Contradiction in definition of pi?

Date: 08/09/97 at 00:16:25
From: keng tong
Subject: Contradiction in definition of pi?

Hi Dr. Math,

Good day to you.

I have learned that the definition of "pi" is the ratio of the 
circumference of a circle to the diameter of the same circle. So 
in Math we write pi = circumference / diameter.

We learned that an irrational number is a number that cannot be 
expressed as a fraction. We learned that pi is an irrational number, 
BUT in the beginning, when we write "pi equals circumference over 
diameter," aren't we expressing pi in terms of a fraction? 

Is that a contradiction? For how can circumference (a finite, rational 
number) divided by diameter (a finite, rational number) give a number 
that is irrational ?

I hope that you can help me to clear my doubt.

Best Wishes!

Date: 08/09/97 at 11:28:13
From: Doctor Anthony
Subject: Re: Contradiction in definition of pi

The answer is that if the circumference is a rational number the 
diameter will be irrational, and vice versa.  

A similar situation arises if you draw an isosceles right 
angled triangle with sides 1, 1, sqrt(2) or, alternatively, 
sqrt(2), sqrt(2), 2.

So  sin(pi/4) = 1/sqrt(2)  or   sqrt(2)/2

Either way the answer is irrational.

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

Date: 08/13/97 at 02:35:23
From: Doctor Mike
Subject: Re: Contradiction in definition of pi
    
   Hello,  I am sending another short answer to emphasize the
   exact formulation of that definition.  The complete form: 
   "an irrational number is a number that cannot be expressed in
   a fraction, where both the numerator and denominator are 
   integers (whole numbers)."  I hope this is now clear for you.
    
   Best regards,

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