Is Pi Infinite?

Date: 01/29/98 at 20:54:12
From: Maggie
Subject: Does pi ever turn into zeros and ones?

I'm immensely curious about pi. How can we know that is is infinite? 
Does it ever turn into a pattern of zeros and ones, like a computer 
code? 

Please help. Thank you.

Date: 01/30/98 at 11:55:28
From: Doctor Wilkinson
Subject: Re: Does pi ever turn into zeros and ones?

A mathematician named Lambert in the nineteenth century showed that pi 
is not a rational number. This means it is not the quotient of two 
integers. You may know, for example, that 22/7 is an approximation to 
pi. (355/113 is a much better one). But Lambert's theorem shows that 
neither of these approximations is exact. Because pi is irrational, 
the decimal expansion of pi cannot come to an end, and it cannot 
repeat either.

In 1882, another mathematician named Lindemann proved much more. He 
showed that you can't write an equation with integral coefficients 
that has pi as a solution. For exmaple, the square root of 10 is a 
rough approximation to pi, but Lindemann's theorem shows that it 
cannot be exact.

Lindemann's theorem also shows that a very famous ancient problem 
called "squaring the circle" is impossible. Squaring the circle means 
finding the side of a square whose area is equal to the area of the 
circle with a given radius, if all you are allowed to use is a compass 
and a straightedge. This problem had puzzled people since the ancient 
Greeks.

In answer to your second question, it is of course possible to 
represent pi in binary notation, as a string of 0's and 1's. Lambert's 
theorem again shows that this string cannot come to an end or repeat.  
But your question seems to refer to the ordinary decimal 
representation. That is, you want to know if the decimal

    3.14159265358979....

eventually leads to just a string of 0's and 1's. The answer to this
question is that nobody knows, but most mathematicians would be 
extremely surprised if this turned out to be so. Pi has been 
calculated to about 50 billion decimal places, and so far all the 
digits from 0 to 9 appear just about equally often.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/30/98 at 12:32:31
From: Doctor Sam
Subject: Re: Does pi ever turn into zeros and ones?

Maggie,

What a terrific question!  Pi is fascinating and I share your 
curiosity about it!

We know that pi is an infinite decimal because of the work of many 
fine mathematicians over the course of several centuries. In 1767 a 
mathematician named Johann Heinrich Lambert proved that pi was an 
irrational number. Irrational numbers are numbers that do not 
terminate or repeat when written out as a decimal.

Does it ever turn into a pattern of zeros and ones? What an 
interesting question. The answer is that no one knows. Millions and 
millions of digits of pi have been calculated using super computers.  
The digits look random, with about the same number of zeros and ones 
and twos and threes and so on, spread throughout. Some modern 
mathematicians suspect that this pattern continues "out to infinity" 
which would mean that it will never become a sequence of just ones and 
zeros or any other digits.

On the other hand, no one really knows.  

The only thing that we know for certain is that pi is irrational: its 
decimal will not terminate or begin repeating.  

If you are interested, you might try to look in some math books in 
your library. Whole books have been written about pi (a good one is 
_The History of Pi_) but lots of books have small chapters about it 
and you might want to start with one of them.

One more thing. I don't know if you have learned about other number 
systems. Since you are interested in computer codes, perhaps you know 
that computer scientists like to write numbers in the binary number 
system, a system which uses only the digits 1 and 0 to write down 
every number. For example, the number 13 can be written as 1101 in 
binary and the fraction 1/2 can be written as the "binary decimal 
number" 0.1

In binary EVERYTHING is written with just ones and zeros. So if pi 
were written in binary it would appear to have just ones and zeros 
also.

Thanks for your question,

-Doctor Sam,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/