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#1 2008-06-01 08:25:44

bossk171
Member
Registered: 2007-07-16
Posts: 301

Irrational number with predictable pattern

I've been thinking about irrational numbers recently (my job doesn't require much brain power) and it occurred to me that there can be rational numbers with predictable patterns. Any number with repeating decimals can be written as a fraction:

.333333... = 1/3

But you can have a non-repeating but predictable number. I vaguely remember reading somewhere that the number

.12345678910111213... (all the natural numbers just lumped together after the decimal place)

has a name. Does anyone know what it is? Also I was thinking:

.2 1 2 11 2 111 2 1111 2 11111 2....(spaces added for emphasis)

Has to be irrational and predictable. (It's a 2 then one 1, then a 2 and two 1s then a 2 and three 1s, etc). Is anyone familiar with this? It's a little basic so it can't be new.

As I think more about it, there are defiantly an infinite amount of predictable irrationals. For example:

.112358132134.... (the Fibonacci sequence scrunched together)

Is a pretty obvious one, but you can make a Fibonacci sequence with any two starting values (4 and 5, or 1 and 3 for example) and because there are an infinite amount of natural numbers, that's an infinite amount of predictable irrational numbers.

So does anyone know a name for the predictable irrationals? Can anyone prove that they're transcendental (or not, I just assume they are)? Does anyone know of any cool ones?


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#2 2008-06-01 08:34:13

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

Re: Irrational number with predictable pattern

I vaguely remember reading somewhere that the number

.12345678910111213... (all the natural numbers just lumped together after the decimal place)

I don't know about any official name, but I call it the "number of everything".  Translating it into binary and then ascii, every single play that Shakespeare wrote is in that number.  Every single mathematical proof/definition/axiom is also in that number.

Cool, but utterly useless.


"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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#3 2008-06-01 12:32:04

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: Irrational number with predictable pattern

I guess that makes sense, because all of the possible strings appear in there. That's really cool, I didn't think of it that way. Does that mean that both pi and e appear in there completely (I'm not just talking 3.14159265358979, I'm talking the whole thing) or does that only include finite works (like Shakespeare)?

And good point about it being useless, are there any predictable pattern irrationals that are useful?


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#4 2008-06-01 13:59:44

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

Re: Irrational number with predictable pattern

Does that mean that both pi and e appear in there completely (I'm not just talking 3.14159265358979, I'm talking the whole thing) or does that only include finite works (like Shakespeare)?

What would happen if pi did appear in there completely?

And good point about it being useless, are there any predictable pattern irrationals that are useful?

What's your definition of useful?  No matter what it is, as far as I know, no.  But that's only as far as I know.


"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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#5 2008-06-01 14:41:13

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: Irrational number with predictable pattern

Ricky wrote:

What would happen if pi did appear in there completely?

Well my next point would be that if both pi and e were contained in that number entirely, then that means one of them has to contain the other, and that logic could be expanded to contain all irrational numbers... That just seems wrong.

As for "useful," I just meant mathematically relevant. I don't have any really strict idea of what the word should mean, just the opposite of what you meant when you said that .1234... was "utterly useless"


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#6 2008-06-13 12:01:54

accc
Member
Registered: 2008-06-13
Posts: 1

Re: Irrational number with predictable pattern

It doesn't cover everything but many such examples are covered by Liouville numbers, all of which are transcendental.

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