Re: Why do we need to know the full number of Pi

The short answer is that we don't need to know the exact value of p, which is a good thing considering that it is impossible to write down all the digits of p. This is because p was proven in 1882 to be a transcendental number. This means that there is no algebraic equation which has p as its root. So for instance, we cannot write p = a/b for some integers a and b, since then p would be the solution to the algebraic equation bx = a.

Even though we can never write down all the digits of p, we can write down a good approximation. The current computational records page indicates that p is now known to over 206 billion digits. Of course, that much precision is never needed in real life. If the Earth was a perfect sphere, knowing the radius of the Earth exactly and p to a mere 17 digits would allow computation of the Earth's circumference down to about the width of a hydrogen atom, and p has been known to this precision since around 1600.

So why do people do it? Well, there is one practical reason. Algorithms for computing p are a great test for a new supercomputer, and the results are easily verified against the known answer. If there is any problem with addition, multiplication, or memory storage, having your new computer figure p out to a billion digits is likely to catch the problem.

The more important reason for the modern interest in computing p is that it is a mathematically interesting problem with solution techniques drawn from many different fields of mathematics. The number p seems so simple, indeed, the circle is one of the simplest geometric figures. Yet the digits of p doggedly refuse to hold any kind of pattern, even after looking at billions of digits. Is it necessary to know this many digits in order to use p in practical problems? Certainly not. However, the search for ever more digits of p requires some very deep mathematical thinking, and that has resulted in some intriguing ideas in computation. I'd suggest looking here for more on p and it's history.

Mark Huber