If and are two distinct rational numbers written in their lowest terms, then , which implies that , which implies that . Therefore, if is a real number and we can find a sequence of rational numbers (in their lowest terms, with denominators tending to infinity) such that (which is equivalent to saying that ), then cannot be rational.♦ Loosely speaking, if you can approximate well by rationals, then is irrational. This turns out to be a very useful starting point for proofs of irrationality.
Let us construct inductively a sequence of rationals that approximate . (This is not necessarily the best proof of the irrationality of but it gives an easy illustration of the technique.) We begin with , and observe that . If were , then we would have , so the "" at the end of this is our error term in the first approximation. Now suppose we have defined and in such a way that . Then set and . (The justification for this choice is that if were then would be too, as can easily be checked.) Then
Thus, we have constructed a sequence of rationals , with denominators tending to infinity, such that for every . But from this we deduce that , and therefore that . Since
To prove the irrationality of , we start with the power-series expansion:
We then set to be . This is a fraction with denominator that divides . It differs from by
Also, this difference is strictly positive and not zero. Therefore, , so is irrational.
The two proofs given so far can easily be, and usually are, presented in other ways that do not mention the basic principle explained in the quick description. However, sometimes that basic principle plays a much more important organizational role: one is given a number to prove irrational, and one attempts to do so by finding a sequence of good rational approximations to .
Another point is that if is irrational then such a sequence always exists: one can take the convergents from the continued-fraction expansion of . But this observation is less helpful than it seems, since for many important irrational numbers (such as ) there does not seem to be a nice formula for the continued-fraction expansion. The point of the method explained here is that it is much more flexible: any sequence of good approximations will do (and sequences are indeed known that prove the irrationality of ).