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### Contradiction in definition of pi?

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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/
```
Associated Topics:
Middle School Pi

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