 1 if the nth digit of the nth string above is 0, and the nth digit of x is 0 if the nth digit of the nth string above is 1...

then x would be an infinite string of 0s and 1s, and thus in R (again, you have to be careful to make sure that it's not all 1s at the end, but this can be done), but because x's nth digit disagrees with the nth digit of the nth string in our table, x isn't in our table; thus we've contradicted ourself and we can conclude that it's impossible to pair up the sets N and R.

Since R is just a small part of the real numbers (being only from 0 to 1 instead of - infinity to + infinity ) then all the real numbers are clearly a bigger set than N. Thus, yes, there's a sense in which the irrationals greatly outnumber the rationals.

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I think that any rational number in it's decimal [or other base] point expansion must terminate in a repeatative string. ie. all zeros, all ones, all 12,12,12, etc.

With any of these a rational number can by multiplication [shifting to the left] by the number of digits in the repetitions and then subtracting the original number it cancels the repeating string and gives the original rational times the base^(number of repeating digits) - 1. Which as a 'finite' digit string be rational.

note: finite digit string is defined as one that terminates in an infinite string of zeros.

for any pattern [repeated sequence] that occurs the next subsequent digit in the expansion is limited to a single value. As we add digits the single pattern possibility as opposed to all other potential patterns eventually becomes 1/infinity. [ b = base; 1/b, 1/b^2, 1/b^3, etc, 1/infinity ]

I suppose you have to balance that against the ability to start a pattern from any limited number of digits immediately to the left of your current working position [or just your current working digit].


Thinking about probabilities of repeating patterns . . .

in six digits XXXXXX [using base 10]

you could have a pattern of one digit X X X X X X ( 10^1 possibilities)
you could have a pattern of two digits XX XX XX ( 10^2 possibilities)
you could have a pattern of three digits XXX XXX ( 10^3 possibilities)

so (10^1+10^2+10^3)/(10^6) = 1110/1000000

which means the probability of a repeating string occuring is slightly more than the squareroot of the total possible permutations.

Is that a reasonable argument that the total number of rational numbers is slightly more than the squareroot of all the irrational numbers?

[of course the squareroot of infinity is still infinity so we shouldn't run out of rational numbers anytime soon ]
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actually, if you want a way to compare the size of the rationals and irrationals, it's about like

rationals of size = N => irrationals of size 2^N (that's two to the N)

why is this?

we'll need to define what's called the power-set of a set. so: if X is a set, then it's power set P(X) is the set of all subsets of X; it's perhaps easiest to get a feel for the definition by looking at an example.

if X = [1,2,3] is a set of three items, then P(X) has as members:

[] (nothing)
[1] (just one)
[2] (just two)
[3] (just three)
[1,2] (all but three)
[2,3] (all but one)
[1,3] (all but two)
[1,2,3] (everything)

or 8 elements, and 8 = 2^3; it's not hard to prove that in general if X is a finite set then P(X) is of size 2^(size of X).

Now, let's let N = the set of integers ( N = [0,1,2,...])...you'll have to take my word for it for now that the size of N and the size of the set of rationals is the same.

Recall that we could identify the real numbers from 0 to 1 with infinite strings of the form 0.(1's and 0's); but, look at this way:

if M is a subset of N, then we can map M onto a string of 0s and 1s like this: the nth digit of M's string is 1 if n is in M, and is 0 otherwise; it's easy to see that this gives each subset M of N a unique string o 0s and 1s, and that each string of 0s and 1s has a subset M of N associated to it, and so the size of the set of all the infinite strings of 0s and 1s and the size of P(N) (the power set of N) is the same.

we already saw that the set of infinite strings of 0s and 1s is the same size as the numbers from 0 to 1 and thus we've (sort-of, I'm skipping over some of the minor details you have to check) shown that the size of the numbers from 0 to 1 and the power set of N is the same, and thus if you wanted to make the comparison the size of the reals (which is, but hasn't been shown to be, the same as the size of the reals from 0 to 1) is 2^(size of the integers = size of the rationals).

(do keep going as long as you like. This is the part of math that I barely understand but really enjoy hearing about.)

en.wikipedia.org...'s_diagonal_argument


there's a good link to Cantor's diagonal argument which helps understand whats going on here.

sorry it took me so long to reply, but no one has entered somethin new so i forgot about it. More links would be helpful.

i think that the easiest way to understand it, atleast for me, is to image that it is infinately possible to get closer to a number without accually being that number.

for example in .9 you are approaching 1 but you arent quite there but you can get closer (.999) and closer (.99999999) still without accually getting the number 1. But while .9 is almost 1 in some since, it can be seen that .99999999 is much closer to that number and so on.

when i first thought of this concept it definately blew my mind, when i started calculus. But it really just depends on the scale which you attempt to look at it. Another example of how something can infanitely exist is if you take something and fold (or tear) it in half. Once you initially fold (tear) it in half and you do it again and again it gets smaller and smaller. Now on a scientific level you could continue to make it smaller and smaller in halves forever. (i guess maybe a limit is the size of an atom or something) This is like the idea of half-life where the presence of a substance becomes less and less over time, but im not as sure about this principle so correct me if im wrong.

Being a horribly nit-picky mathematician/engineer who tries his best to avoid paradoxes (for this instance, that would be the Point Paradox), I'd like to ask you first to define the universe for us.

Are we speaking in terms of an absolute continuous space or an absolute discrete space? If neither, what degree of continuity is there in our universe?

This is important because in an absolute continuum, there is no "start" and "end". "Start" and "End", "On" and "off"... they are all DISCRETE concepts. So we can't eventalking about one number ending and another number starting.

We don't exactly have discrete numbers. We have probability fields I guess. You could draw analogies between this and physical reality (Uncertainty Principle).

EDIT: Wow, I'm tired, I apologize ahead of time in case there are spelling or grammar errors my blurry vision couldn't catch.

[edit on 6-12-2004 by Nox]