# Is pi infinite?

1. Jan 13, 2004

### pallidin

Is pi Infinite?

2. Jan 13, 2004

### chroot

Staff Emeritus
No, $\pi < \infty$.

- Warren

3. Jan 13, 2004

### Integral

Staff Emeritus
Further

$$3.14 < \pi < 3.15$$

4. Jan 13, 2004

### luther_paul

What do u mean pi is infinite? the value itself or the digit? if u are talking about the digits, its infinite because it is an irrational number, meaning unending digits. the value itself is not infinite because it has an exact value (22/7).
its common sense to say that!peace!

5. Jan 13, 2004

### chroot

Staff Emeritus
22/7 is a rational number, and thus has a repeating or terminating decimal expansion. 22/7 is not pi. 22/7 is just a crude approximation sometimes used in place of pi.

- Warren

6. Jan 13, 2004

### pnaj

luther_paul

Sorry to be picky, but ...
... is not actually true (just in case pallidin gets the wrong idea).

A rational number is any number that can be written m/n where m,n are integers.
So,
1/7 = 0.142857142857142857142857..(142857 recurring)
is rational, but has 'unending ' digits.

In fact, technically,
1/1 = 1.000000000000000000000000..(0 recurring)
has infinitely many digits.

But, for an irrational, there will never be a recurring pattern in the expansion, as for the rationals.

Another way to look at it is this.

Suppose you trying to calculate the decimal expansion of 1/7 by hand (or by any other method). Once you'd got to the first recurrence of the sequence (142857) you would stop, because it's for certain that the pattern will recur infinitely.

So far, over 200 billion digits of PI have been calculated ... no pattern yet.

7. Jan 13, 2004

### pnaj

8. Jan 13, 2004

### master_coda

And there never will be a pattern. Just in case anybody wasn't clear about that from the above posts.

9. Jan 14, 2004

### luther_paul

pnaj
thanks for the info... i got that wrong, it must've been nonterminating and non-recurring/nonrepeating...

warren
ur right 22/7 is just a crude estimation of pi, but i used 22/7 to show that pi has an exact value, not infinity.. the topic is about pi being an infinite.

thanks, by the way..

10. Jan 16, 2004

### mouseonmoon

well, there
seems to be a pattern of PI repeating here..(smile)

anyway-what's the 'fartherest out' that's been found for a 'repeater'
(hope this isn't an 'irrational' question--i'm simply guessing someone's played around
with the idea ie. recurring patterns)

or,in different words, what 'number' begins its pattern sequence with the greatest expansion?
or/and, --what rational number has been found with the 'longest pattern'?

what am i asking?....how long can a sequence be?....is there any formula that 'predicts' these patterns etc.....anything along these lines-who's written about this? (or 'wierd' math in general--who was the guy who 'proved' 1+1 didn't equal 2 ?--did anyone ever prove he was wrong??)

why am i asking?....well, why has PI been taken to over 200 billion digits?
Can one really say, 'it will never repeat'?
seriously...

11. Jan 16, 2004

### pnaj

No, you can't say there will never be a pattern, just that you cannot predict that there will be one, even after calculating the first 200 billion digits.

One could positively show that it is terminating ... if you found the pattern.

One cannot positively show that there is no pattern.

Apart from the massively small chance that a pattern does emerge, the main reason for looking seems to be bench-testing of more-and-more powerful computers.

P.

12. Jan 16, 2004

### krab

That ain't right. Pi was proved to be a transcendental number already way back in 1882. Therefore, no pattern. More info.

13. Jan 16, 2004

### HallsofIvy

Assuming that by "pattern" you mean section of digits that then repeats, the rational number m/p where p is prime and m< p will have repeating section with p digits. Since there exists arbitrarily large prime numbers, the repeating section can be as large as you please.

What exactly do you mean by "pattern"? Most of the responses here are using it to mean a section that repeats.
It is fairly easy to show that if the decimal expansion of a number eventually consists of a section that then repeats, that number is rational. It is also fairly easy to show that pi is not rational and so never repeats.

14. Jan 16, 2004

### master_coda

When I said that pi doesn't have a "pattern" I meant that it does not repeat. Almost any other notion of pattern is difficult to define mathematically.

15. Jan 16, 2004

### chroot

Staff Emeritus
Is this true? How about 1/3, which has only one repeating digit? Or 1/7, which has six?

