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[polymath257]

It depends on what you are calculating and why. For example, if the irrationality of the number is important, using pi=3 would be a very idea. If, instead, you want to get an order of magnitude estimate for some physical process, it would be useful.

The point is that pi is NOT the same as 3 or 22/7 or 355/113. It is an irrational number. It is even a transcendental number (not the root of any polynomial with integer coefficients). There are cases where this is an important aspect of the number pi.

But, for example, we do not know whether pi+e is irrational or not. Most mathematicians suspect it is, but nobody has found a proof. There is no way to tell this by an approximation.

Have you attempted to prove this? Has anyone come close?

Well, there is a conjecture due to Lang that given any real numbers x_1 ,...x_n, then the transcendence degree of Q(x_1 ,...x_n, e^{x_1},...e^{x_n}) is at least n. This would give the transcendence of e+pi, but nobody has a proof of this conjecture either.

There are some pretty remarkable results in transcendence theory. It isn't my area of specialty, so I haven't seriously attempted proofs in the area, The issue seems to be the addition of exponentials and logarithms (recall that i*pi is a logarithm of -1).

All transcendence proofs that have been discovered so far are done via contradiction (not surprising given the definitions involved) and amount to getting a contradiction by finding an integer between 0 and 1 using sophisticated approximation techniques.

A very easy example of a transcendental number is the sum of 1/10^(n!). The proof shows it can be approximated by rationals too well to be algebraic!