Why the value of pi is unknown and cannot be known
The Sunday Nation
Nairobi,
13 October 2013
Chris Munene has been wondering about the value of the mathematical quantity known as “pi”. He says that ever since he was in school, he has always know it to be 3.14 or as the fraction 22/7. Then recently, he read somewhere that the pi is not known. “How can this be? Have we been lied to all this time? Please shed some light”, he writes.
Pi is the ratio of the circumference of a circle to the diameter. Thus is it is measurable quantity; all you need to do is draw a circle measure the two dimension and divide them.
However, the answer you get will be greatly dependent on the accuracy of your ruler. You may find that a circle of 10cm diameter has a circumference of about 31.4cm. Dividing these two of course give the value of pi as 3.14.
But if you used 22/7 as the value of pi and calculated the expected circumference, you find that is comes to 31.43cm. Since the ruler only gives measurements to the nearest millimetre, you must round this answer to 31.4cm. This agrees with the experimental value.
Can you then extrapolate this finding and say that a circle of, say, 10m will have a circumference of 31.4m? Not quite: The scale of the standard 10-metre measuring tape divided into millimetres. Thus you should be able to get a much more accurate circumference – at least to the third decimal place of a meter. If you are very careful, you should get something like 31.416m. This yields the value of pi as 3.1416.
Now this does not agree with 22/7 (= 3.1429) from the third decimal place onwards. So, which of the two is more accurate?
The answer is that 22/7 is also an approximation of the value of pi. Indeed, a better fraction is 333/106 (3.1415) or even, 355/113 (3.1416). But, even though they are more accurate, the latter two are quite cumbersome on the tongue! The last one agrees with the experimental value up to the fourth decimal place.
What do we do if we want to get an even more accurate value? Do we draw a larger circle, perhaps 100m in diameter? That wouldn’t be a practical path to take.
Mathematicians have looked at the geometry of a circle and come up with ingenious formulae for evaluating pi to a high degree of accuracy. All the methods have one common feature: A series of fractions added together and each subsequent fraction is smaller than the previous one.
The series never ends; that is, there is always a smaller fraction that can be added. For that reason, the exact value is not only unknown, but it CANNOT be known. There is always room for another decimal place.
There is a website that gives the value of pi to 100,000 decimal place; but soon enough, another one will come up with 100,001 decimals!
Furthermore, it is not possible to write pi as a simple fraction of a number x divided by a number y.
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