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#1 2006-07-21 01:01:05

Kurre
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Registered: 2006-07-18
Posts: 280

pi and other numbers with infinite numbers of decimals

i was just thinking about Pi's decimals. If they continue forever, every number should appear an infinite number of times. and if that is true, every number  combination that exists, should also appear, and if it appear once it should appear an infinite numbers of times. so for example the number combination 4738002863772894374738387388399 appears an infinite number of times in Pi's decimals...
is this true?

Last edited by Kurre (2006-07-21 05:32:24)

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#2 2006-07-21 05:03:24

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

Re: pi and other numbers with infinite numbers of decimals

its possible, but not definate, because pi's digits aernt random, so theres no surety that the combination will be present in an infiniate given number of digits


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#3 2006-07-21 05:28:11

John E. Franklin
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Registered: 2005-08-29
Posts: 3,562

Re: pi and other numbers with infinite numbers of decimals

That's a tough question.  I think you could have an irrational number that never had "666" in it.
Is it possible?  Yes!  But maybe I don't know.


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#4 2006-07-21 05:38:27

Kurre
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Registered: 2006-07-18
Posts: 280

Re: pi and other numbers with infinite numbers of decimals

luca-deltodesco wrote:

its possible, but not definate, because pi's digits aernt random, so theres no surety that the combination will be present in an infiniate given number of digits

no but they doesnt repeat and start over from 14159 (or at least noone has seen any kind of pattern in it as far as i know), so it must be an irrational order all the way...and since there are an infinite amount of numbers.....:p

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#5 2006-07-21 06:25:08

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

Re: pi and other numbers with infinite numbers of decimals

Kurre wrote:

...and since there are an infinite amount of numbers.....:p

infinate amount of numbers doesnt mean infinate amount of combinations, take for example

0.1111111111111111111....111112

i.e. infinate 1's then a 2, infinate numbers, no pattern, but you can only get a combination involving 1's, or 1's with a 2 on the end


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#6 2006-07-21 07:13:52

Ricky
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Registered: 2005-12-04
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Re: pi and other numbers with infinite numbers of decimals

Kurre, as luca pointed out, you make the assumption that pi's digits are random.  That does have to be the case.

For example, there doesn't have to be a 1/10th chance that the next digit will be 8.  There could be a 1/100th chance, and it would still be irrational.  Right now, we are finding that there are long strings of digits where 9 is completely absent.  We are trying to find out if at one point it will drop out all together.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2006-07-21 11:54:02

MathsIsFun
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Registered: 2005-01-21
Posts: 7,551

Re: pi and other numbers with infinite numbers of decimals

9 could stop occuring? Then there would be a finite number of 9s. But how would you prove it?

Mathematics never ceases to amaze.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#8 2006-07-21 21:04:37

numen
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Registered: 2006-05-03
Posts: 115

Re: pi and other numbers with infinite numbers of decimals

luca-deltodesco wrote:

infinate amount of numbers doesnt mean infinate amount of combinations, take for example

0.1111111111111111111....111112

i.e. infinate 1's then a 2, infinate numbers, no pattern, but you can only get a combination involving 1's, or 1's with a 2 on the end

But luca, that doesn't make any sense. If there was an infinite amount of 1's, then they wouldn't stop and suddenly end with a 2, because then the 1's must be finite. neutral


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#9 2006-07-22 03:13:25

All_Is_Number
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Registered: 2006-07-10
Posts: 258

Re: pi and other numbers with infinite numbers of decimals

Ricky wrote:

Kurre, as luca pointed out, you make the assumption that pi's digits are random.  That does have to be the case.

For example, there doesn't have to be a 1/10th chance that the next digit will be 8.  There could be a 1/100th chance, and it would still be irrational.  Right now, we are finding that there are long strings of digits where 9 is completely absent.  We are trying to find out if at one point it will drop out all together.

This is an interesting thread. I'm inclined to believe the original poster's proposition. Making the assumption that pi really is an irrational number (lack of evidence of being rational is not proof that it's irrational), then there are an infinite number of decimal places. Even if the odds were 1 in 10^100 of a nine appearing, there would still be occurrences of nines occurring repetitively, as 1 in 10^100 are much better odds than 1 in infinity.

Imagine a deck of cards. Shuffle the deck and deal. The odds of dealing a royal flush are very, very slim. However, repeat the shuffle/deal process indefinitely, and a royal flush will be dealt repetitively on an infinite number of occasions.

The point I'm trying to make is that no matter how slim the odds of "process A" happening, given an infinite number of attempts, process A will occur an infinite number of times.


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#10 2006-07-22 03:19:48

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: pi and other numbers with infinite numbers of decimals

Interesting question.
It may be extended for another numbers, such as e, phi, sqrt(2), ect.
Last month I read about this theme.
I'll post a link with information.


IPBLE:  Increasing Performance By Lowering Expectations.

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#11 2006-07-22 03:20:44

All_Is_Number
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Registered: 2006-07-10
Posts: 258

Re: pi and other numbers with infinite numbers of decimals

luca-deltodesco wrote:

its possible, but not definate, because pi's digits aernt random, so theres no surety that the combination will be present in an infiniate given number of digits

If pi's digits aren't random, isn't that evidence that pi is in fact not irrational? If they are not random, that implies a pattern exists, even if we don't yet understand it.

