Pi and Irrational Numbers

[wexer9]

Pi is an irrational number, meaning it never ends.
It's infinitely long, therefore it must contain every possible combination of numbers an infinite amount of times.
Does this mean pi contains every piece of software (in 0's and 1's) ever written? And if we could reduce genetic code to numbers, would it contain the genetic code for every possible human being?
It's mind-blowing stuff, if I've got it right.

 

[Thabiguy]

Pi is an irrational number, meaning it never ends.
It's infinitely long, therefore it must contain every possible combination of numbers an infinite amount of times.

Almost surely^WP.

Does this mean pi contains every piece of software (in 0's and 1's) ever written? And if we could reduce genetic code to numbers, would it contain the genetic code for every possible human being?

Almost surely.

It's mind-blowing stuff, if I've got it right.

It is, isn't it? Now consider that there are infinitely many other numbers that also contain all that; in fact, there's infinitely many times more of them than natural numbers...

 

[Alkatran]

Pi is an irrational number, meaning it never ends.
It's infinitely long, therefore it must contain every possible combination of numbers an infinite amount of times.
Does this mean pi contains every piece of software (in 0's and 1's) ever written? And if we could reduce genetic code to numbers, would it contain the genetic code for every possible human being?
It's mind-blowing stuff, if I've got it right.

Just being infinitely long doesn't mean it contains every possible combination of digits. There are infinitely many odd numbers, but none of them are 16. It is actually unknown whether or not the particular sequence that makes up pi contains every possible finite sequence. It's generally believed that this is the case.

But let's keep the "contains the genetic code of every possible human being" talk in perspective. Start counting. Congratulations, you've started a process which will enumerate all possible genetic codes, computer programs, you name it. You just won't get to many of them before a century has gone by and you can't count anymore because you're dead.

 

[shadron]

Pi is an irrational number, meaning it never ends.
It's infinitely long, therefore it must contain every possible combination of numbers an infinite amount of times.

No. It might, and in pi's case it very well might, but it is not a necessity for all irrational numbers. For example, suppose you had an irrational number that expanded into a random sequence of only ones and zeros (there has to be one - rather, an infinity of them -, right?). That certainly would not create the digit sequence '7', right?

Does this mean pi contains every piece of software (in 0's and 1's) ever written? And if we could reduce genetic code to numbers, would it contain the genetic code for every possible human being?

Yup - with their social security numbers attached, as well.

 

[Tim Thompson]

Irrational

Pi is an irrational number, meaning it never ends.

No, "irrational" means that it cannot be expressed as a ratio of integers, hence "irrational". 1/3 Is a rational number, but its decimal equivalent is 0.333333333333333 ... forever. It never ends either, but it is not irrational.

 

[Perpetual Student]

[latex]\pi[/latex] is a transcendental number (which is a subset of the irrationals). Basically that means it cannot be the solution to a polynomial equation with rational coefficients.
No regularly repeating patterns have been discovered. The numbers generated by [latex]\pi[/latex] appear to be totally random. In the first six billion decimal places of [latex]\pi[/latex], each of the digits from 0 through 9 appears about six hundred million times, which would be expected if the digits were random. However, that does not prove it is is totally random and to my knowledge that proof has not yet been done.
So the answer to your questions is a qualified yes, it appears that way.
Here is an interesting link, showing the appearance of sequences of numbers within [latex]\pi[/latex]:

http://www.angio.net/pi/piquery (Try your birthdate.)

 

[JoeTheJuggler]

I'm experiencing deja vu.

Wasn't there just recently a thread on almost this exact topic? I remember I mentioned Carl Sagan's Contact and posted this link to a page where you can search the first 200 million digits of pi.

I didn't see the thread title on the first page of this forum, but it seems to me it was active within the last week or thereabouts.

ETA: D'oh--I see Perpetual Student already posted the same link here.

 

[GreyICE]

When you add it to infinity it makes infinity bigger too!

Not joking.

Well... not really.

 

[Dave Rogers]

I explained recently to my daughter that, if you choose a coding scheme and look far enough along the digits of pi, eventually you probably really will find a series of digits that decodes to the message "Help, I'm trapped in a universe factory!"

http://xkcd.com/10/

Dave

 

[orange31]

Pi has been computed to over a trillion digits without repeating a pattern.

