Associated Topics || Dr. Math Home || Search Dr. Math

### Pi as the Sum of Rational Numbers

```Date: 02/20/2003 at 00:24:53
From: Huang Tang (Joy)
Subject: Pi as a sum of rational numbers

I managed to come up with this equation:

pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... + 1/(4n+1) - 1/(4n+3) + ...

Is is right? If so, since pi is the sum of rational numbers, doesn't
it have to be rational also?
```

```
Date: 02/20/2003 at 02:21:41
From: Doctor Jacques
Subject: Re: Pi as a sum of rational numbers

Hi Joy,

It is true that

Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

However, this does not imply that Pi is rational. What we have here
is not a sum, but the limit of an infinite series, i.e. the limit of
the sequence of partial sums:

1
1 - 1/3
1 - 1/3 + 1/5
etc.

Although each such partial sum is rational, the limit of these sums is
not.

There are other examples. Consider, for example, the square root of 2:

sqrt(2) = 1.414213562373...

We can write this as an infinite series:

1 + 4/10 + 1/100 + 4/1000 .....

In this case, all the terms are rational. However, it is well known
(since Archimedes) that sqrt(2) is irrational.

By the way, you could also have used that argument on Pi itself (or
any irrational number):

Pi = 3.1415926...
Pi = 3 + 1/10 + 4/100 + 1/1000 ...

Does this clarify the issue?

- Doctor Jacques, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search