--All Is Number   <--- believes there are some fundamental mathematic operations and/or patterns that mankind has yet to discover or comprehend.


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#12 2006-07-22 03:22:26

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: pi and other numbers with infinite numbers of decimals

Such numbers are called normal.
Here is it: (I'm not explaining because the explanations are there):
http://en.wikipedia.org/wiki/Normal_number


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#13 2006-07-22 04:31:38

All_Is_Number
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Registered: 2006-07-10
Posts: 258

Re: pi and other numbers with infinite numbers of decimals

krassi_holmz wrote:

Such numbers are called normal.
Here is it: (I'm not explaining because the explanations are there):
http://en.wikipedia.org/wiki/Normal_number

To whom are you responding?


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#14 2006-07-22 04:37:43

John E. Franklin
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Registered: 2005-08-29
Posts: 3,562

Re: pi and other numbers with infinite numbers of decimals

I am of the belief that there are patterns of digits of long lengths that won't exist in pi.
It is just a guess, but an analogy is that if time went on forever, you could not do everything you wanted to do so you have to pick and choose.


igloo myrtilles fourmis

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#15 2006-07-22 05:34:35

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: pi and other numbers with infinite numbers of decimals

I'm talking about the randomination of the numbers (for example pi).
Numbers in which all the combinations of digits are met, are called normal.
So you're disputing if pi is normal.
It is beleived that it is, but it's not proved.


IPBLE:  Increasing Performance By Lowering Expectations.

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#16 2006-07-22 07:04:18

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

Re: pi and other numbers with infinite numbers of decimals

John E. Franklin wrote:

if time went on forever, you could not do everything you wanted to do so you have to pick and choose.

If time went on forever, there will always be time to do at least one more thing. If there is always time to do at least one more thing, there would be no reason to have to pick and choose since you would have time to do everything.


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#17 2006-07-22 07:14:55

Patrick
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Registered: 2006-02-24
Posts: 1,005

Re: pi and other numbers with infinite numbers of decimals

Well, you're assuming pi doesn't have an exact value, but it does - as far as I've understood. It is EXACTLY:

where P is the perimeter of a circle and d is its diameter. So just because we can't seem to find a pattern, it doesn't have to mean the digits are random. Or am I totally off track?


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#18 2006-07-22 07:33:52

All_Is_Number
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Registered: 2006-07-10
Posts: 258

Re: pi and other numbers with infinite numbers of decimals

Patrick wrote:

Well, you're assuming pi doesn't have an exact value, but it does - as far as I've understood. It is EXACTLY:

where P is the perimeter of a circle and d is its diameter. So just because we can't seem to find a pattern, it doesn't have to mean the digits are random. Or am I totally off track?

I don't think anyone is claiming Pi does not have an exact value. It is just an exact value that cannot be shown with a ratio of two integers.


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#19 2006-07-22 08:45:00

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: pi and other numbers with infinite numbers of decimals

"Random" has a meaning implied behind it which I think many have not heard of.

If I have 5 marbles in a bag, each a different color, and I have only a 10% chance of pulling out the blue marble, then the action of pulling out marbles is not a random one.

When we say random, we mean that the chances of all events happening are equal.

Now I believe statistically speaking, the number of digits in pi do not appear an equal number of times.  It's been a while, I'll have to see if I can find that again.  Thus, the digits in pi are not equal.

Now we only know a finite number of the digits.  So many this is only because we are looking at a subset of the digits that they aren't equal.  But when we've calculated it to over 50,000 places, thats normally enough to get a pretty good picture.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#20 2006-07-22 09:08:50

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

Re: pi and other numbers with infinite numbers of decimals

Ricky wrote:

"Random" has a meaning implied behind it which I think many have not heard of.

If I have 5 marbles in a bag, each a different color, and I have only a 10% chance of pulling out the blue marble, then the action of pulling out marbles is not a random one.

When we say random, we mean that the chances of all events happening are equal.

Now I believe statistically speaking, the number of digits in pi do not appear an equal number of times.  It's been a while, I'll have to see if I can find that again.  Thus, the digits in pi are not equal.

Now we only know a finite number of the digits.  So many this is only because we are looking at a subset of the digits that they aren't equal.  But when we've calculated it to over 50,000 places, thats normally enough to get a pretty good picture.

I think the important thing to remember is that since Pi has an infinite number of decimal places without a recognizable pattern, it does not matter if 1 appears more often than 5, since they will both appear an infinite number of times. The same goes for 09821346598712985170279561278934560 and 98435019823450928834087 both of which should appear an infinite number of times. Pi has been calculated to millions of decimal places, but that is still just a drop in the bucket of the infinite number of decimal places the irrational number contains.


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#21 2006-07-22 13:00:02

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: pi and other numbers with infinite numbers of decimals

But, just because the pattern is unrecognizable, we can't be sure that every number will appear only finite number of times.
I'm sure that every numbers occur infinitely, but there's a problem - this must be proved to be true.


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