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/ |
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