- Warren

16. Jan 16, 2004

### lethe

there are lots of irrational numbers that do have patterns. my favorite example is this one:

0.1101001000100001000001.....

one way to define whether a number has a pattern is to show that the number is normal. a number is normal if, loosely speaking, any finite sequence of digits appears about as frequently as it would for a random sequence of digits

it is probable that pi is normal, but not proved, so in fact, there may be a pattern. in fact, people sometimes look for patterns in the digits of pi. we simply don t know if it has any. remember the movie Pi? i think that guy found the torah in the digits of pi, or something (of course, it was a work of fiction).

17. Jan 16, 2004

### pnaj

HallsofIvy,
Did you see the original question?

We were all trying to help someone understand the nature of the difference between a rational number and a irrational one.

So, I wouldn't dispute what you say, of course ... you're right ... but I'm not sure how much help ....
... would really be to someone who might not really know what 'rational' is.

================
Also, krab and master_coda, thanks for pointing out that PI has been proved to be transcendental (and thus not rational), I could have misled someone there ... I wasn't precise enough. What I should have said was ...

Suppose you didn't know whether PI was rational or not.

Then ...
... makes more sense.

What I trying to get our friend to get the gist of was this:

If you look at the decimal expansion of any particular number upto to squillionth digit, and not see a terminating-repeating sequence of digits, that does not mean that is not rational. You gain no information that will help you predict whether it is rational or not.

Cheers,
P.

18. Jan 16, 2004

### master_coda

But I could argue that a number has a pattern as long as we have an algorithm for generating its digits.

Perhaps it's best to avoid the use of the word "pattern", since it seems hard to define properly.

19. Jan 16, 2004

### chroot

Staff Emeritus
There are lots of algorithms to generate the digits of pi......

- Warren

20. Jan 16, 2004

### pnaj

Sorry, probably my fault for introducing the dreaded 'pattern' word.

21. Jan 17, 2004

### mouseonmoon

first, let me thank krab for the Wikipedia link-
love it and been ' lost in delight ' ever since!!

(and speaking of logarithms-John Napier invented 'em-what a character! and math was 'just a hobby' for him-
http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Napier.html

and speakin' of patterns:
The infinite continued fraction expansion of e contains an interesting pattern that can be written as follows:
e= {1;1,1,2,1,1,4,1,1,6,1,1,8,1,1 ...}
(which was described by Richard Feynman as "The most remarkable formula in mathematics"!)

But back to the original post in this thread-is Pi infinite?-

the mind boggles-my does anyway.....

seems there 'could be' an 'infinite' number of 'patterns' in the 'expansion' of Pi (are there absolutely no patterns within the 200 billion digits so far generated?-seems impossible)--yet it 'cannot' 'in fact' 'repeat' (("repeat": as it is proven to be transcendent))...holding in thought the computer
laboring on with the 'expansion' of Pi to 'infinity'....

Has anyone put this question to one of those gifted individuals known as 'idiot savants'? I've heard where some are able to 'instantly' perform calculations of immense complexity--i am not aware of anyone with this ability today.......(perhaps it would be inhumane to attempt such a thing-let me express my vote against such an 'experiment'!)

i've reached my limit...sparkplugs burnt out, confused astoundment, brain cells fused........
excuse me, i need to find the 'Dear Santa' forums....oh Dorothy!!

22. Jan 17, 2004

### krab

Don't be sorry. As Warren points out, with a sufficiently broad definition of pattern, pi does have a pattern. Just no repeating digits.

23. Jan 17, 2004

### lethe

yes, if you make the definition of "pattern" so broad as to be trivial and useless, so that all things are patterns, then pi has a pattern. on the other hand, there exist patterns which are not trivial and are mathematically well defined. one example is repeating sequence of digits, which pi does not contain. there are other patterns which pi may have. it is not known whether it does or does not.

24. Jan 17, 2004

Staff Emeritus
ANd of course pi escapes patterns more general than repeating digits, since it's transcendental, i.e. not the solution of any polynomial equation with rational coefficients.

So both square root of two and pi lack repeating digits, but square root of two will be found to have a more subtle pattern deriving from its definition as a solution of x2 = 2, but you will never be able to find such a pattern in pi.

25. Jan 17, 2004

### moshek

pnaj,

With the eye's of non Euclidian mathematics
that see alway's mathematics as
"One whole organism" ( Hilbert vision 1900)
your original question does pai infinity?"
became significant when we present pi in base 16
and this close a circle of 5,000 years voyager to find pai.

Moshek