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Irrational Pi

Date: 7/15/96 at 9:38:19
From: Louis Newton
Subject: Irrational Pi

Dear Dr. Math,

Why is pi considered an irrational number when it's defined as the 
ratio of the circumference of a circle to its diameter?

Date: 7/15/96 at 12:9:14
From: Doctor Paul
Subject: Re: Irrational Pi

I think before we answer this we should define what it means for a 
number to be irrational.  Then this problem is simple.  An irrational 
number is one that can not be defined as a fraction p/q for any 
integers p and q.  That is, the number can not be written as one whole 
number divided by another whole number.  

Your question is a good one. If Pi is the ratio of the circumference 
to its diameter, then how can it be irrational?  Well you're assuming 
that 'ratio' means two whole numbers the way it does at the horse 
races. At the races, the 'odds' or ratio of money earned to money bet 
is always a nice number like 10:1 or 3:5. Well, with a circle, it 
doesn't work out that way. If you measure the diameter to be x inches 
(where x is a whole number) there is no way that the circumference 
will be a whole number. If it is, then you're not dealing with a 
circle... you've got an oval or some other shape.  

If you find a circle with diameter and circumference to both be whole 
numbers, let me know.  You'll become famous :-)  Don't spend too much 
time on it, though... I promise you won't be able to do it. *grin*

-Doctor Paul,  The Math Forum
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Associated Topics:
High School Transcendental Numbers
Middle School Number Sense/About Numbers

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