



 
Matthew, we have two responses for you: Matthew, If you measure the time two sprinters take to run 100 meters, the answer you will get depends on the instrument you take:
So I would say that the ``real time'' that the sprinters take, if such a thing exists, is an irrational number. It is the same thing with pi: If I measure the circumference/diameter of a small coin with a string, I am happy to get 3. But if I take a larger circle and use better instruments, I get 3.14. With even larger circles and even better instruments, I would get even more decimals correct. And so on. But because pi is a mathematical concept (ratio of circumference to diameter on an idealised circle) rather than a physical event (a race), it is possible to compute it mathematically rather than physically. On this University of St. Andrews web page we read that after centuries of calculations with polygons,
Lambert proved that pi was irrational in 1761. Claude Tardiff.
Hi Matthew. How are you measuring the length of your string and the radius of your circle? You are using a device that only has a certain amount of precision. A ruler might measure 0.05 cm at best, other devices can get much more precise, but no device can get the uncertainty below about 10^{33} cm (and no such device is possible!). So there is in fact a "rounding off" implicit in your measurements. Pi is a number which is irrational, and so numeric representation of it will necessarily be an approximation (3.14, for example, or 22/7 or 3.1415926535897932384626433832795029). However, we don't just have "approximations" or "estimates" of pi. We know its value precisely, we just can't express it in decimal or fraction form, because then it would be rational (by definition). Some valid and absolutely precise expressions of Pi:
Some people continue to calculate more and more decimal places for more accurate approximations of pi. I see someone has posted the first billion digits of pi on the internet. And other people continue to try to memorize a lot of digits of pi as well (although why anyone would want to be able to remember and recite 67000 digits is something I'll never understand). Stephen.  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 