Re: Rationality of Pi

The question was:
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There was a message on LM_Net saying that a math teacher had heard that pi has been found to be a terminating decimal and wanting verification. I am also a math teacher and would be interested in knowing if this is true.
Thanks for your help.
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Unfortunately, no - if pi had been found to be a terminating decimal, then it would be a rational number, but that is not the case.

Last year, Yasumasa Kanada at the University of Tokyo computed the value of Pi to 6,442,450,000 places. Like the "Everready Bunny Rabbit", the value of Pi keeps on going and going and going...if your students recited the value of Pi as calculated by Kanada at the rate of 5 digits per second, without stopping to eat or sleep (unlikely!), they would be reciting for 41 years!

Of course, during that time someone else would have taken Pi to even more digits - and it still will not have terminated.

Some people express Pi as 22/7, which is only correct to three places. 22/7 is a rational number. An irrational number cannot be expressed as a ratio of any two whole numbers, no matter how large the two numbers are. For Pi to be rational, with a value as calculated by Kanada to over 6 billion places, those two whole numbers would have to be very large!

In 1768, Johann Heinrich Lambert proved that Pi is an irrational number.

Most numbers are irrational. If you play an instrument like a guitar, you will notice how much easier it is to have a poorly tuned guitar than a tuned guitar.

In fact, this is exactly how Pythagoras came up with the concept of rational and irrational numbers. Pythagoras was interested in musical harmony, and noticed that strings that sounded good together - musical harmony - had lengths that were related to ratios of simple integers. For example, middle C and the high C an octave higher are related by a factor of 2/1 (we now consider 256 waves/second to be middle C, and 512 waves/second to be the higher C). The notes of a major chord, like the C major chord (C and E and G notes on our scale) are related by the ratios of 1/1, 5/4, and 3/2 respectively. Pythagoras use this concept of ratios of simple integers to develop the musical scale!

Pythagoras was thrilled to find that the ratios of the lengths of the sides of a particular right triangle (3,4,5) were also rational numbers. He and his followers began to believe that all numbers that occurred naturally for him were rational numbers, the ratios of whole numbers, and that his was part of the beauty of the mathematics of nature.

So, imagine how shocked Pythagoras and his followers were when they found their first irrational number. If they made a right triangle with two sides the same length, the ratio of one of the equal sides to the hypotenuse was 2 divided by the square root(2) - and the square root of 2 turned out to be irrational! They even developed a proof that the square root of 2 is irrational.

Back to Johann Heinrich Lambert. He proved that Pi is an irrational number by first proving that the function tan(x) cannot be rational if x is any rational number other than zero.

His argument continued that, if x is pi/4, then tan(pi/4) =1, and 1 is a rational number. So, pi/4 cannot be rational. However, 4 is a rational number - so, pi must not be rational.

A bit more recently, Ivan Niven published, "A simple proof that Pi is irrational" , in the Bulletin of the American Mathematical Society, volume 53 (1947). page 59.

There are several Web sites that discuss Pi in all its irrational glory. If you are interested, you can explore some of these Web sites starting at:
Pi
There is even a site that gives a tongue-in-cheek proof that Pi is rational - perhaps that spawned the message on LM_Net that led to your question.