I thought that was impressive until I became aware of a non-repeating number computed to over 11 trillion digits.

It's called the US national debt, and yes I'd call it an irrational number .

http://zfacts.com/p/461.html

 

[HansMustermann]

Actually, I think that the wording you're looking for is that it's a normal number. That's what makes it basically as good as an infinite sequence of random digits.

 

[drkitten]

Pi has been computed to over a trillion digits without repeating a pattern.

I thought that was impressive until I became aware of a non-repeating number computed to over 11 trillion digits.

It's called the US national debt, and yes I'd call it an irrational number .

http://zfacts.com/p/461.html

There are eleven trillion digits in the national debt?

Hmm. So, you're not only a political neanderthal, and incapable of following a thread -- but you're innumerate as well.

 

[Myriad]

Whether or not the expansion of pi (in whatever specified base) can ever be proven to contain every possible finite sequence of digits, irrational numbers exist that are guaranteed to contain every possible finite sequence of digits. For instance, the number

0.12345678910111213141516171819202122...

It's odd to think that this number is merely a single specific quantity, and yet it contains all the books in Borges' Babylon Library, and infinitely more.

I think it tends to support Wolfram's notion that even though it's generally thought that numbers, that is the quantities that numerals represent, are fundamental entities in mathematics while the symbols (such as digit sequences) that we use to represent them are barely adequate artificial contrivances, it might make more sense to think of that the other way around. Mathematics might actually be about the behavior of sequences of symbols under various rules, while the question of what numbers "actually are" might be beyond any possible understanding.

Respectfully,
Myriad

 

[Solitaire]

An argument over mathematical nomenclature.

 

[CapelDodger]

Does this mean pi contains every piece of software (in 0's and 1's) ever written? And if we could reduce genetic code to numbers, would it contain the genetic code for every possible human being?

As Thabiguy said, almost certainly. Also Moby Dick or the complete works of Shakespeare; much cheaper and less messy than an infinite number of chimps banging away on keyboards.

For any particular sequence you could certainly never be sure that it isn't in there. Unless I'm wrong, of course.

 

[sol invictus]

I think it tends to support Wolfram's notion that even though it's generally thought that numbers, that is the quantities that numerals represent, are fundamental entities in mathematics while the symbols (such as digit sequences) that we use to represent them are barely adequate artificial contrivances, it might make more sense to think of that the other way around. Mathematics might actually be about the behavior of sequences of symbols under various rules, while the question of what numbers "actually are" might be beyond any possible understanding.

But there are ways to define numbers using (say) geometry that have nothing to do with symbols... pi as a geometrical ratio, 0 as the genus of a torus, 1 as the generator of the 3rd homotopy group of a 3-sphere, etc.

I think mathematics is discovered, not invented... and numbers are a fundamental part of that. Our notation for them, not so much.

 

[orange31]

There are eleven trillion digits in the national debt?

Hmm. So, you're not only a political neanderthal, and incapable of following a thread -- but you're innumerate as well.

LOL! You're right about my math skills, only 14 digits so far in the national debt.
(It just seems like 11 trillion).

However, I don't think your quote about 'my politics' is fair....
to the neanderthal.

 

[drkitten]

As Thabiguy said, almost certainly. Also Moby Dick or the complete works of Shakespeare; much cheaper and less messy than an infinite number of chimps banging away on keyboards.

For any particular sequence you could certainly never be sure that it isn't in there. Unless I'm wrong, of course.

Actually, there's enough regularity in pi that it wouldn't astonish me to learn that pi is not, in fact, "normal" and you could prove things didn't appear in it.

For example, there's a theorem from the mid 90s that allows you to compute the x'th digit of pi without computing the previous (x-1) digits. The trick is that it only works in hexadecimal (base 16), not in base 10. But this suggests to me that pi might not be normal in base 16, meaning, of course, that it might not be normal is base-2, either.

 

[sol invictus]

For example, there's a theorem from the mid 90s that allows you to compute the x'th digit of pi without computing the previous (x-1) digits. The trick is that it only works in hexadecimal (base 16), not in base 10. But this suggests to me that pi might not be normal in base 16, meaning, of course, that it might not be normal is base-2, either.

Really?

What's the theorem?

 

[drkitten]

What's the theorem?

http://www.math.hmc.edu/funfacts/ffiles/20010.5.shtml

Neat, huh?