Pi's Last Digit

Date: 7/20/96 at 9:8:23
From: Anonymous
Subject: Pi's Last Digit

I was browsing though the elementary questions and I found the section 
on pi. I know pi is a nonterminating decimal, but if there is a last 
digit (this is a paradox..but who cares...) wouldn't the last digit be 
a 0? Take a decimal, such as 2.263, and add a zero at the end: 2.2630. 
2.263 = 2.2630, so the last digit of 2.263 would be a 0, right? So 
adding a 0 at the end of pi wouldn't change the original value of pi.

Date: 7/26/96 at 18:33:32
From: Doctor Tom
Subject: Re: Pi's Last Digit

But "adding a zero to the end" just doesn't make sense.

Can you tell me how to "add a zero to the end" of the decimal
expansion of 1/3?

1/3 = .3333333.... (the 3's go forever)

Whereever you stick in a zero, you change the value.  For example, 1/3 
is none of the following values:

.330, .3333330, .333333333333333333333333330.

Right?

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

Date: 10/19/98 at 10:50:56
From: Robert Marinier
Subject: Pi's last digit

I read another question about pi's last digit being zero. In your 
counter-argument, you said that wasn't reasonable because adding a zero 
to say, .3333333... (repeating) makes it something other than 1/3. I 
disagree with the premise for your argument. The example you gave was 
rational, whereas pi is irrational. If pi is irrational, then the 
"next" digit in a never-ending list could be anything, but once you get 
out to infinity, the difference between whatever the infinityeth digit 
is and zero is at most infinitely small, and 1 over infinity is 
essentially zero, so the last digit is zero.

Date: 10/19/98 at 14:43:48
From: Doctor Rick
Subject: Re: Pi's last digit

Hello, Robert. The answer to which you refer, uses the example of 1/3 
to make a point which I think you missed. It does not matter whether 
the number you consider is rational or irrational. What matters is that 
the decimal expansion is infinite. Both pi and 1/3 can be written as 
infinite series:

   1/3 = 3/10 + 3/100 + 3/1000 + ...

   pi = 3 + 1/10 + 4/100 + 1/1000 + ...

The point of the answer was that if you "add a zero to the end," you 
are necessarily terminating the series. An infinite series HAS NO LAST 
TERM. Giving it a last term means terminating the series, making it 
finite. 

Doing this to pi has an even greater effect than doing it to 1/3, 
because not only does it make an infinite series finite, it also makes 
an irrational number rational. Every finite series of digits is the 
expansion of a rational number. Since you agree that pi is irrational, 
it can have no last term, zero or not.

Now, let's consider your argument that the difference between the
"infinityeth digit" and zero is infinitely small. To speak correctly, 
we must speak in terms of limits. There is no infinityeth digit, only 
the limit of the Nth digit as N increases infinitely.

In these terms, what I think you are saying is that if we replace the 
Nth digit of pi with 0, in the limit, the difference |pi - pi*| goes 
to zero (where pi* is the altered decimal expansion). This is true - 
but it does not mean that the digit we replaced must be 0. 

The Nth term in the decimal expansion of pi is d_N/10^N where d_N is 
the Nth digit (0 <= d_N < 10). The effect of replacing this digit with 
0 is therefore:

                           -N             -N+1
   |pi - pi*| =  lim  (d 10  ) <= lim  (10    ) = 0
                N->inf  N        N->inf

no matter what d_N is. In other words, a digit far down the line has
infinitesimally little effect on the value of pi, regardless of its 
value, 0 or not. Your observation tells us nothing about any digit of 
pi.

Infinity, infinite series, and infinite decimal expansions are hard to 
think about. Keep thinking! I hope what I've said is helpful.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/